 1906 OPT312.HLP, Generated by A3UTIL.EXE, 1994  (c) PCSCC Inc., 1993,01╢!Yÿ?uÄ╨,pö«╤╢┌0è▀ G\                                                                             ┌─────────────────────────┐                                             │╒════════════════════════╧╕                                            ││    Application File     ├─────────────────────────────────────       ││                         └──────────────────────────────────────      ││                                                                      ││            Interactive Problem Solving in                            ││                                                                      ││                  GEOMETRICAL OPTICS                                  ││                                                                      ││             Using AstroFormulaCalculator 3                           ││                                                                      ││      Application File  Version 3.12    03-01-1994                     │                                                                                        by  P. DuBois Murphy, Ph.D.                                                                                                    (c) Copyright 1981, 1994  PC SuperComputer Capital, Inc.                                                                                                            All rights reserved.                                                                                                     Read and understand  WARRANTY INFORMATION . Type J twice to read.                                                                            Instructions on using help file are at end of file. Type K to read.                                                                                                                                                   ╒══════════╕                                                            │ CONTENTS └──────────────────────────────────────────────────────────                                                                          0.00                  Table of Contents                                                                                                               *    To quickly jump to Sections, position the Section line number      *    on the │JUMP│ line (line 12) and type J or type Lxxx (enter)       *    where xxx is line number. For example, to jump to line 139,        *    type L139 (ent).  Type S or R to skip back/forth to Sections.                                                                         SECTION                   TITLE                              PAGE│LINE                                                                           0.00 ... Table of Contents ................................... 1   29  1.00 ... Introduction ........................................ 2   85  2.00 ... An Introduction to Calculations Using ............... 4  148           AstroFormCalc 3.                                               3.00 ... Electromagnetic Radiation: The IR Source ............ 6  264           (Blackbody)                                                    4.00 ... Circumference of an Ellipse: Calculation by ......... 7  315           Series and Integration                                         5.00 ... Behavior of the Aspherical Refraction Surface ....... 9  369           (Cartesian Ovoid).                                             6.00 ... Behavior of the Spherical Refraction Surface ........ 9  394           (Paraxial Rays).                                               7.00 ... Thin Lenses, Lensmaker's Formula, Gaussian ......... 10  460           Formula and Newtonian Formula.                                 8.00 ... Thin lenses, Focal Planes and Lateral or ........... 12  552           Transverse Magnification.                                      9.00 ... Compound Thins Lenses, Effective Focal Lengths ..... 14  620           and Dioptric Power.                                            10.0 ... Thick Lenses, Principal Planes and the Paraxial .... 15  656           Approximation.                                                 11.0 ... Compound Thick Lenses, Principal Planes and ........ 15  682           Effective Focal Lengths.                                       12.0 ... Reflections from Planar, Aspherical and ............ 16  717           Spherical Mirrors.                                             13.0 ... Finding Best Values of Transistor Parameters ....... 17  781           from Experimental Data by Nonlinear Least                               Squares.                                                       14.0 ... Shooting a Marble into a Moving Ferris Wheel ....... 23 1021           Car: Solution of Simultaneous Nonlinear                                 equations                                                      15.0 ... Resource Allocation in an Instrument Company: ...... 28 1278           Solving Simultaneous Linear equations                          16.0 ... Glossary of Terms .................................. 32 1458  17.0 ... Product License and Warranty. ...................... 33 1494  18.0 ... User Support. ...................................... 34 1560  19.0 ... Product Information and How to Order Complete ...... 35 1580           Package.                                                       20.0 ... Software Included in Astro 3 Registration .......... 37 1670  21.0 ... Order/Registration Form ............................ 39 1755  22.0 ... Instructions for Manually Installing Package. ...... 40 1820  23.0 ... Interested in Becoming the Author of Your Own ...... 41 1859           Interactive Application File?                                  24.0 ... Help File: Summary of File Viewing Commands ........ 41 1880                                                                        ╒═══════════════╕                                                       │ INTRODUCTION  └──────────────────────────────────────────────────────                                                                         1.00          Introduction                                              ────────────────────────────────────────────────────────────────────────                                                                                 To change the display colors to high-intensity white on blue            type Q when the highlight bar is lighted on the EQU below.                                                                             'EQU                                                                    'COL HWHITE BLUE                                                        'HELP                                                                                                                                                    Geometrical optics deals with the behavior of light and closely         related radiations like UV and IR in situations where the wave          nature of the light is negligible.                                                                                                              This document is an on-line application file which discusses            the theory and presents solved problems in Geometrical Optics           which includes lenses and mirrors.  Excellent references, which         can serve both as a supplement to an existing course in Optics          or as a stand-alone study or self-help guides, are the books:                                                                                                                                                                    "Optics"                                                                 Eugene Hecht, Ph.D.                                                     Schaum Outline of Theory and Problems                                   McGraw-Hill Book Company, New York                                      1975                                                                                                                                           "The Optics Problem Solver"                                              Dr. M. Foigel, Chief Editor                                             Research and Education Association                                      61 Ethel Road West, Piscataway, New Jersey, 08864                       Revised Printing 1990                                                                                                                 Personal computers (PC's) and their automation and calculation          capabilities were not available before 1981.  Proper use of the         PC can reduce or eliminate much of the tedium associated with           reducing theory into algebraic expressions and then into useful         day-to-day applications.                                                                                                                        This PC application file is intended to provide students of             Optics with help in mastering the execution of the necessary            algebra and math calculations upon which Optics is heavily              based.  The application is driven by a versatile                        spreadsheet/calculator call AstroFormulaCalculator 3 which              exists as the file ACAL3.EXE.  This program is specifically             tailored for analysis based on algebraic expressions, formulas          or equations.  One of its main functions is to eliminate the            "programming-type overhead" in working with known formulas and          to allow meaningful learning to proceed rapidly.                                                                                                AstroFormCalc 3 can be used with any traditional outline manual         or text to develop equation templates that solve specific               problems.                                                                                                                                       This file is organized by topics with are summarized in the             Table of Contents.  The use of AstroFormCalc 3 is discussed in          the on-line help file which is accessed by typing (F1) from the         equations display.  This entire file can be searched using the          auto-search command "A".                                                                                                               ╒═══════════════════╕                                                   │ HOW TO CALCULATE  └───────────────────────────────────────────────────                                                                        2.00     An Introduction to Calculations Using AstroFormCalc 3.         ────────────────────────────────────────────────────────────────────────                                                                                 This Section provides an introduction to numbers, number                formats, variables, variable names, mathematical operations,            equations and related topics.  In general, in order to avoid            ambiguities, one must always express numbers, mathematical              expression and equations in an acceptable way when entering             them into a computer program.  These requirements lead to the           formulation of a set of rules which are summarized in the               on-line Help File for the ASTRO 3 program.  This help file can          be reached from the main equation display by typing the (F1)            key.                                                                                                                                            For example:  It is very common to have to evaluate an                  expression which might be written as:                                                                                                                                   456.789 x 0.345                                                                                                         As written, one would interpret this expression as:  Find the           product that results from multiplying the decimal 456.789 by            the decimal 0.345.  Because the letter symbols are most often           associated with variable names, the use of "x" as a                     mathematical operator for multiplication and the                        equally-acceptable use of "x" as a variable name in an equation         such as "y=x" would result in confusion for the                         expression-interpreter of the program.  As a result, the                multiplication operator is commonly the "*" symbol and the              "by-the-rules" way to enter the above expression would be               456.789 * 0.345 (enter) In the language of "computerese",               "(enter)" means to type the enter key of the keyboard and               usually results in some sort of computer action.                                                                                                The following examples illustrate some common ways to write             mathematical expressions and the "by-the-rules" way they are            entered into the ASTRO 3, and many other, computer programs.            These examples exist as equations templates that can be                 "called-up" by scrolling the screen until the highlighted bar           appears over the appropriate 'EQU line.                                                                                                         After the templates are loaded, follow the instructions and             always type (F2) from the main equation display to return to            this application helpfile immediately after the 'EQU reference.                                                                        Topic # 1: ENTERING NUMBERS and EXPRESSIONS                                                                                                     Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                                                                                         24.1 x 56.7       24.1*56.7      1366.47        * means multipl.                                                                                     5                                                                       -            5/180          2.77E-2 or    / means division             180                          2.77*10^-2                                                                                                        6 ∙ 3 ∙ 4      6*3*4          72           * not ∙ for multipl.                                                                                 2(-7)5         2*-7*5         -70            implied multipl.                                                        not allowed.                                                                                       25÷5           25/5           5            / is division operat.                                                                                                                                                         2      5                                                             14 -  + 8 -  (14+2/7)+(8+5/9)   22.84           express compound           7      9                                     fractions as sums                                                                                                                                                                                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTCWRIT      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 2: ENTERING EXPONENTS and SQUARE ROOTS                                                                                                  Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                    ___                                                                    √144         (space) sqr(144)  12         use sqr(..) function                                                    to take square root.           2.3                                                                   10             10^2.3           199.5       ^ means raise to power               4                                                              5.67 x 10      5.67e4            56700      e means base 10                 4     3                                                             3x10 -2x10     3e4-2e3 or        28000      calculation in                             3*10^4-2*10^3                scientific notation.            2 -1                                                               (2x10 )        (2e2)^-1          0.005    no implied exponentiation                                                                              ___                                                                    √ 6                                                                     ------    (space) sqr(6)/(4*sqr(3))   .3535  must match parentheses     4√3                                                                                                                                                                                                                                                                                                                                                                'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTCEXPO      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The remainder of the on-line manual studies many scientific             concepts which when grouped together are commonly called                "Optics".                                                                                                                              ╒════════════╕                                                          │ RADIATION  └─────────────────────────────────────────────────────────                                                                         3.00       Electromagnetic Radiation: The IR Source (Blackbody)         ────────────────────────────────────────────────────────────────────────                                                                                 Electromagnetic radiation propagates through space in                   wavefronts of fluctuating magnetic and electric fields.  The            frequency of this radiation identifies it as radio waves,               microwaves, infrared, visible UV and so on.  The first equation         template identifies the kind of radiation from its frequency or         wavelength. Geometrical optics is most applicable to radiation          in and around the visible which includes ultra-violet (UV) and          infrared (IR) radiations.                                                                                                              Topic # 3: ELECTROMAGNETIC SPECTRUM, FREQUENCY, WAVELENGTH                                                                                      Problem # 3.1: Enter frequency in Hertz or wavelength in meters. One or  the other should be entered as 0.                                       (a) What radiation has a wavelength of 400 nm (400*10^-9 m)?            (b) What radiation has a frequency of 400 THz (400*10^14 Hz)?                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTCSLWV      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          An IR source (blackbody) is the "perfect" absorber of all               incident electromagnetic radiation.  It is also the "perfect"           radiator because it is in equilibrium at a fixed temperature            and so must emit all its incident energy.  However, the                 wavelength distribution of its emitted radiation follows                Planck's Law, Setfan's Law and Wien's Law and is usually in the         infrared region, hence the term "blackbody".  Understanding             these Laws is fundamental to understanding radiation in                 general. However, most the distribution of illuminance from a           visible source rarely shows the IR-source distribution.                                                                                Topic # 4: RADIATION, POWER EMITTED vs ABSORBED, STEFAN'S LAW, BLACKBODY                                                                        Problem # 4.1: A household-type 'radiator' has an emissivity of 0.6 an   area of 2.0 m^2                                                         (a) At what rate is radiation absorbed by the radiator when the walls   of the room are at 27°C?                                                (b) At what rate is radiation emitted from the radiator when its        temperature is 50°C?                                                    (c) What is the net radiation rate from radiator?                                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC1946      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ SERIES and INTEGRATION └──────────────────────────────────────────────                                                                        4.00       Circumference of an Ellipse: Calculation by Series and                Integration                                                    ────────────────────────────────────────────────────────────────────────                                                                                 Calculations involving integration and summation of formulas            are relevant to all math-based theories.  The following                 examples illustrate these methods.                                                                                                              The circumference of an ellipse with axes lengths of "a" and            "b" as measured from its origin cannot be expressed in a simple         "closed" form by a formula.  The approximate value is                   calculated as:                                                                                                                                         l ≈ 2 * pi * sqr( (a^2 + b^2)/2 )                                                                                                        Thus if a = 50 and b = 10, the circumference, l ≈ 226.5.  The           circumference can be express "exactly" by use of the elliptic           integral of the second kind E(k,Φ):                                                                                                                    l = 4 * a * E(k,Φ)   k= sqr( abs(a^2-b^2) )/a                                                                                            There are at least two methods to obtain a higher degree of             accuracy:  the first is to use a series expansion for E(k,Φ)            and the second is to numerically integrate E(k,Φ).  The                 following two examples show how AstroFormCalc 3 can be used to          sum up a series or to accurately integrate an equation.  The            methods are very general and are applicable to many problems in         Optics as well as virtually all other disciplines.  It should           be noted that the "approximation" formula, given above, is only         accurate to about 8%.                                                                                                                  Topic # 5: SERIES, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                        Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the series             approximation.                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTCELSE      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 6: INTEGRATION, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                   Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the elliptic integral  method.                                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTCELIN      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ ASPHERICAL REFRACTION  └──────────────────────────────────────────────                                                                        5.00       Behavior of the Aspherical Refraction Surface (Cartesian              Ovoid).                                                        ────────────────────────────────────────────────────────────────────────                                                                                 The following problem studies the behavior of a spherical wave          diverging from a point source and impinging on an aspherical            refracting surface known as a Cartesian Ovoid.  These surfaces          include the ellipsoid, hyperboloid and paraboloid.                                                                                     Topic # 7: ASPHERICAL REFRACTING SURFACES, CARTESIAN OVOID                                                                                      Problem # 7.1: Write an expression for the Cartesian ovoid with image    and object distances of 15 and 12 cm, respectively. Ovoid made of       glass Ng=1.5 and surrounded by air Na=1.                                (a) Calculate the ray length from ovoid surface to its image if that    point is 14 cm from the source object.                                  (b) What does a negative distance mean?                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC402       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ SPHERICAL REFRACTION   └──────────────────────────────────────────────                                                                        6.00       Behavior of the Spherical Refraction Surface (Paraxial                Rays).                                                         ────────────────────────────────────────────────────────────────────────                                                                                 The basis of most geometrical optics is the spherical                   refracting surface which is of higher symmetry than its                 aspherical cousins, discussed above.  This higher symmetry              allows many basic formulas to be derived.  The following                problems deal with spherical surfaces.                                                                                                 Topic # 8: SPHERICAL REFRACTING SURFACES, PARAXIAL RAYS                                                                                         Problem # 8.1: The input end of a wide glass (Ng=1.6) fiber is ground    to a convex hemisphere of 1.5 cm radius.                                (a) If a point source is located 4 cm to the left of the hemisphere's   vertex, where will its image be?                                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC405       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 9: SPHERICAL REFRACTING SURFACES, PARAXIAL RAYS                                                                                         Problem # 9.1: The input end of a wide glass (Ng=1.6) fiber with convex  hemisphere of 1.5 cm radius is dipped in water (Nw=1.32).               (a) If a point source is located 4 cm to the left of the vertex, where  will its image be?                                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC406       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 10: SPHERICAL REFRACTING SURFACES, OBJECT FOCAL LENGTH                                                                                  Problem # 10.1: An interface separates air on the left and glass (Ng=1.  44) on the right.                                                       (a) What is the object focal length if its radius of curvature is 6     cm?                                                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC408       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 11: SPHERICAL REFRACTING SURFACES, IMAGE FOCAL LENGTH                                                                                   Problem # 11.1: An interface separates air on the left and glass (Ng=1.  44) on the right.                                                       (a) What is the image focal length if its radius of curvature is 6 cm?                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC408A      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 12: SPHERICAL REFRACTING SURFACES, OBJECT FOCAL LENGTH                                                                                  Discussion: What is the radius of curvature of a glass fiber (Ng=1.48)   required to focus light rays 95 cm in front of its spherical interface  in a non-aqueous media of isopropanol (Nm=1.38)?                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC409       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════╕                                                         │ THIN LENSES └─────────────────────────────────────────────────────────                                                                        7.00       Thin Lenses, Lensmaker's Formula, Gaussian Formula and                Newtonian Formula.                                             ────────────────────────────────────────────────────────────────────────                                                                                 The majority of calculations in geometrical optics make two             basic assumptions:  1) the impinged-upon surface is a sphere            (spherical approximation) and 2) the impinging light strikes            that surface at small angles with respect to its optical axis           (paraxial approximation). In general practice, however, one             usually invokes these formulas for any lens calculation.                The following problems deal with "thin" lenses. Such a lens             is infinitely thin and all distances are measured from their            interior principal plane which passes through its center.                                                                                       An illustration of a "thin" lens is shown below.                        Its center is marked by "||".  All distances are measured               from it.                                                                                                                                                    ss                                                                     s  s                                                                    s  s                     M[T] + means erect image     ╦ Y[O]           s    s                                                 ║······o·········s·||·s········o··········o  S source point             S     F[O]       s    s       F[I]        ║  P image point                                s  s               Y[I] ╩  S[O] Source distance       |·X[O]·|···f····· s  s·····f···|····X[I]··|  S[I] Image distance                           ss                                                   |--------S[O]------|-----------S[I]-------|                                                                                                   Sign Convention:  X[O] + means its left of F[O]        focal_length < 0                   X[I] + means its right of F[I]           means                   Y[I], Y[O] + means above optical axis        diverging lens.                                                                                  The following problems illustrate a number of calculations              involving thin lenses.                                                                                                                 Topic # 13: THIN SPHERICAL LENS EQUATION, LENSMAKER'S FORMULA                                                                                   Discussion: The source and image points are located 5 and 7 cm,          respectively, from a spherical concave lens with source radius of -3    cm and image radius of 4 cm. What should the index of refraction of     the glass be?                                                                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC410       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 14: THIN SPHERICAL LENS EQUATION, LENSMAKER'S FORMULA                                                                                   Problem # 14.1: A source is 20 cm from a glass (ng=1.6) convex-plano     lens with a radius of curvature of 4 cm. Where is the image located     (a) when its flat surface faces source and                              (b) when its flat surface is away from source?                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC411       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 15: THIN SPHERICAL LENS EQUATION, FOCAL LENGTH vs N[G], N[L]                                                                            Discussion: Calculate the focal length of a bi-concave thin lens made    of glass (Ng=1.6) and immersed in isopropanol (Na=1.35). The source     radius is -9 and the image radius is 18 cm.                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC413       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 16: LENS EQUATION, FOCAL LENGTH vs INDEX REFRACTION, RADIUS CURV                                                                        Problem # 16.1: A bi-concave (R1=-8 cm, R2=12 cm) thin lens is made of   hepa-fluor (Ng=1.37) and is immersed in solvent alpha-k-1 (Nm=1.77).    (a) What is its focal length?                                           (b) is it converging or diverging?                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC414       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 17: MEASURING INDEX REFRACTION in MEDIUM from FOCAL LENGTHS                                                                             Problem # 17.1: A bi-convex lens of index 1.6 has a focal length of 60   cm in air and 240 cm when immersed in a transparent fluid.              (a) What is the index of refraction of the transparent fluid?                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC416       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ THIN LENS IMAGING └───────────────────────────────────────────────────                                                                        8.00       Thin lenses, Focal Planes and Lateral or Transverse                   Magnification.                                                 ────────────────────────────────────────────────────────────────────────                                                                                 Predicting or estimating the location of an object, also called         a source or target, and the location of its image are very              important.  The following problems deal with this topic.                                                                               Topic # 18: THIN LENS IMAGERY, CONVEX, TRANSVERSE MAGNIFICATION                                                                                 Problem # 18.1: A thin bi-convex lens of index 1.6 has equal radii of    40 cm in air.                                                           (a) Compute the focal length.                                           (b) If the image is to be projected full-size on a white wall, what     are the image and object distances?                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC419       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 19: THIN LENS IMAGERY, CHARACTERISTICS for REAL OBJECTS,         GAUSSIAN                                                                                                                                       Problem # 19.1: A 10 cm tall statue of Curly Howard is sitting 100 cm    from a positive (converging) lens of 30 cm focal length.                (a) Describe Curly's image.                                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC420       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 20: THIN LENS IMAGERY, CHARACTERISTICS for REAL OBJECTS,         NEWTONIAN                                                                                                                                      Problem # 20.1: A 9 cm tall statue of Mo Howard is sitting 100 cm from   a positive (converging) lens of 30 cm focal length.                     (a) Describe Mo's image.                                                                                                                       'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC421       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 21: THIN LENS IMAGERY, MAGNIFICATION, OBJECT, IMAGE DISTANCES                                                                           Problem # 21.1: An object positioned 38 inches to the right of a         positive lens is imaged 8 inches to its left.                           (a) Where will the object appear if its image is moved to 2 inches      from the lens?                                                          (b) Describe it.                                                                                                                               'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC423       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 22: THIN LENS IMAGERY, FIND F from R1, R2, BI-CONVEX, GAUSSIAN                                                                          Problem # 22.1: A thin bi-convex glass (N[g]=1.6) with radii of 20 and   70 cm casts a half-sized image of a plant pot on a white wall.          (a) What is its focal length?                                           (b) What are the lens to plant and lens to wall distances?                                                                                     'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC425       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒══════════════════════╕                                                │ COMPOUND THIN LENSES └────────────────────────────────────────────────                                                                        9.00       Compound Thins Lenses, Effective Focal Lengths and                    Dioptric Power.                                                ────────────────────────────────────────────────────────────────────────                                                                                 Lens combinations are very useful.  The following problems deal         with combinations of "thin" lenses.                                                                                                                                                                            Topic # 23: COMPOUND THIN LENSES, MAGNIFICATION, FRONT BACK FOCAL        LENGTHS                                                                                                                                        Problem # 23.1: A compound lens consists of two bi-convex lenses of      focal lengths 15 and 30 cm, respectively, separated by 60 cm.           (a) Describe the image generated by a statue 10 cm tall placed 12 cm    in front of the first lens.                                             (b) What separation distance is required for unity magnification.                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC427       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 24: COMPOUND THIN LENSES, MAGNIFICATION, MICROSCOPE                                                                                     Problem # 24.1: A low cost microscope is fabricated out of a bottom      lens of 2 cm focal length, a 10 cm cardboard tube and a top lens with   a 8 cm focal length.                                                    (a) Locate the image of an ant 3 cm from the bottom lens.               (b) Describe the ant.                                                                                                                          'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC431       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒══════════════╕                                                        │ THICK LENSES └────────────────────────────────────────────────────────                                                                        10.0 Thick Lenses, Principal Planes and the Paraxial Approximation.     ────────────────────────────────────────────────────────────────────────                                                                                 The basic thin lens formalism and equations can be applied to           "thick" lenses if effective focal lengths and principal planes          (from which distances are measured) are properly defined.  The          single principal plane of the thin lens is split into two               principal planes H1 and H2 separated by the lens thickness D.           The next problem deal with this topic.                                                                                                 Topic # 25: THICK LENS, GAUSSIAN FORMULA, PARAXIAL APPROXIMATION                                                                                Problem # 25.1: A double convex lens has radii of 6 and 24 cm, a         thickness of 3 cm and index of 1.6.                                     (a) Locate both the principal and focal points and compute the image    distance for an object 15 cm in front of V1.                            (b) determine the magnification of the image and                        (c) determine the f.f.l. and b.f.l.                                                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC442       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════════╕                                               │ COMPOUND THICK LENSES └───────────────────────────────────────────────                                                                        11.0       Compound Thick Lenses, Principal Planes and                           Effective Focal Lengths.                                       ────────────────────────────────────────────────────────────────────────                                                                                 As with "thin" lenses, "thick" lenses can also be used in               combinations. It should be noted that if the thickness of the           individual lenses, as measured between their principal planes,          approaches 0.0, then the solutions are the same as for the              "thin" lens combinations discussed above.  The following                problems deals with this topic.                                                                                                        Topic # 26: LENS COMBINATIONS, THICK LENSES, GAUSSIAN FORMULA                                                                                   Discussion: Two identical bi-convex lenses are separated by 30 mm. Each  lens has a radius of 50 and 30 mm, a thickness of 20 mm and an index    of 1.6. Calculate individual and system focal lengths.                                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC443       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 27: LENS COMBINATIONS, THICK LENSES, GAUSSIAN FORMULA                                                                                   Discussion: A bi-concave lens (radius=-50 mm, index=1.6) is positioned   150 mm up a plastic tube in front of a plano-convex lens (radius 50 mm, index 1.6). Calculate the image location of a 9 mm carpenter ant 90 mm  in front of the device.                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC446       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════╕                                                         │ MIRRORS     └────────────────────────────────────────────────────────                                                                         12.0       Reflections from Planar, Aspherical and Spherical                     Mirrors.                                                       ────────────────────────────────────────────────────────────────────────                                                                                 Lenses are refracting surfaces and mirrors are reflecting               surfaces.  There exists a correspondence between the equations          that predict the behavior of mirrors and those of lenses if one         takes into account sign conventions, etc.  A spherical mirror           is shown below.  Mirror surface is represented by 's'.                                                                                                           s                                                                   · o·                   S = source object                         ·  °    s °                   V = vertex of sphere               ·  °          s   ° P     C          P = image (between F and C)S ·°· · · · · · · · ·o· · ·°o · · o          R = 2 * F                  |                   V|s     |  ·°                                       |                    | s   ·|°  R      S[O] = distance S to vertex      |                    |  s°  |         S[I] = distance vertex to image   |-------S[O]---------|-S[I]-|    C = center of sphere R = radius curvat.                                                                        Sign Convention: S[O] + means its left of vertex V, focus + means concv                  S[I] + means its left of vertex V, focus - means convex                 R    + means center C is right of V                                                                                                     The following problems deal with mirrors.                                                                                              Topic # 28: SPHERICAL REFRACTING MIRRORS, PARAXIAL RAYS                                                                                         Problem # 28.1: A convex mirror (17 cm radius) casts the image of a      bird on a screen 20 cm distant.                                         (a) Where is the bird located and                                       (b) what will it look like?                                                                                                                    'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC460       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 29: SPHERICAL REFRACTING MIRRORS, IMAGE vs MAGNIFICATION                                                                                Problem # 29.1: A spherical mirror casts an inverted half-sized image    of an object that is located 110 in from its vertex.                    (a) Where will the image be located?                                                                                                           'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC461       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 30: SPHERICAL REFRACTING MIRRORS, IMAGE vs MAGNIFICATION                                                                                Problem # 30.1: An eraser 5 cm tall sits 150 cm in front of a concave    mirror of focal length 400 cm.                                          (a) Describe the image.                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE OPTC495       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The remainder of this files discusses some general analysis             applicable to any discipline which can be easily setup and              performed using other members of the ASTRO 3 Family.                                                                                   ╒═══════════════════════╕                                               │ FIT EQUATIONS TO DATA └──────────────────────────────────────────────                                                                         13.0       Finding Best Values of Transistor Parameters                          from Experimental Data by Nonlinear Least Squares.             ────────────────────────────────────────────────────────────────────────                                                                                 A very common problem in Engineering and Science is to estimate         the best values of the coefficients of a mathematical model             given a set of experimental data.  In the example discussed             here, the area of application is Electrical Engineering and the         current problem is to extract the best values of parameters             which characterize the behavior of a transistor.  By fitting            the theoretical equation, discussed below, to the experimental          data, one attempts to determine the DC current gain, BETA0, the         critical frequency, FC, and the transition frequency, FT.                                                                                       In this example, the values of the DC current gain, BETA0, and          the critical frequency, FC, for a transistor will be estimated          by fitting the theoretical equation of BETA as a function of            frequency F to an experimental data set in which the current            gain BETA is measured at 5 different frequencies.  The                  transition frequency, FT, is derived from the DC current gain           BETA0 and FC and is not directly extracted.                                                                                                     The current gain, BETA, and transition frequency, FT, of a              small signal rf transistor can be approximated as a function of         the measurement frequency, F, by the equations:                                                                                                      BETA=BETA0/SQR(1+F^2/FC^2)                                                                                                                      FT=BETA0*FC                                                                                                                                where:                                                                                                                                          BETA = current gain at frequency F                                                                                                              BETA0 = low frequency current gain, CE mode, zero AC                    potential between collector and emitter                                                                                                         FC = transistor cut-off frequency.  Current gain drops to               0.707 (-3dB) times the low frequency value                                                                                                      FT = transistor unity-gain frequency                                                                                                            The equipment used in these measurements operates to 10 MHz;            therefore, it is usually not possible to measure the unity-gain         frequency, FT, directly.  The data set has been generated by            measuring the current gain, BETA, vs.  frequency, F, from 1 to          10 MHz.  The experimentally measured data points are:                                                                                                F (MHz)         BETA                                                                                                                              1             105                                                       2.5           90                                                        5             80                                                        7.5           60                                                        10            50                                                                                                                         The Astro 3 program AstroDataFitter 3 (AFIT3.EXE) is used to            set up and solve this problem.  The "front end" of this program         is identical to that of AstroFormulaCalculator 3.  However,             AFIT3 is specialized to solve the general nonlinear regression          problems and the general polynomial regression which include            the popular linear least-squares methods.                                                                                                       After the equations, a comment and data are entered into AFIT3,         the equation display should appear similar to what is shown             below:                                                                                                                                 ......................................................................  --- Comment -->  Calculate BETA0 (DC current gain) and FC (critical     frequency) for a transistor by fitting BETA vs. F (MHz) to theoretical  equation.                                                                                                                                                    BETA=BETA0/SQR(1+F^2/FC^2)                   BETA =  0                                                                                                  FT=BETA0*FC                            FT    0                                                                                                                                                     11:14:05         *** Accuracy ?  Warning # 8 ***          10-20-1980                                                                                                                                                       BETA0 => 0                    F :  0                   FC =  0       ......................................................................                                                                                   The "Comment", if any, is displayed at the top.  The equations          BETA and FT, in the order in which they are entered, are                displayed from top down.  The time and date are displayed in            the middle along with the "Accuracy ?  Warning #8" Comment.             This frequently-seen warning Comment is used to inform the user         that the displayed values of the dependent variables BETA and           FT may not be related to the values based on the equations and          the independent variables.  To actually calculate the dependent         variables, one would enter a value for an independent variable          or type C.  Finally the 3 independent variables BETA0, F an FC          are listed at the bottom in alphabetical order.  The blinking           cursor ">" is positioned on BETA0.                                                                                                              Data points consist of two numbers:  a data coordinate and a            data value.  In the present problem, data are to be entered in          (F,BETA) pairs.  To designate F as the variable associated with         the data coordinate, move the cursor to F and type O.  The              equal sign "=" will be replaced by a colon ":".  A colon is             used to signify that variable F is now a data coordinate.  Only         one of the two equations, BETA, is to be associated with the            data value.  The other equation must be made "invisible" or             inactive.  Move the cursor to FT and type I.  The equal sign            after FT will disappear.  At this time, data points are defined         as (F,BETA) pairs.                                                                                                                              Next data are entered in (F,BETA) pairs, type D.  The screen            will appear as shown below and enter the data.                                                                                         .....................................................................                                                                                 --- Data point #  1  of  1                                                                                                                          Coord: F(1) =  0 <--                                                                                                                            Value: BETA(1) =  0                                                                                                                   .....................................................................                                                                                    Finally initial "guesses" for the two active independent                variables BETA0 and FC are set and the problem is solved.                                                                                       The Program will optimize the values of the active independent          variables BETA0 and FC, whose names are followed by the "=", in         order to attempt to minimize the sum of squared errors between          the calculated value of the single active dependent variable            (with the data coordinates of each data point substituted into          the coordinate variable) and the data value of each data point.         In this example, each of the five data coordinates will be              substituted into F, BETA will be calculated and then the                squared differences between BETA and the data value of each of          the five points will be summed and minimized.                                                                                                   The program will optimized BETA0 and FC to minimize the ERROR           and the iteration counter will "roll off" as the fit proceeds.          Note that the value of the coordinate variable F stays                  constant.  After about 64 iterations, the fit will converge as          the estimated significant figures approaches 16.  In this case          SIG FIGS reaches 14.  The ERROR will tend to minimize but it            will not approach 0.0 unless the fit is "perfect".  The screen          should appear as shown below.                                                                                                          ......................................................................  --- Comment -->  Calculate BETA0 (DC current gain) and FC (critical     frequency) for a transistor by fitting BETA vs. F (MHz) to theoretical  equation.                                                                                                                                                    BETA=BETA0/SQR(1+F^2/FC^2)               BETA =  104.1315                                                                                               FT=BETA0*FC                        FT    568.0856                                                                                                                                                  11:24:15  Iteration # 64   ERROR = 6.3589147  SIG FIGS = 14 10-20-1980                                                                                                                                                     BETA0 =  104.1315             F :  0                 FC =  5.455464  ......................................................................                                                                                   Obtaining a summary of results.  Following optimization, one            may want to list a summary of the results.  Type R.  The screen         should appear as below.                                                                                                                ..................................................................                                                                               #    F    BETA     CALCULATED      ERROR       SYMBOLIC ERROR                                                                                   1    1    105       102.425       2.575026              +++++                                                                                   2    2.5  90        94.66504     -4.66504       ---------                                                                                       3    5    80        76.76686      3.23314               +++++++                                                                                 4    7.5  60        61.25394     -1.253941            ---                                                                                       5    10   50        49.87006      .1299422              0                                                                                       #    F    BETA     CALCULATED      ERROR       SYMBOLIC ERROR                                                                                  ...................................................................                                                                                      The "results" screen list the data point number (#), the value          of the data coordinate (F), the value of the data value (BETA),         the calculated value of BETA based on the equation and the              optimized values of BETA0 and FC, the ERROR which is the actual         value of BETA less its calculated value and finally the                 SYMBOLIC ERROR which is a visual representation of the                  uncertainty in the fit.  For each point, the actual value of            BETA can be thought of as being in the middle under the C of            SYMBOLIC and would correspond to a 0 if the fit were                    more-or-less perfect.  "+" mean the fitted curve exceeds the            data value at this point and the number of "+" is a relative            measure of how much.  "-" means the fitted curve underestimates         the data value at this point and the number of "-" is a                 relative measure of how much.                                                                                                                   Plotting the fitted equation and experimental data.  The                equation and data can be plotted to verify the "fit" visually.          Type P.  Type Y (enter) in response to the prompt to include            data on plot.  The screen should appear similar to below:                                                                              .......................................................................                                                                         105.-█                                                                      |a a                                                                    |      a                                                                |         a                                                             |             a                                                      91.-            █  a                                                       |                   a                                                   |                     a                                                 |                         a                                             |                           a   █                                    77.-                               a                                       |                                 a                                     |                                     a                                 |                                        a                              |                                           a                        63.-                                               a                       |                                                  █a                   |                                                       a               |                                                           a  a        |                                                                a   49.-                                                                       ||--------------|--------------|--------------|--------------|-----      1             2.96           4.91           6.87           8.83                                    BETA[a]  vs. F                          .......................................................................                                                                                  The first equation BETA is represented by "a".  The 5 data              points as (TIME, BETA) points are plotted as "█".                                                                                               AstroDataFitter 3, and its comprehensive on-line help file with         many interactive examples, is available with registration for           AstroFormulaCalculator 3.  See below for details.                                                                                      ╒═══════════════════════════╕                                           │ NONLINEAR SIMUL EQUATIONS └───────────────────────────────────────────                                                                        14.0 Shooting a Marble into a Moving Ferris Wheel Car: Solution of               Simultaneous Nonlinear equations                               ────────────────────────────────────────────────────────────────────────                                                                                 In this example, a user will calculate the required angle of            elevation of his marble shooter in order for him to shoot his           marble into a moving ferris-wheel car.  A set of simultaneous           nonlinear equations must be solved in order to make                     predictions.                                                                                                                                    The marble shooter is located at the origin of a X-Y coordinate         system.  The ferris wheel is located a distance DWHEEL from the         shooter and has a radius RWHEEL about its center which is a             height of HWHEEL above the ground.  The wheel turns at a speed          of REVS revolutions per second.  A diagram of the problem is            shown below.                                                                                                                           .....................................................................                                                                                                                            FERRIS WHEEL                           path of marble                                           |                                                 |-/--|                |              *   *   *   *   *   *      RWHEEL  |/   |                |            *                        *           /    |                |          *                             *       /|    |                |        *                                 *___/_ |    |                |      *                                    _car_ |    |   HWHEEL       |   /*                                            |    |                |  / /                                            |    |                | / /                                             |    |                |/ /   <-ALPHA (angle of elevation)               |    |                |_/_______________________________________________|____|_____________  x=0                                                  x=DWHEEL           ......................................................................                                                                                   If all distances are in units of feet, all times are in                 seconds, there is no wind to divert the marble and air                  resistance is neglected, then the equations of motion for the           marble and ferris wheel can be shown to be:                                                                                                     XWHEEL = -RWHEEL*SIN(6.28*REVS*T)+DWHEEL                                                                                                        YWHEEL = RWHEEL*COS(6.28*REVS*T)+HWHEEL                                                                                                         XMARBLE = (LSHOOTER+VMARBLE*T)*COSD(ALPHA)                                                                                                      YMARBLE = (LSHOOTER+VMARBLE*T)*SIND(ALPHA)-16.08*T^2                                                                                            DELTAX = XWHEEL-XMARBLE                                                                                                                         DELTAY = YWHEEL-YMARBLE                                                                                                                         where:                                                                                                                                          XWHEEL = horizontal position of ferris-wheel car at time T                                                                                      RWHEEL = radius of wheel                                                                                                                        T = time in seconds reference to the ferris-wheel car being             located at the extreme top of the wheel at T=0                                                                                                  DWHEEL = distance from the bottom of the marble shooter to              directly below the center of the wheel.  The bottom of the              shooter is the origin.                                                                                                                          YWHEEL = vertical position of the ferris wheel car at time T                                                                                    HWHEEL = height of the center of the wheel above ground                                                                                         XMARBLE = horizontal position of marble during its flight               from the marble shooter.  Its initial speed is VMARBLE and              angle of elevation is ALPHA.                                                                                                                    LSHOOTER = length of the barrel of the marble shooter                                                                                           VMARBLE = exit speed of marble in feet/second from marble               shooter                                                                                                                                         ALPHA = angle of elevation of marble shooter in degrees above           the ground                                                                                                                                      YMARBLE = vertical position of marble during its flight from            the marble shooter.  Its initial speed is VMARBLE and angle             of elevation is ALPHA.                                                                                                                          DELTAX = difference in horizontal positions of ferris wheel             car and marble                                                                                                                                  DELTAY = difference in vertical position of ferris wheel car            and marble                                                                                                                                      In this example, the barrel of the marble shooter is 4 feet             long and initially has its elevation angle at 45 degrees.  It           is located 400 feet from directly below the center of the               wheel.  The ferris wheel rotates at 1 revolution in 20 seconds          (.05 revs/sec) and its radius is 40 feet.  The center of the            wheel is 45 feet above the ground.  With the built-in                   propulsion mechanism, the user can shoot the marble such that           its exit speed can reach a maximum speed of 150 feet/sec.                                                                                       If the user shoots the marble out of the shooter at T=0, what           must its angle of elevation be in order for the marble to land          in the car?  At what time T will it land in the car?                                                                                            The user needs to determine ALPHA and T.  The six equations             which govern the motions form a system of nonlinear equations.          The values of ALPHA and T can be determine by numerical methods         if reasonable initial "guesses" for ALPHA and T are supplied.                                                                                   The Astro 3 program AstroEquationSolver 3 (ASOL3.EXE) is used           to set up and solve this problem.  The "front end" of this              program is identical to that of AstroFormulaCalculator 3.               However ASOL3 is specialized to solve general nonlinear                 simultaneous equations and the special case of linear                   simultaneous equations.                                                                                                                         After the equations, a comment and data are entered into ASOL3,         the equation display should appear similar to what is shown             below:                                                                                                                                 ......................................................................  --- Comment -->  Shoot marble into ferris wheel car.  T is time in      seconds.  Alpha is angle of elevation of shooter.                                                                                                      XWHEEL=-RWHEEL*SIN(6.28*REVS*T)+DWHEEL           XWHEEL    0                                                                                     YWHEEL=RWHEEL*COS(6.28*REVS*T)+HWHEEL           YWHEEL    0                                                                                   XMARBLE=(LSHOOTER+VMARBLE*T)*COSD(ALPHA)         XMARBLE    0                                                                              YMARBLE=(LSHOOTER+VMARBLE*T)*SIND(ALPHA)-16.08*T^2    YMARBLE    0                                                                                             DELTAX=XWHEEL-XMARBLE                   DELTAX =  0                                                                                             DELTAY=YWHEEL-YMARBLE                   DELTAY =  0                                                                             11:24:06        *** Accuracy ?  Warning # 8  ***        12-18-1980                                                                                 ALPHA =  0               DWHEEL    0               HWHEEL    0                                                                               LSHOOTER    0                 REVS    0               RWHEEL    0                                                                                      T =  0              VMARBLE  > 0                                 ......................................................................                                                                                   Designate the active equations.  There are six equations;               however, only the last two, DELTAX and DELTAY, determine when           the marble lands in the car (at DELTAX=0 and DELTAY=0).  The            first four equations should be inactivated because they                 calculate intermediate results.  Note:  Inactive equations do           not display the "=" following the name of the dependent                 variable.                                                                                                                                       Designate the active independent variables.  The Program will           attempt to solve for the values of the active independent               variables only.  In this case they are ALPHA and T.  Only ALPHA         and T should have their names followed by "=".  The screen              should appear as shown above.                                                                                                                   At this time, the Program is set up to solve for the values of          ALPHA and T using equations DELTAX and DELTAY.  The marble will         reach the car when DELTAX=0 and DELTAY=0.  The Program will use         numerical methods to determine the values of ALPHA and T that           satisfy these equations starting with "reasonable" initial              values supplied by the user.                                                                                                                    One must now set DELTAX and DELTA to the desired values which           are to be solved for using the nonlinear method.  Set DELTAX to         0.0 and Set DELTAY to 0.0.  The Program will optimize the               values of the active independent variables, whose names are             followed by the "="( ALPHA and T), in order to attempt to               minimize the squared error between the calculated values of the         active dependent variables DELTAX and DELTAY and the desired            values of 0.0 and 0.0, respectively.  The desired values are            always the values of the active dependent variables at the              instant the optimization starts.  Following the optimization            for ALPHA and T, the converged (minimum error) solution should          appear similar to that shown below.                                                                                                    ......................................................................  --- Comment -->  Shoot marble into ferris wheel car.  T is time in      seconds.  Alpha is angle of elevation of shooter.                                                                                                      XWHEEL=-RWHEEL*SIN(6.28*REVS*T)+DWHEEL       XWHEEL    369.5772                                                                                  YWHEEL=RWHEEL*COS(6.28*REVS*T)+HWHEEL       YWHEEL    70.97021                                                                                XMARBLE=(LSHOOTER+VMARBLE*T)*COSD(ALPHA)      XMARBLE    369.5772                                                                          YMARBLE=(LSHOOTER+VMARBLE*T)*SIND(ALPHA)-16.08*T^2    YMARBLE    70.97                                                                                         DELTAX=XWHEEL-XMARBLE               DELTAX = -1.46E-05                                                                                          DELTAY=YWHEEL-YMARBLE                DELTAY => 1.40E-05                                                                              Iteration # 76   ERROR = 2.030347824645198D-05   SIG FIGS = 4                                                                                 ALPHA =  27.54638        DWHEEL    400             HWHEEL    45                                                                              LSHOOTER    4                 REVS    .05             RWHEEL    40                                                                                     T =  2.752203       VMARBLE    150                               ......................................................................                                                                                   The elevation angle is ALPHA=27.55 degrees.  The marble will            land in the car 2.75 seconds after launch and the car will be           369.6 feet from the marble shooter and 71 feet above the                ground.                                                                                                                                         It may be informative to plot the y-coordinates of the marble           and the ferris wheel car in order to determine if the marble            actually landed in the car or merely struck it and bounced              away.  For the marble to land in the car it must fall from              above into the car.  The results of the plot should be similar          to what is shown below.                                                                                                                .......................................................................  77.3-bb                                                                     |     *  bd   d   d                                                     |dd         b  b      d   d                                             |                b b          d  d                                      |                     b  b         d                                74.3-                          b b        d                                 |                               b  b      d                             |                                    b b    d          marble falls     |                                         b     d      into car at      |                                             b  bd  <-this time.   71.3-                                                  bd                   |                                                     db b              |                                                       d   b           |                                                         d     b       |                                                           d       68.2-                                                             d         |                                                               d       |                                                                       |                                                                       |                                                                   65.2-                                                                       ||--------------|--------------|--------------|--------------|----       2             2.22           2.43           2.65           2.87                         YWHEEL[b] YMARBLE[d]  vs. T                       .......................................................................                                                                                  The active equations are plotted with the symbols "b" and "d"           which represent YWHEEL and YMARBLE, respectively.  It can be            seen from the plot that the "d" points rise above the "b" at            about 2.15 seconds where the marble's height is about 76.7              feet.  Thus, the marble is on its way down when it reaches the          car and it will land in it!                                                                                                                     AstroEquationSolver 3, and its comprehensive on-line help file          with many interactive examples, is available with registration          for AstroFormulaCalculator 3.  See below for details.                                                                                  ╒════════════════════════╕                                              │ SIMUL LINEAR EQUATIONS └─────────────────────────────────────────────                                                                         15.0       Resource Allocation in an Instrument Company:  Solving                Simultaneous Linear equations                                  ────────────────────────────────────────────────────────────────────────                                                                                 This example will show how the resource planners in a small             instrument company can estimate what their current space                allocations must be in order for their company to grow to a             predicted size over a ten year period.  The company planners            will use a mathematical model generated by a consulting firm.           This model has been constructed from known growth patterns of           similarly sized companies.  One must solve simultaneous linear          equations in order to make predictions.                                                                                                         The instrument company allocates space for 6 major functions            which are:  1) demonstration laboratories, 2) research and              engineering labs, 3) office space, 4) storage, 5) warehouse and         6) parking lots.  The current allocations of space are                  predicted to change over a 10 year period by a consulting firm          who has studied the developments of similarly sized companies.          They have supplied the company planners with a Space Allocation         Table which is tabulated below.                                                                                                                                SPACE ALLOCATION TABLE                                                                                                                   Allocations at end of 10 year period                                                                                                             C means current; F, future                                                                                                             D - Demonstration labs, R - Research and engineering labs,                                                                                      O - Office space, S - Storage, W - Warehouse, P - Parking                                                                                                                                                                      CD      CR     CO     CS     CW     CP                                                                                                   FD    .90     .05    .07    .05     0      .05                                                                                                  FR    .04     .90    .05    .05     0      .05                                                                                                  FO    .05     .03    .75    .05    .20     .05                                                                                                  FS    .01     .01    .03    .51    .05      0                                                                                                   FW    0       .01    .05    .30    .60     .05                                                                                                  FP    0        0     .05    .04    .15     .80                               ---      ---    ---    ---    ---     ---                                1        1      1      1      1        1                                                                                                  Each column represents the future distribution of the given             space.  For example, the first column represents the                    demonstration lab space.  If the current allocation is 1 unit,          then 90 % will remain as demonstration labs after 10 years; 4 %         will be converted to research labs; 5 %, to office space; 1 %,          to storage; 0 %, to warehouse space and 0 % to parking.  A              similar breakdown holds for the future space allocations                represented by columns 2 through 6.  Note that all columns must         add up to 1 (or 100 %).                                                                                                                         The company planners want to organize the current space in              accordance with the space-conversion factors so that their              resources can grow to their expectations 10 years from now.             Such early planning is hoped to eventually minimize expensive           conversions and remodeling as the revenue and size of the               instrument company grow.                                                                                                                        The information in the Table can be arranged into 6 equations           which are formulated by summing the conversion factors along            each row.  These six equations are:                                                                                                               FD = .9*CD+.05*CR+.07*CO+.05*CS+.05*CP                                  FR = .04*CD+.9*CR+.05*CO+.05*CS+.05*CP                                  FO = .05*CD+.03*CR+.75*CO+.05*CS+.20*CW+.05*CP                          FS = .01*CD+.01*CR+.03*CO+.51*CS+.05*CW                                 FW = .01*CR+.05*CO+.3*CS+.6*CW+.05*CP                                   FP = .05*CO+.04*CS+.15*CW+.80*CP                                                                                                              Ten years from now, the company planners want the future                percentages of space to be allocated:                                                                                                             FD = 10 %                                                               FR = 10 %                                                               FO = 20 %                                                               FS = 5 %                                                                FW = 40 %                                                               FP = 15 %                                                                                                                                     What should the current space allocations be in order for the           future space allocations to reach the predicted levels?                                                                                         The 6 equations described above form a group of simultaneous            linear equations.  The equations can be solved for the current          values (CD, CR, CO, CS, CW and CP) given the future values (FD,         FR, FO, FS, FW and FP).  After the equations, comment and               values for the dependent variables are entered into                     AstroEquationSolver 3, the equation display should look similar         to what is shown below.                                                                                                                ......................................................................  --- Comment -->  Space allocations --- D demo labs,R research,O office, S storage,W warehouse,P parking--- C current,F future                                                                                                   FD=.9*CD+.05*CR+.07*CO+.05*CS+.05*CP                FD =  10                                                                                    FR=.04*CD+.9*CR+.05*CO+.05*CS+.05*CP                FR =  10                                                                                FO=.05*CD+.03*CR+.75*CO+.05*CS+.20*CW+.05*CP            FO =  20                                                                                    FS=.01*CD+.01*CR+.03*CO+.51*CS+.05*CW               FS =  5                                                                                      FW=.01*CR+.05*CO+.3*CS+.6*CW+.05*CP                FW =  40                                                                                       FP=.05*CO+.04*CS+.15*CW+.80*CP                   FP => 15                                                                                                                                                    13:52:54      *** Accuracy ?  Warning # 8  ***            12-20-1980                                                                                  CD =  0                   CO =  0                   CP =  0                                                                                     CR =  0                   CS =  0                   CW =  0       ......................................................................                                                                                   To solve this problem, one supplies values for the dependent            variables FD ....FP and the Program calculates the values of            the independent variables CD..CW.  This type of calculation is          "inverted" with respect to most calculations.  These equations          are linear with respect to the unknowns CD..CW.  However, they          can also be solved by nonlinear methods, discussed above, which         are more general.  For the time being, the equations will be            solved as linear equations.  In the event they were not linear          equations, the Program will inform the user with a warning              comment and the nonlinear method should then be tried.                                                                                          The future percentages are set to their desired values as as            shown above.  Notice that the values of CD..CW are all zero and         the warning comment #8 informs the user that there may be, and          is not in this case, any relationship between the displayed             values of FD...FP and the values of CD..CW.                                                                                                     To solve for CD..CW using linear methods, Some conditions must          be satisfied, one of which is that the number of active                 independent and independent variables must be equal.  In this           example, the numbers are both 6.  The Program will solve the            simultaneous equations and display the current allocation               percentages required in order to achieve the future ones at the         end of the 10 year period.  Notice that the calculated values           of both the current and future allocations always sum to 100.           The screen should now look similar to what is shown below.                                                                             ......................................................................  --- Comment -->  Space allocations --- D demo labs,R research,O office, S storage,W warehouse,P parking--- C current,F future                                                                                                   FD=.9*CD+.05*CR+.07*CO+.05*CS+.05*CP                FD =  10                                                                                    FR=.04*CD+.9*CR+.05*CO+.05*CS+.05*CP                FR =  10                                                                                FO=.05*CD+.03*CR+.75*CO+.05*CS+.20*CW+.05*CP            FO =  20                                                                                    FS=.01*CD+.01*CR+.03*CO+.51*CS+.05*CW               FS =  5                                                                                      FW=.01*CR+.05*CO+.3*CS+.6*CW+.05*CP                FW =  40                                                                                       FP=.05*CO+.04*CS+.15*CW+.80*CP                   FP => 15                                                                                                                                                    13:58:43                                                   12-20-1980                                                                                 CD =  9.458092            CO =  7.996075            CP =  6.118691                                                                              CR =  9.757506            CS =  2.683652            CW =  63.98599........................................................................                                                                                 AstroEquationSolver 3, and its comprehensive on-line help file          with many interactive examples, is available with registration          for AstroFormulaCalculator 3.  See below for details.                                                                                  ╒═══════════╕                                                           │ GLOSSARY  └───────────────────────────────────────────────────────────                                                                        16.0     Glossary of Terms                                              ────────────────────────────────────────────────────────────────────────                                                                               ║ Note, Use the Jump command to locate a glossary reference.    ║       ║ Scroll the screen until the reference is in the middle at line║       ║ 12.  The Jump box will light up.  Type J to locate.  Repeat   ║       ║ typing A, as needed, to continue the search.                  ║                                                                                                                                                Cartesian ovoid.  See OPTC402.                                                                                                                Compound thick lenses.  See OPTC443.                                                                                                          Compound thin lenses.  See OPTC427.                                                                                                           Integration.  See OPTCELIN.                                                                                                                   IR Source (blackbody).  See OPTC1946.                                                                                                         Lens Imagery.  See OPTC419.                                                                                                                   Mirrors.  See OPTC495.                                                                                                                        Series.  See OPTCELSE.                                                                                                                        Spherical refracting surfaces.  See OPTC405.                                                                                                  Radiation.  See OPTCSLWV.                                                                                                                     Thick Lenses.  See OPTC442.                                                                                                                   Thin Lenses.  See OPTC410.                                                                                                                    ╒═════════╕                                                             │ LICENSE └─────────────────────────────────────────────────────────────                                                                        17.0         Product License and Warranty.                              ────────────────────────────────────────────────────────────────────────                                                                                       Software License Agreement  (Registered Users)                                                                                            If you are a registered user, PCSCC grants you the right to use         this Software for a single-user computer or a workstation not           attached to a network.  You may not rent or lease this Software         without the expressed written permission of PCSCC.  You may not         decompile, disassemble, reverse engineer, copy, create a                derivative work, or otherwise use this Software except as               stated in this agreement.  The license authorizes you to create         and use an extra copy of the Software on a home or laptop               computer, as long as the extra copy is never loaded at the same         time this Software is loaded on the primary computer on which           you use this Software.                                                                                                                          If any provision of this agreement is found to be unlawful,             void, or unenforceable, then that provision shall be severed            from this Agreement and will not affect the validity and                enforceability of any of the remaining provisions.  This                agreement shall be governed by the laws of the State of                 Delaware.                                                                                                                                       The Software and accompanying materials are provided with               Restricted Rights.  Use, duplication, or disclosure by the              Government is subject to restrictions as set forth in                   subparagraph (c) (1) (ii) of The Rights in Technical Data and           Computer Software clause at DFARS 252.227-7013, or                      subparagraphs (c) (1) and (2) of the Commercial Computer                Software--Restricted Rights at 48 CFR 52.227-19, as applicable.         Contractor/manufacturer is PC SuperComputer Capital,                    Incorporated, at the address listed below.                                                                                                      Non registered users must comply with the Shareware policy              described elsewhere in this software Pacakage.                                                                                                  Type T to return to the Table of Contents.                                                                                                                       Warranty Information                                                                                                           PC SuperComputer Capital, Inc.  (PCSCC) hereby disclaims all            warranties relating to this Software, whether express or                implied, including without limitation any implied warranties of         merchantability or fitness for a particular purpose.  PCSCC             will not be liable for any special, incidental, consequential,          indirect or similar damages due to loss of data, loss of                profits, lost savings or any other reason, even if PCSCC or an          agent of PCSCC has been advised of the possibility of such              damages.  In no event shall PCSCC's liability for any damages           ever exceed the price paid for the license to use Software,             regardless of the form of the claim.  The person using the              Software bears all risk as to the quality and performance of            the Software.  This Software is licensed AS IS.                                                                                                 Any action or proceeding brought by either party against the            other arising out of or related to this agreement shall be              brought in a STATE or FEDERAL COURT of competent jurisdiction           located in Fairfield County, Connecticut.  The parties hereby           consent to in personam jurisdiction of said courts.                                                                                             Type T to return to the Table of Contents.                                                                                             ╒══════════════╕                                                        │ USER SUPPORT └────────────────────────────────────────────────────────                                                                        18.0                    User Support.                                   ────────────────────────────────────────────────────────────────────────                                                                                 This program has undergone over 5 years of "beta testing"               around the world.  The algorithms have been in use since 1976,          are proven and believed highly reliable.  However,                      comprehensive programs of this nature may develop what are              called "bugs".  User support is provided to all registered              users for a minimum of 3 months without charge.  Registered             users can receive such help via phone/mail.  Registration also          enrolls the user for notification of "bug" fixes, upgrades, new         versions, new products and many more benefits.  For the phone           number/mail address see below or position the jump line here            and type J.  ∙Registration∙.  Support will be provided to             non-registered users and users beyond the free support period           on a time and materials basis.                                                                                                         ╒═══════════════╕                                                       │ PRODUCT  INFO └───────────────────────────────────────────────────────                                                                        19.0         Product Information and How to Order Complete Package.     ────────────────────────────────────────────────────────────────────────                                                                                AstroFormulaCalculator 3  Product Description  7-1-1993                                                                                          AstroFormCalc 3 is a unique and innovative mathematics tool             which offers productivity enhancements not found in any other           software package either retail or shareware.  This hands-on             program combines the functions of both a spreadsheet and                calculator; but it is better.  Why?  Because it is right                between the two:  Easier than a spreadsheet and more powerful           than a calculator.  With its special format-free interface,             formula or equation-based iterations are much easier to set up          than in a spreadsheet.  It is designed to be "smarter" which            makes it very user-friendly for Professionals of all levels,            especially for the infrequent or "computer-shy" user.  Broad            applications exist in many disciplines where calculations with          formulas, equations or other math recipes are required such as          in Economics, Genetics, Chemistry, Electrical Engineering,              Engineering Mechanics, Fluid Mechanics & Hydraulics, Finance,           Physics, Statistics and even Nursing.  Features include:                                                                                        DOS or Windows as non-Windows application.  Runs on all PC              compatibles.  DOS 2.1+, Monochrome display, 200K RAM, floppy or         disk.                                                                                                                                           DOS command line support for any direct calculation such as             addition, multiplication, subtraction and division.                                                                                             Format-free entry of up to 63 equations with at least 256               independent variables and 256 constants.  Smart and                     "hassle-free" interpreter.                                                                                                                      Equations and formulas are written according to the BASIC               conventions.  Versatile library of built-in functions.                                                                                          Advanced calculator functions including many automated                  numerical methods such as:                                                                                                                      ∙ Newton-Raphson for solving x in y=f(x)                                ∙ Advanced simplex for solving x at W=Y=Z in W=f(x), Y=g(x),              Z=h(x)                                                                ∙ Numerical integration (two methods)                                   ∙ Numerical differentiation                                             ∙ Summation on integer index                                            ∙ Finding maxima and minima                                             ∙ Plotting equations                                                    ∙ Saving/retrieving data files with full file management                  support                                                               ∙ Screen dumps as DOS text files for inclusion in documents                                                                                     Highly-structured documentation, on-line with Table of                  Contents, Glossary and search function.  Interactive with               program.                                                                                                                                        ∙ On-line helpfile.  Comprehensive summary of                             commands/functions with step-by-step examples on uses of all            advanced functions.                                                   ∙ On-line user manual.  A "learn by example" document which               illustrates the universal nature of problem-solving with many           step-by-step examples from Science, Engineering and Finance.                                                                                  Full support for ACTION21 and ACTION11 data sets.                                                                                               Comments from users:                                                                                                                     ∙ "... calculate circuit board drilling and routing speeds...easy to       use, versatile and powerful with good documentation."                                                                    J. L.  New York     ∙ "...used the integration capability to determine defect density...do     not have any tool (either PC or host based) that does for me what       ...(AstroFormCalc 3)...does."                                                                                            J. R.  Virginia     ∙ "...for Electrical Engineering practice it is better than a              spreadsheet and a calculator..."                                                                                         C. B.  Florida      ∙ "...used to do frequency spectrum work and calculations on circuits,     ...of real value and a productivity tool."                                                                               E. S.  California   ∙ "...one of a handful of scarcely available packages...in the types       of numeric problems they solve."                                                                                         W. C.  Chile                                                                                       This productivity-enhancing tool has been developed by the team         at PC SuperComputer Capital, Inc.  Order the complete version           of AstroFormulaCalculator 3 today!                                                                                                     ╒════════════════╕                                                      │ OTHER PRODUCTS └──────────────────────────────────────────────────────                                                                        20.0              Software Included in Astro 3 Registration             ────────────────────────────────────────────────────────────────────────                                                                                 Register for AstroFormulaCalculator 3 and receive, in addition          to the most recent version of ACAL3.EXE and ACAL3HLP.HLP, these         3 powerful math analysis programs:  AstroEquationSolver 3,              AstroDataFitter 3 and AstroShadowBox 3.  Each has its own               extensive and interactive on-line helpfile written just like            the one your reading now.  A print program is included which            prints theses helpfiles to the standard printer or to a DOS             text file.  All Astro 3 programs are written with the same              high-level user-friendly front-end requiring minimal effort to          setup and execute relatively elaborate calculations.  All data          sets and helpfiles are compatible among members of the family.          A brief summary of the other members of the Astro 3 shareware           family included with the registration package follows:                                                                                    ∙ AstroEquationSolver 3 - Solves both linear and nonlinear                    simultaneous equations with up to 30 unknowns.  Characterized           by extremely easy setup.  Uses active/inactive method of                AstroFormulaCalculator 3 to select variables.  Strictly high            level.  No reference to matrices at all!.  Includes program             ASOL3.EXE and extensive on-line help file ASOL3HLP.HLP that             includes many learn-and-use examples.                                                                                                     ∙ AstroDataFitter 3 - Fit data to any algebraic expression.                   Nonlinear optimization.  Fully-automated polynomial regression          with automatic "equation generator".  Easy to use, even by a            novice.  Will generate interpolated tables from fitted                  equations.  Up to 30 "active" variables.  Up to 500 data                points.  "Smart" algorithm requires minimal user input to solve         problems.  DOS text plots of "fit".  Point-by-point listing of          fitted data.  Residual variance, correlation coefficient,               coefficient of determination and much more!  Includes program           AFIT3.EXE and extensive on-line help file AFIT3HLP.HLP that             includes many illustrated examples.                                                                                                       ∙ AstroShadowBox 3 - The pop-up resident member (TSR) of the                  Astro 3 team.  Provides math support for other DOS programs.            Pop-up via hot key, surround calculation in "shadow box", and           type "(enter)" to pass result down through keyboard buffer or           pop up, enter calculation and pass result down.  Great for              adding powerful and interactive math capabilities to DOS text           editors such as Brief (tm, Borland), PE2 (tm, IBM) or                   WordPerfect (tm, WordPerfect).  Extensive features also include         screen dumps with appends as DOS text files or as parsed                columns in 123 (tm, Lotus) .WK1 files.  Requires as little as           15K of main memory.  Includes program ABOX3.EXE and on-line             help file ABOX3HLP.HLP                                                                                                                          If you like Engineering, Science or Finance and you have ideas          that you could transform into an interactive helpfile like              SCIFIN.HLP (The Astro 3 Science Manual) or PHYCAL.HLP ( The             Astro 3 Physics Manual) or FINCAL.HLP (The Astro 3 Finance              Manual), then you should write one and market it!  If you write         it in a commercial spreadsheet format, you will not likely be           able to distribute a fully functional package which is the mark         of a true Shareware program and the policy of the Association           of Shareware Professionals!  However, the AstroUtility 3                program provides the tools needed to generate an Astro 3                interactive application file with a minimum of effort AND               REGISTERED USERS OF THE AstroUtility 3 PROGRAM RECEIVE A LIMTED         DISTRIBUTION LICENSE WHICH ALLOWS INCLUSION OF ANY OR ALL OF            THE ASTRO 3 PROGRAMS IN YOUR PACKAGE!                                                                                                     ∙  AstroUtility 3  -   Support for authors to construct their own             on-line application help files.  Develop equation files in your         favorite editor and translate them into .EQU format for final           "test and touch-up".  Likewise, develop an application file in          your favorite editor, automatically include information from            previously developed .EQU files, and then compile a                     full-feature .HLP with automatic Table of Contents, line                numbers and so on.  .HLP files support all apostrophe commands          and many other useful features such as CalcMerge with lets you          dynamically stuff values calculated in the main equation                display into the application file.  As with the other Astro 3           programs, AstroUtility 3 comes with its own interactive and             extensive on-line help file which can be printed, if needed.            Application files can contain up to 64,000 lines!                                                                                               See below for instructions on ∙Registration∙.                                                                                          ╒══════════════╕                                                        │ REGISTRATION └───────────────────────────────────────────────────────                                                                         21.0                 Order/Registration Form                            ────────────────────────────────────────────────────────────────────────                                                                                 If a printer is available, position the order form (part 1) on          the screen and type the "(print screen)" key to make a copy.            Then, advance the text and position the order form (part 2) on          the screen and type "(print screen)" again to finish.                                                                                                  ASTRO 3 Products Order/Registration Form  (part 1)                                                                                  Name:________________________________  Title:________________                                                                                   Company:_____________________________________________________                                                                                   Address:____________________________ City:___________________                                                                                   State:___________   Zip:_____________  Country:______________                                                                                   Date:_____/_____/_____           (Fill out form completely)                                                                                     ---------------------------------------------------------------         Product                     #     Price/ea     S&H       Total          ---------------------------------------------------------------         AstroFormulaCalculator 3   ___    _$29.95_    _$5.00_   _______         Complete Package includes                                               ACAL3, ASOL3, AFIT3 + ABOX3                                                                                                                     AstroUtility 3 + ASTRO3    ___    _$99.95_    _$5.00_   _______         distribution license                                                ------------------------------------------------------------------------------------------------------------------------------------------------                ASTRO3 Products Order/Registration Form  (part 2)                                                                                   * Canadian Orders, Add $2.00 per copy                   _______                                                                                 * International Orders, Add $6.00 per copy              _______                                                                                 * CT Residents add appropriate sales tax                _______                                                                                                                  TOTAL ENCLOSED      __________                                                                                 Disk Size: [ ] 5-1/4"   [ ] 3-1/2"   [ ] Check  [ ] Visa  [ ] MC                                                                                Card# __________________________________     Exp. Date ________                                                                                 Signature ______________________________                                                                                                        How did you hear about our Products?    [ ] BBS   [ ] Friend            [ ] Other _____________________________________________________                                                                                               Mail Completed form with Payment to:                                                                                                                  PCSCC, Inc.   Dept. OPTC                                                   P. O. Box 173                                                        Botsford,  CT   06404  USA                                                                                                  Support ONLY,  1-203-270-9079     E-mail to CIS, 72263,1657                                                                                          [Mar 1994]  Prices subject to change without notice.                                                                                                  Type T to return to the Table of Contents.                                                                                       ╒═════════════════════╕                                                 │ MANUALLY INSTALLING └─────────────────────────────────────────────────                                                                        22.0           Instructions for Manually Installing Package.            ────────────────────────────────────────────────────────────────────────                                                                             The following are minimal instructions for manually installing                                AstroFormulaCalculator 3.                                                                                              Minimum distribution files:  README.1ST, ACAL3.ICO, ACAL3.EXE,          ACAL3HLP.HLP and OPTCAL.HLP with its *.EQU files.                                                                                               These files will probably be bundled in a compressed file such as       OPTCAL.ZIP, OPTCAL.SDN, or something similar.  Please unload.                                                                                   Create a DOS directory named ASTRO3 and change to it:                   cd\ (enter)    md\astro3 (enter)  cd \astro3 (enter)                                                                                            Copy ACAL3.EXE, ACAL3HLP.HLP, ACAL3.ICO and (optionally) OPTCAL.HLP     and all *.EQU file into the directory ASTRO3                                                                                                    Add these statements to AUTOEXEC.BAT in the base directory:             SET ASTRO3=C:\ASTRO3                                                    append to existing PATH statement  ;C:\ASTRO3;                                                                                                  Running Under Windows as a Non-Windows Application                                                                                             Follow the installation instructions and install on the hard drive C.   Start WINDOWS (tm, MicroSoft).  Click on FILE then click on PROPERTIES. Enter the following information for each of the fields.  Description:   AstroFormCalc3, Command:  ACAL3.EXE, Working Directory:  \ASTRO3.       Click on CHANGE ICON.  Enter Filename:  C:\astro3\acal3.ico.  Click on  OK to exit and click on OK to exit FILE.  The AstroFormCalc 3 icon      should appear in the accessory window.  Click on it to run ACAL3.                                                                                To start:  cd\astro3 (enter)    acal3 optcal.hlp (enter)                Read the optics manual.  Access on-line help (file: ACAL3HLP.HLP) from  equations display with (F1) key.                                                                                                              ╒═════════════╕                                                         │ AUTHOR ???? └─────────────────────────────────────────────────────────                                                                        23.0            Interested in Becoming the Author of Your Own                         Interactive Application File?                             ────────────────────────────────────────────────────────────────────────                                                                                 Many technologies are based on math/algebra.                            AstroFormulaCalculator 3 and its interactive, problem-solving           application files offer distinct advantages to the Author.  The         *.EQU templates are easy to develop and PCSCC offers complete           support for compiling these into the interactive application            files such as the help file ACAL3HLP.HLP or OPTCAL.HLP (The             Astro 3 Science Manual) or PHYCAL.HLP (The Astro 3 Physics              Manual) or FINCAL.HLP (The Astro 3 Finance Manual).                     Application files can be written for any level of user from the         very beginner to the Ph.D.  and distributed with any or all of          the ASTRO 3 family of programs.  See the OTHER PRODUCTS               section above for more details.  To automatically jump type "J"         if the jump bar is highlighted on those words.                                                                                         ╒═══════════╕                                                           │ HELP FILE └───────────────────────────────────────────────────────────                                                                        24.0          Help File: Summary of File Viewing Commands               ────────────────────────────────────────────────────────────────────────                                                                            A - Auto-search of entire text for user-supplied string. Search is          in uppercase and is case insensitive. If string found,                  type A to continue or to enter new string, type (space) A.          B,N - Move back or ahead 18 lines.  Also use (backspace) for "B".       E, (esc) - Exit from help file to equations display.                    F,T,K - Jump to the first (beginning), T.O.C or last (end) line.        G - Skip to Glossary of terms or summary of functions, if present.      J,(ent) - Search line on │JUMP│, Reference is after . Jump to                    line number or perform autosearch ahead if text reference.    L,P - Jump to line/page. Type Lxxx(ent) or Pyy(ent) to jump to                line #xxx or page #yy. With + or -, ahead or back 56 lines.       S,R - Skip ahead (S) or back (R) in help file to major headings         W - Skip to Summary of useful calculations or warnings, if present      1 to 9 - Typing 1 to 9 advances the text by 1 to 9 lines.               (ctrl Q) - Exit from help file.  Same as command "E".                   Edit keys - Use edit keys: (Home) (up arrow) (PgUp) (left arrow)                    (right arrow) (End) (down arrow) (PgDn).                                                                                             Entry into the application file is at/near the last position.           Type: T to return to Table of Contents or E to exit the file.      