/ Amiga Tools 3 / Amiga Tools 3.iso / grafik / raytracing / rayshade-4.0.6.3 / doc / guide / texture.tex

< prev

next >

MacOS 8.1

Win98

DOS

view JSON data

view as text

+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 63 68 61 70 74 65 72 | 20 7b 54 65 78 74 75 72 |\chapter| {Textur|
|00000010| 69 6e 67 7d 0a 0a 54 65 | 78 74 75 72 65 73 20 61 |ing}..Te|xtures a|
|00000020| 72 65 20 75 73 65 64 20 | 74 6f 20 6d 6f 64 69 66 |re used |to modif|
|00000030| 79 20 74 68 65 20 61 70 | 70 65 61 72 61 6e 63 65 |y the ap|pearance|
|00000040| 20 6f 66 20 61 6e 20 6f | 62 6a 65 63 74 20 74 68 | of an o|bject th|
|00000050| 72 6f 75 67 68 20 74 68 | 65 0a 75 73 65 20 6f 66 |rough th|e.use of|
|00000060| 20 70 72 6f 63 65 64 75 | 72 61 6c 20 66 75 6e 63 | procedu|ral func|
|00000070| 74 69 6f 6e 73 2e 20 20 | 41 20 74 65 78 74 75 72 |tions.  |A textur|
|00000080| 65 20 6d 61 79 20 6d 6f | 64 69 66 79 20 61 6e 79 |e may mo|dify any|
|00000090| 20 73 75 72 66 61 63 65 | 20 63 68 61 72 61 63 74 | surface| charact|
|000000a0| 65 72 69 73 74 69 63 2c | 0a 73 75 63 68 20 61 73 |eristic,|.such as|
|000000b0| 20 64 69 66 66 75 73 65 | 20 63 6f 6c 6f 72 2c 20 | diffuse| color, |
|000000c0| 72 65 66 6c 65 63 74 69 | 76 69 74 79 2c 20 6f 72 |reflecti|vity, or|
|000000d0| 20 74 72 61 6e 73 70 61 | 72 65 6e 63 79 2c 20 6f | transpa|rency, o|
|000000e0| 72 20 69 74 20 6d 61 79 | 20 0a 6d 6f 64 69 66 79 |r it may| .modify|
|000000f0| 20 74 68 65 20 73 75 72 | 66 61 63 65 20 6e 6f 72 | the sur|face nor|
|00000100| 6d 61 6c 20 28 60 60 62 | 75 6d 70 20 6d 61 70 70 |mal (``b|ump mapp|
|00000110| 69 6e 67 27 27 29 20 69 | 6e 20 6f 72 64 65 72 20 |ing'') i|n order |
|00000120| 74 6f 20 67 69 76 65 20 | 74 68 65 0a 61 70 70 65 |to give |the.appe|
|00000130| 61 72 61 6e 63 65 20 6f | 66 20 61 20 72 6f 75 67 |arance o|f a roug|
|00000140| 68 20 73 75 72 66 61 63 | 65 2e 0a 0a 41 6e 79 20 |h surfac|e...Any |
|00000150| 6e 75 6d 62 65 72 20 6f | 66 20 74 65 78 74 75 72 |number o|f textur|
|00000160| 65 73 20 6d 61 79 20 62 | 65 20 61 73 73 6f 63 69 |es may b|e associ|
|00000170| 61 74 65 64 20 77 69 74 | 68 20 61 6e 20 6f 62 6a |ated wit|h an obj|
|00000180| 65 63 74 2e 20 20 49 66 | 20 6d 6f 72 65 20 74 68 |ect.  If| more th|
|00000190| 61 6e 0a 6f 6e 65 20 74 | 65 78 74 75 72 65 20 69 |an.one t|exture i|
|000001a0| 73 20 73 70 65 63 69 66 | 69 65 64 2c 20 74 68 65 |s specif|ied, the|
|000001b0| 79 20 61 72 65 20 61 70 | 70 6c 69 65 64 20 69 6e |y are ap|plied in|
|000001c0| 20 74 68 65 20 6f 72 64 | 65 72 20 67 69 76 65 6e | the ord|er given|
|000001d0| 2e 20 20 54 68 69 73 20 | 61 6c 6c 6f 77 73 0a 6f |.  This |allows.o|
|000001e0| 6e 65 20 74 6f 20 63 6f | 6d 70 6f 73 65 20 74 65 |ne to co|mpose te|
|000001f0| 78 74 75 72 69 6e 67 20 | 66 75 6e 63 74 69 6f 6e |xturing |function|
|00000200| 73 20 61 6e 64 20 63 72 | 65 61 74 65 2c 20 66 6f |s and cr|eate, fo|
|00000210| 72 20 65 78 61 6d 70 6c | 65 0a 61 20 74 69 6c 65 |r exampl|e.a tile|
|00000220| 64 20 6d 61 72 62 6c 65 | 20 67 72 6f 75 6e 64 20 |d marble| ground |
|00000230| 70 6c 61 6e 65 20 75 73 | 69 6e 67 20 74 68 65 20 |plane us|ing the |
|00000240| 7b 5c 65 6d 20 63 68 65 | 63 6b 65 72 7d 20 61 6e |{\em che|cker} an|
|00000250| 64 20 7b 5c 65 6d 20 6d | 61 72 62 6c 65 7d 0a 74 |d {\em m|arble}.t|
|00000260| 65 78 74 75 72 65 73 2e | 0a 0a 54 65 78 74 75 72 |extures.|..Textur|
|00000270| 65 73 20 61 72 65 20 61 | 73 73 6f 63 69 61 74 65 |es are a|ssociate|
|00000280| 64 20 77 69 74 68 20 6f | 62 6a 65 63 74 73 20 62 |d with o|bjects b|
|00000290| 79 20 66 6f 6c 6c 6f 77 | 69 6e 67 20 74 68 65 20 |y follow|ing the |
|000002a0| 6f 62 6a 65 63 74 20 73 | 70 65 63 69 66 69 63 61 |object s|pecifica|
|000002b0| 74 69 6f 6e 0a 62 79 20 | 61 20 6e 75 6d 62 65 72 |tion.by |a number|
|000002c0| 20 6f 66 20 6c 69 6e 65 | 73 20 6f 66 20 74 68 65 | of line|s of the|
|000002d0| 20 66 6f 72 6d 3a 0a 0a | 5c 62 65 67 69 6e 7b 63 | form:..|\begin{c|
|000002e0| 65 6e 74 65 72 7d 0a 7b | 5c 74 74 20 74 65 78 74 |enter}.{|\tt text|
|000002f0| 75 72 65 7d 20 7b 5c 65 | 6d 20 6e 61 6d 65 7d 20 |ure} {\e|m name} |
|00000300| 24 3c 24 7b 5c 65 6d 20 | 54 65 78 74 75 72 69 6e |$<${\em |Texturin|
|00000310| 67 20 41 72 67 75 6d 65 | 6e 74 73 7d 24 3e 24 20 |g Argume|nts}$>$ |
|00000320| 5b 7b 5c 65 6d 20 54 72 | 61 6e 73 66 6f 72 6d 61 |[{\em Tr|ansforma|
|00000330| 74 69 6f 6e 73 7d 5d 0a | 5c 65 6e 64 7b 63 65 6e |tions}].|\end{cen|
|00000340| 74 65 72 7d 0a 0a 54 72 | 61 6e 73 66 6f 72 6d 61 |ter}..Tr|ansforma|
|00000350| 74 69 6f 6e 73 20 6d 61 | 79 20 62 65 20 61 70 70 |tions ma|y be app|
|00000360| 6c 69 65 64 20 74 6f 20 | 74 68 65 20 74 65 78 74 |lied to |the text|
|00000370| 75 72 65 20 69 6e 20 6f | 72 64 65 72 20 74 6f 2c |ure in o|rder to,|
|00000380| 20 66 6f 72 20 65 78 61 | 6d 70 6c 65 2c 0a 73 68 | for exa|mple,.sh|
|00000390| 72 69 6e 6b 20 6f 72 20 | 67 72 6f 77 20 66 65 61 |rink or |grow fea|
|000003a0| 74 75 72 65 20 73 69 7a | 65 2c 20 63 68 61 6e 67 |ture siz|e, chang|
|000003b0| 65 20 74 68 65 20 6f 72 | 69 65 6e 74 61 74 69 6f |e the or|ientatio|
|000003c0| 6e 20 6f 66 20 66 65 61 | 74 75 72 65 73 2c 20 61 |n of fea|tures, a|
|000003d0| 6e 64 0a 63 68 61 6e 67 | 65 20 74 68 65 20 70 6f |nd.chang|e the po|
|000003e0| 73 69 74 69 6f 6e 20 6f | 66 20 66 65 61 74 75 72 |sition o|f featur|
|000003f0| 65 73 2e 0a 0a 53 65 76 | 65 72 61 6c 20 6f 66 20 |es...Sev|eral of |
|00000400| 74 68 65 20 74 65 78 74 | 75 72 69 6e 67 20 66 75 |the text|uring fu|
|00000410| 6e 63 74 69 6f 6e 73 20 | 74 61 6b 65 20 74 68 65 |nctions |take the|
|00000420| 20 6e 61 6d 65 20 6f 66 | 20 61 20 63 6f 6c 6f 72 | name of| a color|
|00000430| 6d 61 70 20 61 73 20 61 | 6e 0a 61 72 67 75 6d 65 |map as a|n.argume|
|00000440| 6e 74 2e 20 20 41 20 63 | 6f 6c 6f 72 6d 61 70 20 |nt.  A c|olormap |
|00000450| 69 73 20 32 35 36 2d 6c | 69 6e 65 20 41 53 43 49 |is 256-l|ine ASCI|
|00000460| 49 20 66 69 6c 65 2c 20 | 77 69 74 68 20 65 61 63 |I file, |with eac|
|00000470| 68 20 6c 69 6e 65 20 63 | 6f 6e 74 61 69 6e 69 6e |h line c|ontainin|
|00000480| 67 0a 74 68 72 65 65 20 | 73 70 61 63 65 2d 73 65 |g.three |space-se|
|00000490| 70 61 72 61 74 65 64 20 | 76 61 6c 75 65 73 20 72 |parated |values r|
|000004a0| 61 6e 67 69 6e 67 20 66 | 72 6f 6d 20 30 20 74 6f |anging f|rom 0 to|
|000004b0| 20 32 35 35 2e 20 20 45 | 61 63 68 20 6c 69 6e 65 | 255.  E|ach line|
|000004c0| 20 67 69 76 65 73 0a 74 | 68 65 20 72 65 64 2c 20 | gives.t|he red, |
|000004d0| 67 72 65 65 6e 2c 20 61 | 6e 64 20 62 6c 75 65 20 |green, a|nd blue |
|000004e0| 76 61 6c 75 65 73 20 66 | 6f 72 20 61 20 73 69 6e |values f|or a sin|
|000004f0| 67 6c 65 20 65 6e 74 72 | 79 20 69 6e 20 74 68 65 |gle entr|y in the|
|00000500| 20 63 6f 6c 6f 72 6d 61 | 70 2e 0a 0a 5c 73 65 63 | colorma|p...\sec|
|00000510| 74 69 6f 6e 20 7b 54 65 | 78 74 75 72 69 6e 67 20 |tion {Te|xturing |
|00000520| 46 75 6e 63 74 69 6f 6e | 73 7d 0a 0a 5c 62 65 67 |Function|s}..\beg|
|00000530| 69 6e 7b 64 65 66 6b 65 | 79 7d 7b 62 6c 6f 74 63 |in{defke|y}{blotc|
|00000540| 68 7d 7b 7b 5c 65 6d 20 | 42 6c 65 6e 64 46 61 63 |h}{{\em |BlendFac|
|00000550| 74 6f 72 20 73 75 72 66 | 61 63 65 7d 7d 0a 50 72 |tor surf|ace}}.Pr|
|00000560| 6f 64 75 63 65 73 20 61 | 20 6d 69 6c 64 6c 79 20 |oduces a| mildly |
|00000570| 69 6e 74 65 72 65 73 74 | 69 6e 67 20 62 6c 6f 74 |interest|ing blot|
|00000580| 63 68 79 2d 6c 6f 6f 6b | 69 6e 67 20 73 75 72 66 |chy-look|ing surf|
|00000590| 61 63 65 2e 0a 7b 5c 65 | 6d 20 42 6c 65 6e 64 46 |ace..{\e|m BlendF|
|000005a0| 61 63 74 6f 72 7d 20 69 | 73 20 75 73 65 64 20 74 |actor} i|s used t|
|000005b0| 6f 20 63 6f 6e 74 72 6f | 6c 20 74 68 65 20 69 6e |o contro|l the in|
|000005c0| 74 65 72 70 6f 6c 61 74 | 69 6f 6e 20 62 65 74 77 |terpolat|ion betw|
|000005d0| 65 65 6e 0a 74 68 65 20 | 64 65 66 61 75 6c 74 20 |een.the |default |
|000005e0| 73 75 72 66 61 63 65 20 | 63 68 61 72 61 63 74 65 |surface |characte|
|000005f0| 72 69 73 74 69 63 73 20 | 61 6e 64 20 74 68 65 20 |ristics |and the |
|00000600| 63 68 61 72 61 63 74 65 | 72 69 73 74 69 63 73 20 |characte|ristics |
|00000610| 6f 66 0a 74 68 65 20 67 | 69 76 65 6e 20 73 75 72 |of.the g|iven sur|
|00000620| 66 61 63 65 2e 20 20 41 | 20 76 61 6c 75 65 20 6f |face.  A| value o|
|00000630| 66 20 30 20 72 65 73 75 | 6c 74 73 20 69 6e 20 61 |f 0 resu|lts in a|
|00000640| 20 72 6f 75 67 68 6c 79 | 20 35 30 2d 35 30 20 6d | roughly| 50-50 m|
|00000650| 69 78 0a 6f 66 20 74 68 | 65 20 74 77 6f 20 73 75 |ix.of th|e two su|
|00000660| 72 66 61 63 65 73 2e 20 | 20 48 69 67 68 65 72 20 |rfaces. | Higher |
|00000670| 76 61 6c 75 65 73 20 72 | 65 73 75 6c 74 20 69 6e |values r|esult in|
|00000680| 20 61 20 67 72 65 61 74 | 20 70 6f 72 74 69 6f 6e | a great| portion|
|00000690| 20 6f 66 0a 74 68 65 20 | 64 65 66 61 75 6c 74 20 | of.the |default |
|000006a0| 73 75 72 66 61 63 65 20 | 63 68 61 72 61 63 74 65 |surface |characte|
|000006b0| 72 69 73 74 69 63 73 2e | 0a 5c 65 6e 64 7b 64 65 |ristics.|.\end{de|
|000006c0| 66 6b 65 79 7d 0a 0a 5c | 62 65 67 69 6e 7b 64 65 |fkey}..\|begin{de|
|000006d0| 66 6b 65 79 7d 7b 62 75 | 6d 70 7d 7b 7b 5c 65 6d |fkey}{bu|mp}{{\em|
|000006e0| 20 73 63 61 6c 65 7d 7d | 0a 41 70 70 6c 79 20 61 | scale}}|.Apply a|
|000006f0| 20 72 61 6e 64 6f 6d 20 | 62 75 6d 70 20 6d 61 70 | random |bump map|
|00000700| 2e 20 20 54 68 65 20 70 | 6f 69 6e 74 20 6f 66 20 |.  The p|oint of |
|00000710| 69 6e 74 65 72 73 65 63 | 74 69 6f 6e 20 69 73 20 |intersec|tion is |
|00000720| 70 61 73 73 65 64 20 74 | 6f 0a 7b 5c 65 6d 20 44 |passed t|o.{\em D|
|00000730| 4e 6f 69 73 65 28 29 7d | 2e 0a 54 68 65 20 72 65 |Noise()}|..The re|
|00000740| 74 75 72 6e 65 64 20 6e | 6f 72 6d 61 6c 69 7a 65 |turned n|ormalize|
|00000750| 64 20 76 65 63 74 6f 72 | 20 69 73 20 77 65 69 67 |d vector| is weig|
|00000760| 68 74 65 64 20 62 79 20 | 7b 5c 65 6d 20 73 63 61 |hted by |{\em sca|
|00000770| 6c 65 7d 0a 61 6e 64 20 | 74 68 65 20 72 65 73 75 |le}.and |the resu|
|00000780| 6c 74 20 69 73 20 61 64 | 64 65 64 20 74 6f 20 74 |lt is ad|ded to t|
|00000790| 68 65 20 6e 6f 72 6d 61 | 6c 20 76 65 63 74 6f 72 |he norma|l vector|
|000007a0| 20 61 74 20 74 68 65 20 | 70 6f 69 6e 74 20 6f 66 | at the |point of|
|000007b0| 20 69 6e 74 65 72 73 65 | 63 74 69 6f 6e 2e 0a 5c | interse|ction..\|
|000007c0| 65 6e 64 7b 64 65 66 6b | 65 79 7d 0a 55 73 69 6e |end{defk|ey}.Usin|
|000007d0| 67 20 61 6e 20 69 6d 61 | 67 65 20 74 65 78 74 75 |g an ima|ge textu|
|000007e0| 72 65 20 61 70 70 6c 69 | 65 64 20 74 6f 20 74 68 |re appli|ed to th|
|000007f0| 65 20 60 60 62 75 6d 70 | 27 27 20 63 6f 6d 70 6f |e ``bump|'' compo|
|00000800| 6e 65 6e 74 20 6f 66 66 | 65 72 73 20 61 20 6d 6f |nent off|ers a mo|
|00000810| 72 65 0a 64 69 72 65 63 | 74 20 77 61 79 20 74 6f |re.direc|t way to|
|00000820| 20 63 6f 6e 74 72 6f 6c | 20 74 68 65 20 6d 6f 64 | control| the mod|
|00000830| 69 66 69 63 61 74 69 6f | 6e 20 6f 66 20 73 75 72 |ificatio|n of sur|
|00000840| 66 61 63 65 20 6e 6f 72 | 6d 61 6c 73 20 28 73 65 |face nor|mals (se|
|00000850| 65 20 62 65 6c 6f 77 29 | 2e 0a 0a 5c 62 65 67 69 |e below)|...\begi|
|00000860| 6e 7b 64 65 66 6b 65 79 | 7d 7b 63 68 65 63 6b 65 |n{defkey|}{checke|
|00000870| 72 7d 7b 24 3c 24 7b 5c | 65 6d 20 53 75 72 66 61 |r}{$<${\|em Surfa|
|00000880| 63 65 7d 24 3e 24 7d 0a | 41 70 70 6c 69 65 73 20 |ce}$>$}.|Applies |
|00000890| 61 20 33 44 20 63 68 65 | 63 6b 65 72 62 6f 61 72 |a 3D che|ckerboar|
|000008a0| 64 20 74 65 78 74 75 72 | 65 2e 20 20 45 76 65 72 |d textur|e.  Ever|
|000008b0| 79 20 70 6f 69 6e 74 20 | 74 68 61 74 20 66 61 6c |y point |that fal|
|000008c0| 6c 73 20 77 69 74 68 69 | 6e 20 61 6e 0a 60 60 65 |ls withi|n an.``e|
|000008d0| 76 65 6e 27 27 20 75 6e | 69 74 20 63 75 62 65 20 |ven'' un|it cube |
|000008e0| 77 69 6c 6c 20 62 65 20 | 61 73 73 69 67 6e 65 64 |will be |assigned|
|000008f0| 20 74 68 65 20 63 68 61 | 72 61 63 74 65 72 69 73 | the cha|racteris|
|00000900| 74 69 63 73 20 6f 66 20 | 74 68 65 20 6e 61 6d 65 |tics of |the name|
|00000910| 64 20 73 75 72 66 61 63 | 65 0a 61 70 70 6c 69 65 |d surfac|e.applie|
|00000920| 64 20 74 6f 20 69 74 2c | 20 77 68 69 6c 65 20 70 |d to it,| while p|
|00000930| 6f 69 6e 74 73 20 74 68 | 61 74 20 66 61 6c 6c 20 |oints th|at fall |
|00000940| 77 69 74 68 69 6e 20 60 | 60 6f 64 64 27 27 20 63 |within `|`odd'' c|
|00000950| 75 62 65 73 20 77 69 6c | 6c 20 68 61 76 65 0a 69 |ubes wil|l have.i|
|00000960| 74 73 20 75 73 75 61 6c | 20 73 75 72 66 61 63 65 |ts usual| surface|
|00000970| 20 63 68 61 72 61 63 74 | 65 72 69 73 74 69 63 73 | charact|eristics|
|00000980| 2e 20 20 42 65 20 77 61 | 72 79 20 6f 66 20 73 74 |.  Be wa|ry of st|
|00000990| 72 61 6e 67 65 20 65 66 | 66 65 63 74 73 20 64 75 |range ef|fects du|
|000009a0| 65 0a 74 6f 20 72 6f 75 | 6e 64 6f 66 66 20 65 72 |e.to rou|ndoff er|
|000009b0| 72 6f 72 20 74 68 61 74 | 20 6f 63 63 75 72 20 77 |ror that| occur w|
|000009c0| 68 65 6e 20 61 20 70 6c | 61 6e 61 72 20 63 68 65 |hen a pl|anar che|
|000009d0| 63 6b 65 72 65 64 20 73 | 75 72 66 61 63 65 20 6c |ckered s|urface l|
|000009e0| 69 65 73 0a 69 6e 20 61 | 20 70 6c 61 6e 65 20 6f |ies.in a| plane o|
|000009f0| 66 20 63 6f 6e 73 74 61 | 6e 74 20 69 6e 74 65 67 |f consta|nt integ|
|00000a00| 72 61 6c 20 76 61 6c 75 | 65 20 28 65 2e 67 2e 2c |ral valu|e (e.g.,|
|00000a10| 20 24 7a 3d 30 24 29 20 | 69 6e 20 74 65 78 74 75 | $z=0$) |in textu|
|00000a20| 72 65 20 73 70 61 63 65 | 2e 0a 49 6e 20 73 75 63 |re space|..In suc|
|00000a30| 68 20 63 61 73 65 73 2c | 0a 73 69 6d 70 6c 79 20 |h cases,|.simply |
|00000a40| 74 72 61 6e 73 6c 61 74 | 65 20 74 68 65 20 74 65 |translat|e the te|
|00000a50| 78 74 75 72 65 20 74 6f | 20 65 6e 73 75 72 65 20 |xture to| ensure |
|00000a60| 74 68 61 74 20 74 68 65 | 20 70 6c 61 6e 61 72 20 |that the| planar |
|00000a70| 73 75 72 66 61 63 65 20 | 69 73 20 6e 6f 74 0a 63 |surface |is not.c|
|00000a80| 6f 69 6e 63 69 64 65 6e | 74 20 77 69 74 68 20 61 |oinciden|t with a|
|00000a90| 6e 20 69 6e 74 65 67 72 | 61 6c 20 70 6c 61 6e 65 |n integr|al plane|
|00000aa0| 20 69 6e 20 74 65 78 74 | 75 72 65 20 73 70 61 63 | in text|ure spac|
|00000ab0| 65 0a 28 65 2e 67 2e 2c | 20 7b 5c 74 74 20 74 72 |e.(e.g.,| {\tt tr|
|00000ac0| 61 6e 73 6c 61 74 65 20 | 30 20 30 20 30 2e 31 7d |anslate |0 0 0.1}|
|00000ad0| 29 2e 0a 5c 65 6e 64 7b | 64 65 66 6b 65 79 7d 0a |)..\end{|defkey}.|
|00000ae0| 0a 5c 62 65 67 69 6e 7b | 64 65 66 6b 65 79 7d 7b |.\begin{|defkey}{|
|00000af0| 63 6c 6f 75 64 7d 7b 7b | 5c 65 6d 20 73 63 61 6c |cloud}{{|\em scal|
|00000b00| 65 20 48 20 24 5c 6c 61 | 6d 62 64 61 24 20 6f 63 |e H $\la|mbda$ oc|
|00000b10| 74 61 76 65 73 20 63 74 | 68 72 65 73 68 20 6c 74 |taves ct|hresh lt|
|00000b20| 68 72 65 73 68 20 74 73 | 63 61 6c 65 7d 7d 0a 09 |hresh ts|cale}}..|
|00000b30| 54 68 69 73 20 74 65 78 | 74 75 72 65 20 69 73 20 |This tex|ture is |
|00000b40| 61 20 76 61 72 69 61 6e | 74 20 6f 6e 20 47 65 6f |a varian|t on Geo|
|00000b50| 66 66 20 47 61 72 64 6e | 65 72 27 73 20 65 6c 6c |ff Gardn|er's ell|
|00000b60| 69 70 73 6f 69 64 2d 74 | 65 78 74 75 72 69 6e 67 |ipsoid-t|exturing|
|00000b70| 0a 09 61 6c 67 6f 72 69 | 74 68 6d 2e 20 20 49 74 |..algori|thm.  It|
|00000b80| 20 73 68 6f 75 6c 64 20 | 62 65 20 61 70 70 6c 69 | should |be appli|
|00000b90| 65 64 20 74 6f 20 75 6e | 69 74 20 73 70 68 65 72 |ed to un|it spher|
|00000ba0| 65 73 20 63 65 6e 74 65 | 72 65 64 0a 09 61 74 20 |es cente|red..at |
|00000bb0| 74 68 65 20 6f 72 69 67 | 69 6e 2e 20 20 54 68 65 |the orig|in.  The|
|00000bc0| 73 65 20 73 70 68 65 72 | 65 73 20 6d 61 79 2c 20 |se spher|es may, |
|00000bd0| 6f 66 20 63 6f 75 72 73 | 65 2c 20 62 65 20 74 72 |of cours|e, be tr|
|00000be0| 61 6e 73 66 6f 72 6d 65 | 64 0a 09 61 74 20 77 69 |ansforme|d..at wi|
|00000bf0| 6c 6c 20 74 6f 20 66 6f | 72 6d 20 74 68 65 20 61 |ll to fo|rm the a|
|00000c00| 70 70 72 6f 70 72 69 61 | 74 65 6c 79 2d 73 68 61 |ppropria|tely-sha|
|00000c10| 70 65 64 20 63 6c 6f 75 | 64 20 6f 72 20 74 72 65 |ped clou|d or tre|
|00000c20| 65 2e 0a 0a 09 41 20 73 | 61 6d 70 6c 65 20 6f 66 |e....A s|ample of|
|00000c30| 20 6e 6f 72 6d 61 6c 69 | 7a 65 64 20 7b 5c 65 6d | normali|zed {\em|
|00000c40| 20 66 42 6d 7d 20 28 73 | 65 65 20 74 68 65 20 7b | fBm} (s|ee the {|
|00000c50| 5c 65 6d 20 66 62 6d 7d | 20 74 65 78 74 75 72 65 |\em fbm}| texture|
|00000c60| 29 20 69 73 0a 09 67 65 | 6e 65 72 61 74 65 64 0a |) is..ge|nerated.|
|00000c70| 09 61 74 20 74 68 65 20 | 70 6f 69 6e 74 20 6f 66 |.at the |point of|
|00000c80| 20 69 6e 74 65 72 73 65 | 63 74 69 6f 6e 2e 20 20 | interse|ction.  |
|00000c90| 54 68 69 73 20 73 61 6d | 70 6c 65 20 69 73 20 75 |This sam|ple is u|
|00000ca0| 73 65 64 20 74 6f 0a 09 | 6d 6f 64 75 6c 61 74 65 |sed to..|modulate|
|00000cb0| 20 74 68 65 20 73 75 72 | 66 61 63 65 20 74 72 61 | the sur|face tra|
|00000cc0| 6e 73 70 61 72 65 6e 63 | 79 2e 20 20 54 68 65 20 |nsparenc|y.  The |
|00000cd0| 66 69 6e 61 6c 20 74 72 | 61 6e 73 70 61 72 65 6e |final tr|ansparen|
|00000ce0| 63 79 0a 09 69 66 20 61 | 20 66 75 6e 63 74 69 6f |cy..if a| functio|
|00000cf0| 6e 20 6f 66 20 74 68 65 | 20 73 61 6d 70 6c 65 20 |n of the| sample |
|00000d00| 76 61 6c 75 65 2c 20 74 | 68 65 0a 09 74 68 65 20 |value, t|he..the |
|00000d10| 70 72 6f 78 69 6d 69 74 | 79 20 6f 66 20 74 68 65 |proximit|y of the|
|00000d20| 20 70 6f 69 6e 74 20 6f | 66 20 69 6e 74 65 72 73 | point o|f inters|
|00000d30| 65 63 74 69 6f 6e 20 74 | 6f 20 74 68 65 20 65 64 |ection t|o the ed|
|00000d40| 67 65 20 6f 66 0a 09 74 | 68 65 20 73 70 68 65 72 |ge of..t|he spher|
|00000d50| 65 20 28 61 73 20 73 65 | 65 6e 20 66 72 6f 6d 20 |e (as se|en from |
|00000d60| 74 68 65 20 72 61 79 20 | 6f 72 69 67 69 6e 29 2c |the ray |origin),|
|00000d70| 20 61 6e 64 20 74 68 72 | 65 65 20 70 61 72 61 6d | and thr|ee param|
|00000d80| 65 74 65 72 73 0a 09 74 | 6f 20 63 6f 6e 74 72 6f |eters..t|o contro|
|00000d90| 6c 20 74 68 65 20 6f 76 | 65 72 61 6c 6c 20 60 60 |l the ov|erall ``|
|00000da0| 64 65 6e 73 69 74 79 2e | 27 27 20 20 54 68 65 20 |density.|''  The |
|00000db0| 70 72 6f 78 69 6d 69 74 | 79 20 6f 66 20 74 68 65 |proximit|y of the|
|00000dc0| 20 70 6f 69 6e 74 0a 09 | 74 6f 20 74 68 65 20 73 | point..|to the s|
|00000dd0| 70 68 65 72 65 20 65 64 | 67 65 20 69 73 20 64 65 |phere ed|ge is de|
|00000de0| 74 65 72 6d 69 6e 65 64 | 20 62 79 20 65 76 61 6c |termined| by eval|
|00000df0| 75 61 74 69 6e 67 20 61 | 20 7b 5c 65 6d 20 6c 69 |uating a| {\em li|
|00000e00| 6d 62 7d 20 66 75 6e 63 | 74 69 6f 6e 2c 0a 09 77 |mb} func|tion,..w|
|00000e10| 68 69 63 68 20 76 61 72 | 69 65 73 20 66 72 6f 6d |hich var|ies from|
|00000e20| 20 30 20 6f 6e 20 74 68 | 65 20 6c 69 6d 62 20 74 | 0 on th|e limb t|
|00000e30| 6f 20 31 20 61 74 20 74 | 68 65 20 63 65 6e 74 65 |o 1 at t|he cente|
|00000e40| 72 20 6f 66 20 74 68 65 | 20 73 70 68 65 72 65 2e |r of the| sphere.|
|00000e50| 0a 5c 5b 0a 74 72 61 6e | 73 70 20 3d 20 31 2e 20 |.\[.tran|sp = 1. |
|00000e60| 2d 20 5c 66 72 61 63 7b | 66 42 6d 20 2d 20 63 74 |- \frac{|fBm - ct|
|00000e70| 68 72 65 73 68 20 2d 20 | 28 6c 74 68 72 65 73 68 |hresh - |(lthresh|
|00000e80| 20 2d 20 63 74 68 72 65 | 73 68 29 28 31 20 2d 20 | - cthre|sh)(1 - |
|00000e90| 6c 69 6d 62 29 7d 7b 74 | 73 63 61 6c 65 7d 0a 5c |limb)}{t|scale}.\|
|00000ea0| 5d 0a 5c 65 6e 64 7b 64 | 65 66 6b 65 79 7d 0a 0a |].\end{d|efkey}..|
|00000eb0| 5c 62 65 67 69 6e 7b 64 | 65 66 6b 65 79 7d 7b 66 |\begin{d|efkey}{f|
|00000ec0| 62 6d 7d 7b 7b 5c 65 6d | 20 6f 66 66 73 65 74 20 |bm}{{\em| offset |
|00000ed0| 73 63 61 6c 65 20 48 20 | 24 5c 6c 61 6d 62 64 61 |scale H |$\lambda|
|00000ee0| 24 20 6f 63 74 61 76 65 | 73 20 74 68 72 65 73 68 |$ octave|s thresh|
|00000ef0| 7d 0a 5b 7b 5c 65 6d 20 | 63 6f 6c 6f 72 6d 61 70 |}.[{\em |colormap|
|00000f00| 7d 5d 7d 0a 47 65 6e 65 | 72 61 74 65 20 61 20 73 |}]}.Gene|rate a s|
|00000f10| 61 6d 70 6c 65 20 6f 66 | 20 64 69 73 63 72 65 74 |ample of| discret|
|00000f20| 69 7a 65 64 20 66 72 61 | 63 74 69 6f 6e 61 6c 20 |ized fra|ctional |
|00000f30| 42 72 6f 77 6e 69 61 6e | 20 6d 6f 74 69 6f 6e 20 |Brownian| motion |
|00000f40| 28 66 42 6d 29 20 61 6e | 64 0a 75 73 65 73 20 69 |(fBm) an|d.uses i|
|00000f50| 74 20 74 6f 20 73 63 61 | 6c 65 20 74 68 65 20 64 |t to sca|le the d|
|00000f60| 69 66 66 75 73 65 20 61 | 6e 64 20 61 6d 62 69 65 |iffuse a|nd ambie|
|00000f70| 6e 74 20 63 6f 6d 70 6f | 6e 65 6e 74 20 6f 66 20 |nt compo|nent of |
|00000f80| 61 6e 20 6f 62 6a 65 63 | 74 27 73 20 73 75 72 66 |an objec|t's surf|
|00000f90| 61 63 65 2e 0a 7b 5c 65 | 6d 20 53 63 61 6c 65 7d |ace..{\e|m Scale}|
|00000fa0| 20 69 73 20 75 73 65 64 | 20 74 6f 20 73 63 61 6c | is used| to scal|
|00000fb0| 65 20 74 68 65 20 76 61 | 6c 75 65 0a 72 65 74 75 |e the va|lue.retu|
|00000fc0| 72 6e 65 64 20 62 79 20 | 74 68 65 20 66 42 6d 20 |rned by |the fBm |
|00000fd0| 66 75 6e 63 74 69 6f 6e | 2e 20 20 7b 5c 65 6d 20 |function|.  {\em |
|00000fe0| 4f 66 66 73 65 74 7d 20 | 61 6c 6c 6f 77 73 20 6f |Offset} |allows o|
|00000ff0| 6e 65 20 74 6f 20 63 6f | 6e 74 72 6f 6c 20 74 68 |ne to co|ntrol th|
|00001000| 65 20 6d 69 6e 69 6d 75 | 6d 0a 76 61 6c 75 65 20 |e minimu|m.value |
|00001010| 6f 66 20 74 68 65 20 66 | 42 6d 20 66 75 6e 63 74 |of the f|Bm funct|
|00001020| 69 6f 6e 2e 20 20 7b 5c | 65 6d 20 48 7d 20 69 73 |ion.  {\|em H} is|
|00001030| 20 74 68 65 20 7b 5c 65 | 6d 20 48 6f 6c 64 65 72 | the {\e|m Holder|
|00001040| 20 65 78 70 6f 6e 65 6e | 74 7d 0a 75 73 65 64 20 | exponen|t}.used |
|00001050| 69 6e 20 74 68 65 20 66 | 42 6d 20 66 75 6e 63 74 |in the f|Bm funct|
|00001060| 69 6f 6e 20 28 61 20 76 | 61 6c 75 65 20 6f 66 20 |ion (a v|alue of |
|00001070| 30 2e 35 20 77 6f 72 6b | 73 20 77 65 6c 6c 29 2e |0.5 work|s well).|
|00001080| 20 20 24 5c 6c 61 6d 62 | 64 61 24 20 69 73 0a 75 |  $\lamb|da$ is.u|
|00001090| 73 65 64 20 74 6f 20 63 | 6f 6e 74 72 6f 6c 20 7b |sed to c|ontrol {|
|000010a0| 5c 65 6d 20 6c 61 63 75 | 6e 61 72 69 74 79 7d 2c |\em lacu|narity},|
|000010b0| 20 61 6e 64 20 73 70 65 | 63 69 66 69 65 73 20 74 | and spe|cifies t|
|000010c0| 68 65 20 74 68 65 20 66 | 72 65 71 75 65 6e 63 79 |he the f|requency|
|000010d0| 0a 64 69 66 66 65 72 65 | 6e 63 65 20 62 65 74 77 |.differe|nce betw|
|000010e0| 65 65 6e 20 73 75 63 63 | 65 73 73 69 76 65 20 73 |een succ|essive s|
|000010f0| 61 6d 70 6c 65 73 20 6f | 66 20 74 68 65 20 66 42 |amples o|f the fB|
|00001100| 6d 20 62 61 73 69 73 20 | 66 75 6e 63 74 69 6f 6e |m basis |function|
|00001110| 20 28 61 0a 76 61 6c 75 | 65 20 6f 66 20 32 2e 30 | (a.valu|e of 2.0|
|00001120| 20 77 69 6c 6c 20 73 75 | 66 66 69 63 65 29 2e 20 | will su|ffice). |
|00001130| 20 7b 5c 65 6d 20 4f 63 | 74 61 76 65 73 7d 20 73 | {\em Oc|taves} s|
|00001140| 70 65 63 69 66 69 65 73 | 20 74 68 65 20 6e 75 6d |pecifies| the num|
|00001150| 62 65 72 20 6f 66 0a 6f | 63 74 61 76 65 73 20 28 |ber of.o|ctaves (|
|00001160| 73 61 6d 70 6c 65 73 29 | 20 74 6f 20 74 61 6b 65 |samples)| to take|
|00001170| 20 6f 66 20 74 68 65 20 | 66 42 6d 20 62 61 73 69 | of the |fBm basi|
|00001180| 73 20 66 75 6e 63 74 69 | 6f 6e 20 28 69 6e 20 74 |s functi|on (in t|
|00001190| 68 69 73 20 63 61 73 65 | 2c 20 4e 6f 69 73 65 28 |his case|, Noise(|
|000011a0| 29 29 2e 0a 42 65 74 77 | 65 65 6e 20 66 69 76 65 |))..Betw|een five|
|000011b0| 20 61 6e 64 20 73 65 76 | 65 6e 20 6f 63 74 61 76 | and sev|en octav|
|000011c0| 65 73 20 75 73 75 61 6c | 6c 79 20 77 6f 72 6b 73 |es usual|ly works|
|000011d0| 20 77 65 6c 6c 2e 20 20 | 7b 5c 65 6d 20 54 68 72 | well.  |{\em Thr|
|000011e0| 65 73 68 7d 20 69 73 20 | 75 73 65 64 0a 74 6f 20 |esh} is |used.to |
|000011f0| 73 70 65 63 69 66 79 20 | 61 20 6c 6f 77 65 72 20 |specify |a lower |
|00001200| 62 6f 75 6e 64 20 6f 6e | 74 68 65 20 6f 75 74 70 |bound on|the outp|
|00001210| 75 74 20 6f 66 20 74 68 | 65 20 66 42 6d 20 66 75 |ut of th|e fBm fu|
|00001220| 6e 63 74 69 6f 6e 2e 20 | 20 41 6e 79 0a 76 61 6c |nction. | Any.val|
|00001230| 75 65 20 6c 6f 77 65 72 | 20 74 68 61 6e 20 7b 5c |ue lower| than {\|
|00001240| 65 6d 20 74 68 72 65 73 | 68 7d 20 69 73 20 73 65 |em thres|h} is se|
|00001250| 74 20 74 6f 20 7a 65 72 | 6f 2e 0a 0a 49 66 20 61 |t to zer|o...If a|
|00001260| 20 7b 5c 65 6d 20 63 6f | 6c 6f 72 6d 61 70 7d 20 | {\em co|lormap} |
|00001270| 69 73 20 6e 61 6d 65 64 | 2c 20 61 20 32 35 36 2d |is named|, a 256-|
|00001280| 65 6e 74 72 79 20 63 6f | 6c 6f 72 6d 61 70 20 69 |entry co|lormap i|
|00001290| 73 20 72 65 61 64 20 66 | 72 6f 6d 20 74 68 65 20 |s read f|rom the |
|000012a0| 6e 61 6d 65 64 0a 66 69 | 6c 65 2c 20 61 6e 64 20 |named.fi|le, and |
|000012b0| 74 68 65 20 73 61 6d 70 | 6c 65 20 6f 66 20 66 42 |the samp|le of fB|
|000012c0| 6d 20 69 73 20 73 63 61 | 6c 65 64 20 62 79 20 32 |m is sca|led by 2|
|000012d0| 35 35 20 61 6e 64 20 69 | 73 20 75 73 65 64 20 61 |55 and i|s used a|
|000012e0| 73 20 61 6e 20 69 6e 64 | 65 78 20 69 6e 74 6f 0a |s an ind|ex into.|
|000012f0| 74 68 65 20 63 6f 6c 6f | 72 6d 61 70 2e 20 20 54 |the colo|rmap.  T|
|00001300| 68 65 20 72 65 73 75 6c | 74 69 6e 67 20 63 6f 6c |he resul|ting col|
|00001310| 6f 72 6d 61 70 20 65 6e | 74 72 79 0a 69 73 20 75 |ormap en|try.is u|
|00001320| 73 65 64 20 74 6f 20 73 | 63 61 6c 65 20 74 68 65 |sed to s|cale the|
|00001330| 20 61 6d 62 69 65 6e 74 | 20 61 6e 64 20 64 69 66 | ambient| and dif|
|00001340| 66 75 73 65 20 63 6f 6d | 70 6f 6e 65 6e 74 73 20 |fuse com|ponents |
|00001350| 6f 66 20 74 68 65 0a 6f | 62 6a 65 63 74 27 73 20 |of the.o|bject's |
|00001360| 73 75 72 66 61 63 65 2e | 0a 5c 65 6e 64 7b 64 65 |surface.|.\end{de|
|00001370| 66 6b 65 79 7d 0a 0a 5c | 62 65 67 69 6e 7b 64 65 |fkey}..\|begin{de|
|00001380| 66 6b 65 79 7d 7b 66 62 | 6d 62 75 6d 70 7d 7b 7b |fkey}{fb|mbump}{{|
|00001390| 5c 65 6d 20 6f 66 66 73 | 65 74 20 73 63 61 6c 65 |\em offs|et scale|
|000013a0| 20 48 20 24 5c 6c 61 6d | 62 64 61 24 20 6f 63 74 | H $\lam|bda$ oct|
|000013b0| 61 76 65 73 7d 7d 0a 53 | 69 6d 69 6c 61 72 20 74 |aves}}.S|imilar t|
|000013c0| 6f 20 74 68 65 20 7b 5c | 65 6d 20 66 62 6d 7d 20 |o the {\|em fbm} |
|000013d0| 74 65 78 74 75 72 65 2e | 20 20 52 61 74 68 65 72 |texture.|  Rather|
|000013e0| 20 74 68 61 6e 20 6d 6f | 64 69 66 79 69 6e 67 20 | than mo|difying |
|000013f0| 74 68 65 20 63 6f 6c 6f | 72 20 6f 66 0a 61 20 73 |the colo|r of.a s|
|00001400| 75 72 66 61 63 65 2c 20 | 74 68 69 73 20 74 65 78 |urface, |this tex|
|00001410| 74 75 72 65 20 61 63 74 | 73 20 61 73 20 61 20 62 |ture act|s as a b|
|00001420| 75 6d 70 20 6d 61 70 2e | 0a 5c 65 6e 64 7b 64 65 |ump map.|.\end{de|
|00001430| 66 6b 65 79 7d 0a 0a 5c | 62 65 67 69 6e 7b 64 65 |fkey}..\|begin{de|
|00001440| 66 6b 65 79 7d 7b 67 6c | 6f 73 73 7d 7b 7b 5c 65 |fkey}{gl|oss}{{\e|
|00001450| 6d 20 67 6c 6f 73 73 69 | 6e 65 73 73 7d 7d 0a 47 |m glossi|ness}}.G|
|00001460| 69 76 65 73 20 72 65 66 | 6c 65 63 74 69 76 65 20 |ives ref|lective |
|00001470| 73 75 72 66 61 63 65 73 | 20 61 20 67 6c 6f 73 73 |surfaces| a gloss|
|00001480| 79 20 61 70 70 65 61 72 | 61 6e 63 65 2e 20 54 68 |y appear|ance. Th|
|00001490| 69 73 20 74 65 78 74 75 | 72 65 20 70 65 72 74 75 |is textu|re pertu|
|000014a0| 72 62 73 0a 74 68 65 20 | 6f 62 6a 65 63 74 27 73 |rbs.the |object's|
|000014b0| 20 73 75 72 66 61 63 65 | 20 6e 6f 72 6d 61 6c 20 | surface| normal |
|000014c0| 73 75 63 68 20 74 68 61 | 74 20 74 68 65 20 6e 6f |such tha|t the no|
|000014d0| 72 6d 61 6c 20 60 60 73 | 61 6d 70 6c 65 73 27 27 |rmal ``s|amples''|
|000014e0| 20 61 20 63 6f 6e 65 20 | 6f 66 0a 75 6e 69 74 20 | a cone |of.unit |
|000014f0| 68 65 69 67 68 74 20 77 | 69 74 68 20 72 61 64 69 |height w|ith radi|
|00001500| 75 73 20 24 31 2e 20 2d | 20 67 6c 6f 73 73 69 6e |us $1. -| glossin|
|00001510| 65 73 73 24 2e 20 20 41 | 20 76 61 6c 75 65 20 6f |ess$.  A| value o|
|00001520| 66 20 31 20 72 65 73 75 | 6c 74 73 0a 69 6e 20 70 |f 1 resu|lts.in p|
|00001530| 65 72 66 65 63 74 20 6d | 69 72 72 6f 72 2d 6c 69 |erfect m|irror-li|
|00001540| 6b 65 20 72 65 66 6c 65 | 63 74 69 6f 6e 73 2c 20 |ke refle|ctions, |
|00001550| 77 68 69 6c 65 20 61 20 | 76 61 6c 75 65 20 6f 66 |while a |value of|
|00001560| 20 30 20 72 65 73 75 6c | 74 73 0a 69 6e 20 65 78 | 0 resul|ts.in ex|
|00001570| 74 72 65 6d 65 6c 79 20 | 66 75 7a 7a 79 20 72 65 |tremely |fuzzy re|
|00001580| 66 6c 65 63 74 69 6f 6e | 73 2e 20 20 46 6f 72 20 |flection|s.  For |
|00001590| 62 65 73 74 20 72 65 73 | 75 6c 74 73 2c 20 6a 69 |best res|ults, ji|
|000015a0| 74 74 65 72 65 64 20 73 | 61 6d 70 6c 69 6e 67 0a |ttered s|ampling.|
|000015b0| 73 68 6f 75 6c 64 20 62 | 65 20 75 73 65 64 20 74 |should b|e used t|
|000015c0| 6f 20 72 65 6e 64 65 72 | 20 73 63 65 6e 65 73 20 |o render| scenes |
|000015d0| 74 68 61 74 20 6d 61 6b | 65 20 75 73 65 20 6f 66 |that mak|e use of|
|000015e0| 20 74 68 69 73 20 74 65 | 78 74 75 72 65 2e 0a 5c | this te|xture..\|
|000015f0| 65 6e 64 7b 64 65 66 6b | 65 79 7d 0a 0a 5c 62 65 |end{defk|ey}..\be|
|00001600| 67 69 6e 7b 64 65 66 6b | 65 79 7d 7b 6d 61 72 62 |gin{defk|ey}{marb|
|00001610| 6c 65 7d 7b 5b 7b 5c 65 | 6d 20 63 6f 6c 6f 72 6d |le}{[{\e|m colorm|
|00001620| 61 70 7d 5d 7d 0a 47 69 | 76 65 73 20 61 20 73 75 |ap}]}.Gi|ves a su|
|00001630| 72 66 61 63 65 20 61 20 | 6d 61 72 62 6c 65 2d 6c |rface a |marble-l|
|00001640| 69 6b 65 20 61 70 70 65 | 61 72 61 6e 63 65 2e 20 |ike appe|arance. |
|00001650| 20 54 68 65 20 74 65 78 | 74 75 72 65 20 69 73 20 | The tex|ture is |
|00001660| 69 6d 70 6c 65 6d 65 6e | 74 65 64 20 61 73 0a 72 |implemen|ted as.r|
|00001670| 6f 75 67 68 6c 79 20 70 | 61 72 61 6c 6c 65 6c 20 |oughly p|arallel |
|00001680| 61 6c 74 65 72 6e 61 74 | 69 6e 67 20 76 65 69 6e |alternat|ing vein|
|00001690| 73 20 6f 66 20 6d 61 72 | 62 6c 65 2c 20 65 61 63 |s of mar|ble, eac|
|000016a0| 68 20 6f 66 20 77 68 69 | 63 68 20 69 73 0a 73 65 |h of whi|ch is.se|
|000016b0| 70 61 72 61 74 65 64 20 | 62 79 20 31 2f 37 20 6f |parated |by 1/7 o|
|000016c0| 66 20 61 20 75 6e 69 74 | 20 61 6e 64 20 72 75 6e |f a unit| and run|
|000016d0| 73 20 70 65 72 70 65 6e | 64 69 63 75 6c 61 72 20 |s perpen|dicular |
|000016e0| 74 6f 20 74 68 65 20 5a | 20 61 78 69 73 2e 0a 49 |to the Z| axis..I|
|000016f0| 66 20 61 20 63 6f 6c 6f | 72 6d 61 70 20 69 73 20 |f a colo|rmap is |
|00001700| 6e 61 6d 65 64 2c 20 74 | 68 65 20 73 75 72 66 61 |named, t|he surfa|
|00001710| 63 65 27 73 20 61 6d 62 | 69 65 6e 74 20 61 6e 64 |ce's amb|ient and|
|00001720| 20 64 69 66 66 75 73 65 | 20 63 6f 6c 6f 72 73 0a | diffuse| colors.|
|00001730| 77 69 6c 6c 20 62 65 20 | 73 63 61 6c 65 64 20 75 |will be |scaled u|
|00001740| 73 69 6e 67 20 74 68 65 | 20 52 47 42 20 76 61 6c |sing the| RGB val|
|00001750| 75 65 73 20 69 6e 20 74 | 68 65 20 63 6f 6c 6f 72 |ues in t|he color|
|00001760| 6d 61 70 2e 20 20 49 66 | 20 6e 6f 20 63 6f 6c 6f |map.  If| no colo|
|00001770| 72 6d 61 70 20 69 73 0a | 67 69 76 65 6e 2c 20 74 |rmap is.|given, t|
|00001780| 68 65 20 64 69 66 66 75 | 73 65 20 61 6e 64 20 61 |he diffu|se and a|
|00001790| 6d 62 69 65 6e 74 20 63 | 6f 6d 70 6f 6e 65 6e 74 |mbient c|omponent|
|000017a0| 73 20 61 72 65 20 73 69 | 6d 70 6c 79 20 73 63 61 |s are si|mply sca|
|000017b0| 6c 65 64 20 62 79 20 74 | 68 65 0a 76 61 6c 75 65 |led by t|he.value|
|000017c0| 20 6f 66 20 74 68 65 20 | 6d 61 72 62 6c 65 20 66 | of the |marble f|
|000017d0| 75 6e 63 74 69 6f 6e 2e | 20 20 4f 6e 65 20 6d 61 |unction.|  One ma|
|000017e0| 79 20 74 72 61 6e 73 66 | 6f 72 6d 20 74 68 65 20 |y transf|orm the |
|000017f0| 74 65 78 74 75 72 65 20 | 74 6f 0a 63 6f 6e 74 72 |texture |to.contr|
|00001800| 6f 6c 20 74 68 65 20 64 | 65 6e 73 69 74 79 20 61 |ol the d|ensity a|
|00001810| 6e 64 20 6f 72 69 65 6e | 74 61 74 69 6f 6e 20 6f |nd orien|tation o|
|00001820| 66 20 74 68 65 20 6d 61 | 72 62 6c 65 20 76 65 69 |f the ma|rble vei|
|00001830| 6e 73 2e 0a 5c 65 6e 64 | 7b 64 65 66 6b 65 79 7d |ns..\end|{defkey}|
|00001840| 0a 0a 5c 62 65 67 69 6e | 7b 64 65 66 6b 65 79 7d |..\begin|{defkey}|
|00001850| 7b 73 6b 79 7d 7b 7b 5c | 65 6d 20 73 63 61 6c 65 |{sky}{{\|em scale|
|00001860| 20 48 20 24 5c 6c 61 6d | 62 64 61 24 20 6f 63 74 | H $\lam|bda$ oct|
|00001870| 61 76 65 73 20 63 74 68 | 72 65 73 68 20 6c 74 72 |aves cth|resh ltr|
|00001880| 65 73 68 7d 7d 0a 09 53 | 69 6d 69 6c 61 72 20 74 |esh}}..S|imilar t|
|00001890| 6f 20 74 68 65 20 7b 5c | 65 6d 20 66 62 6d 7d 20 |o the {\|em fbm} |
|000018a0| 74 65 78 74 75 72 65 2e | 20 20 52 61 74 68 65 72 |texture.|  Rather|
|000018b0| 20 74 68 61 6e 20 6d 6f | 64 69 66 79 69 6e 67 20 | than mo|difying |
|000018c0| 74 68 65 0a 09 63 6f 6c | 6f 72 20 6f 66 20 61 20 |the..col|or of a |
|000018d0| 73 75 72 66 61 63 65 2c | 20 74 68 69 73 20 74 65 |surface,| this te|
|000018e0| 78 74 75 72 65 20 6d 6f | 64 75 6c 61 74 65 73 20 |xture mo|dulates |
|000018f0| 69 74 73 20 74 72 61 6e | 73 70 61 72 65 6e 63 79 |its tran|sparency|
|00001900| 2e 0a 09 7b 5c 65 6d 20 | 63 74 68 72 65 73 68 7d |...{\em |cthresh}|
|00001910| 20 69 73 20 74 68 65 20 | 76 61 6c 75 65 20 6f 66 | is the |value of|
|00001920| 20 74 68 65 20 7b 5c 65 | 6d 20 66 42 6d 7d 20 66 | the {\e|m fBm} f|
|00001930| 75 6e 63 74 69 6f 6e 20 | 61 62 6f 76 65 0a 09 77 |unction |above..w|
|00001940| 68 69 63 68 20 74 68 65 | 20 73 75 72 66 61 63 65 |hich the| surface|
|00001950| 20 69 73 20 74 6f 74 61 | 6c 6c 79 20 6f 70 61 71 | is tota|lly opaq|
|00001960| 75 65 2e 20 20 42 65 6c | 6f 77 20 7b 5c 65 6d 20 |ue.  Bel|ow {\em |
|00001970| 6c 74 68 72 65 73 68 7d | 2c 0a 09 74 68 65 20 73 |lthresh}|,..the s|
|00001980| 75 72 66 61 63 65 20 69 | 73 20 74 6f 74 61 6c 6c |urface i|s totall|
|00001990| 79 20 74 72 61 6e 73 70 | 61 72 65 6e 74 2e 0a 5c |y transp|arent..\|
|000019a0| 65 6e 64 7b 64 65 66 6b | 65 79 7d 0a 0a 5c 62 65 |end{defk|ey}..\be|
|000019b0| 67 69 6e 7b 64 65 66 6b | 65 79 7d 7b 73 74 72 69 |gin{defk|ey}{stri|
|000019c0| 70 65 7d 7b 24 3c 24 7b | 5c 65 6d 20 53 75 72 66 |pe}{$<${|\em Surf|
|000019d0| 61 63 65 7d 24 3e 24 20 | 7b 5c 65 6d 20 73 69 7a |ace}$>$ |{\em siz|
|000019e0| 65 20 62 75 6d 70 7d 20 | 24 3c 24 4d 61 70 70 69 |e bump} |$<$Mappi|
|000019f0| 6e 67 24 3e 24 7d 0a 09 | 41 70 70 6c 79 20 61 20 |ng$>$}..|Apply a |
|00001a00| 60 60 72 61 69 73 65 64 | 27 27 20 73 74 72 69 70 |``raised|'' strip|
|00001a10| 65 20 70 61 74 74 65 72 | 6e 20 74 6f 20 74 68 65 |e patter|n to the|
|00001a20| 20 73 75 72 66 61 63 65 | 2e 0a 09 54 68 65 20 73 | surface|...The s|
|00001a30| 75 72 66 61 63 65 20 70 | 72 6f 70 65 72 74 69 65 |urface p|ropertie|
|00001a40| 73 20 75 73 65 64 20 74 | 6f 20 63 6f 6c 6f 72 20 |s used t|o color |
|00001a50| 74 68 65 20 73 74 72 69 | 70 65 20 61 72 65 20 74 |the stri|pe are t|
|00001a60| 68 6f 73 65 0a 09 6f 66 | 20 74 68 65 20 67 69 76 |hose..of| the giv|
|00001a70| 65 6e 20 73 75 72 66 61 | 63 65 2e 20 20 54 68 65 |en surfa|ce.  The|
|00001a80| 20 77 69 64 74 68 20 6f | 66 20 74 68 65 20 73 74 | width o|f the st|
|00001a90| 72 69 70 65 2c 20 61 73 | 20 63 6f 6d 70 61 72 65 |ripe, as| compare|
|00001aa0| 64 0a 09 74 6f 20 74 68 | 65 20 75 6e 69 74 20 69 |d..to th|e unit i|
|00001ab0| 6e 74 65 72 76 61 6c 2c | 20 69 73 20 67 69 76 65 |nterval,| is give|
|00001ac0| 6e 20 62 79 20 7b 5c 65 | 6d 20 73 69 7a 65 7d 2e |n by {\e|m size}.|
|00001ad0| 20 20 54 68 65 20 6d 61 | 67 6e 69 74 75 64 65 0a |  The ma|gnitude.|
|00001ae0| 09 6f 66 20 7b 5c 65 6d | 20 62 75 6d 70 7d 20 63 |.of {\em| bump} c|
|00001af0| 6f 6e 74 72 6f 6c 73 20 | 74 68 65 20 65 78 74 65 |ontrols |the exte|
|00001b00| 6e 74 20 74 6f 20 77 68 | 69 63 68 20 74 68 65 20 |nt to wh|ich the |
|00001b10| 62 75 6d 70 20 61 70 70 | 65 61 72 73 0a 09 74 6f |bump app|ears..to|
|00001b20| 20 62 65 20 64 69 73 70 | 6c 61 63 65 64 20 66 72 | be disp|laced fr|
|00001b30| 6f 6d 20 74 68 65 20 72 | 65 73 74 20 6f 66 20 74 |om the r|est of t|
|00001b40| 68 65 20 73 75 72 66 61 | 63 65 2e 20 20 49 66 20 |he surfa|ce.  If |
|00001b50| 6e 65 67 61 74 69 76 65 | 2c 0a 09 74 68 65 20 73 |negative|,..the s|
|00001b60| 74 72 69 70 65 20 77 69 | 6c 6c 20 61 70 70 65 61 |tripe wi|ll appea|
|00001b70| 72 20 74 6f 0a 09 73 69 | 6e 6b 20 69 6e 74 6f 20 |r to..si|nk into |
|00001b80| 74 68 65 20 73 75 72 66 | 61 63 65 3b 20 69 66 20 |the surf|ace; if |
|00001b90| 70 6f 73 69 74 69 76 65 | 2c 20 69 74 20 77 69 6c |positive|, it wil|
|00001ba0| 6c 20 61 70 70 65 61 72 | 20 74 6f 20 73 74 61 6e |l appear| to stan|
|00001bb0| 64 0a 09 6f 75 74 20 6f | 66 20 74 68 65 20 73 75 |d..out o|f the su|
|00001bc0| 72 66 61 63 65 2e 0a 5c | 65 6e 64 7b 64 65 66 6b |rface..\|end{defk|
|00001bd0| 65 79 7d 0a 4d 61 70 70 | 69 6e 67 20 66 75 6e 63 |ey}.Mapp|ing func|
|00001be0| 74 69 6f 6e 73 20 61 72 | 65 20 64 65 73 63 72 69 |tions ar|e descri|
|00001bf0| 62 65 64 20 62 65 6c 6f | 77 2e 0a 0a 5c 62 65 67 |bed belo|w...\beg|
|00001c00| 69 6e 7b 64 65 66 6b 65 | 79 7d 7b 77 6f 6f 64 7d |in{defke|y}{wood}|
|00001c10| 7b 7d 0a 47 69 76 65 73 | 20 61 20 73 75 72 66 61 |{}.Gives| a surfa|
|00001c20| 63 65 20 61 20 77 6f 6f | 64 2d 6c 69 6b 65 20 61 |ce a woo|d-like a|
|00001c30| 70 70 65 61 72 61 6e 63 | 65 2e 20 20 54 68 65 20 |ppearanc|e.  The |
|00001c40| 66 65 61 74 75 72 65 20 | 73 69 7a 65 20 6f 66 20 |feature |size of |
|00001c50| 74 68 69 73 20 74 65 78 | 74 75 72 65 0a 69 73 20 |this tex|ture.is |
|00001c60| 61 70 70 72 6f 78 69 6d | 61 74 65 6c 79 20 24 30 |approxim|ately $0|
|00001c70| 2e 30 31 24 20 6f 66 20 | 61 20 75 6e 69 74 2c 20 |.01$ of |a unit, |
|00001c80| 6d 61 6b 69 6e 67 20 69 | 74 20 6f 66 74 65 6e 20 |making i|t often |
|00001c90| 6e 65 63 65 73 73 61 72 | 79 20 74 6f 0a 73 63 61 |necessar|y to.sca|
|00001ca0| 6c 65 20 74 68 65 20 74 | 65 78 74 75 72 65 20 69 |le the t|exture i|
|00001cb0| 6e 20 6f 72 64 65 72 20 | 74 6f 20 61 63 68 69 65 |n order |to achie|
|00001cc0| 76 65 20 74 68 65 20 64 | 65 73 69 72 65 64 20 61 |ve the d|esired a|
|00001cd0| 70 70 65 61 72 61 6e 63 | 65 2e 0a 5c 65 6e 64 7b |ppearanc|e..\end{|
|00001ce0| 64 65 66 6b 65 79 7d 0a | 0a 5c 73 65 63 74 69 6f |defkey}.|.\sectio|
|00001cf0| 6e 7b 54 65 78 74 75 72 | 65 20 74 79 70 65 73 20 |n{Textur|e types |
|00001d00| 61 64 64 65 64 20 69 6e | 20 34 2e 30 2e 36 2e 33 |added in| 4.0.6.3|
|00001d10| 7d 0a 0a 5c 62 65 67 69 | 6e 7b 64 65 66 6b 65 79 |}..\begi|n{defkey|
|00001d20| 7d 7b 67 72 61 64 69 65 | 6e 74 7d 7b 7b 5c 65 6d |}{gradie|nt}{{\em|
|00001d30| 20 73 75 72 66 31 7d 20 | 7b 5c 65 6d 20 73 75 72 | surf1} |{\em sur|
|00001d40| 66 32 7d 0a 09 09 09 20 | 5b 7b 5c 65 6d 20 47 72 |f2}.... |[{\em Gr|
|00001d50| 61 64 53 68 61 70 65 7d | 5d 0a 09 09 09 20 5b 7b |adShape}|].... [{|
|00001d60| 5c 65 6d 20 73 74 61 72 | 74 20 73 74 6f 70 7d 5d |\em star|t stop}]|
|00001d70| 0a 09 09 09 20 5b 7b 5c | 65 6d 20 47 72 61 64 54 |.... [{\|em GradT|
|00001d80| 79 70 65 7d 5d 0a 09 09 | 09 20 5b 7b 5c 74 74 20 |ype}]...|. [{\tt |
|00001d90| 72 61 6e 64 6f 6d 7d 5d | 7d 0a 0a 54 68 69 73 20 |random}]|}..This |
|00001da0| 74 65 78 74 75 72 65 20 | 62 6c 65 6e 64 73 20 74 |texture |blends t|
|00001db0| 77 6f 20 73 75 72 66 61 | 63 65 73 20 62 65 74 77 |wo surfa|ces betw|
|00001dc0| 65 65 6e 20 74 77 6f 20 | 73 68 61 70 65 73 20 69 |een two |shapes i|
|00001dd0| 6e 20 73 70 61 63 65 2e | 20 20 54 68 65 0a 64 65 |n space.|  The.de|
|00001de0| 66 61 75 6c 74 20 69 73 | 20 74 6f 20 62 6c 65 6e |fault is| to blen|
|00001df0| 64 20 62 65 74 77 65 65 | 6e 20 74 77 6f 20 70 6c |d betwee|n two pl|
|00001e00| 61 6e 65 73 2c 20 73 74 | 61 72 74 69 6e 67 20 61 |anes, st|arting a|
|00001e10| 20 24 7a 20 3d 20 30 24 | 20 61 6e 64 20 73 74 6f | $z = 0$| and sto|
|00001e20| 70 69 6e 67 20 61 74 20 | 24 7a 20 3d 0a 31 24 2e |ping at |$z =.1$.|
|00001e30| 20 54 68 65 20 64 65 66 | 61 75 6c 74 20 69 73 20 | The def|ault is |
|00001e40| 61 20 6c 69 6e 65 61 72 | 20 62 6c 65 6e 64 20 62 |a linear| blend b|
|00001e50| 65 74 77 65 65 6e 20 70 | 6c 61 6e 65 73 20 73 6f |etween p|lanes so|
|00001e60| 20 74 68 61 74 20 61 74 | 20 24 7a 20 3d 20 2e 37 | that at| $z = .7|
|00001e70| 35 24 20 74 68 65 0a 73 | 75 72 66 61 63 65 20 77 |5$ the.s|urface w|
|00001e80| 69 6c 6c 20 62 65 20 24 | 2e 32 35 20 2a 20 73 75 |ill be $|.25 * su|
|00001e90| 72 66 31 20 2b 20 2e 37 | 35 2a 73 75 72 66 32 24 |rf1 + .7|5*surf2$|
|00001ea0| 2e 0a 0a 20 20 20 20 20 | 20 20 20 54 68 65 20 7b |...     |   The {|
|00001eb0| 5c 65 6d 20 47 72 61 64 | 53 68 61 70 65 7d 20 6f |\em Grad|Shape} o|
|00001ec0| 70 74 69 6f 6e 20 73 70 | 65 63 69 66 69 65 73 20 |ption sp|ecifies |
|00001ed0| 74 68 65 20 60 60 73 68 | 61 70 65 27 27 20 6f 66 |the ``sh|ape'' of|
|00001ee0| 20 74 68 65 20 76 6f 6c | 75 6d 65 20 74 6f 0a 77 | the vol|ume to.w|
|00001ef0| 68 69 63 68 20 74 68 65 | 20 74 65 78 74 75 72 65 |hich the| texture|
|00001f00| 20 77 69 6c 6c 20 62 65 | 20 61 70 70 6c 69 65 64 | will be| applied|
|00001f10| 2e 20 20 7b 5c 74 74 20 | 70 6c 61 6e 61 72 7d 20 |.  {\tt |planar} |
|00001f20| 28 74 68 65 20 64 65 66 | 61 75 6c 74 29 20 64 65 |(the def|ault) de|
|00001f30| 73 69 67 6e 61 74 65 73 | 20 74 68 61 74 0a 74 68 |signates| that.th|
|00001f40| 65 20 62 6c 65 6e 64 20 | 77 69 6c 6c 20 62 65 20 |e blend |will be |
|00001f50| 62 65 74 77 65 65 6e 20 | 74 77 6f 20 70 6c 61 6e |between |two plan|
|00001f60| 65 73 20 70 65 72 70 65 | 6e 64 69 63 75 6c 61 72 |es perpe|ndicular|
|00001f70| 20 74 6f 20 24 7a 24 20 | 61 78 69 73 2e 20 20 7b | to $z$ |axis.  {|
|00001f80| 5c 74 74 20 72 61 64 69 | 61 6c 7d 0a 64 65 73 69 |\tt radi|al}.desi|
|00001f90| 67 6e 61 74 65 73 20 74 | 68 61 74 20 74 68 65 20 |gnates t|hat the |
|00001fa0| 62 6c 65 6e 64 20 20 77 | 69 6c 6c 20 62 65 20 62 |blend  w|ill be b|
|00001fb0| 65 74 77 65 65 6e 20 74 | 77 6f 20 63 79 6c 69 6e |etween t|wo cylin|
|00001fc0| 64 65 72 73 20 70 61 72 | 72 61 6c 65 6c 20 74 6f |ders par|ralel to|
|00001fd0| 20 74 68 65 20 24 7a 24 | 0a 61 78 69 73 2e 20 7b | the $z$|.axis. {|
|00001fe0| 5c 74 74 20 73 70 68 65 | 72 69 63 61 6c 7d 20 64 |\tt sphe|rical} d|
|00001ff0| 65 73 69 67 6e 61 74 65 | 73 20 74 68 61 74 20 74 |esignate|s that t|
|00002000| 68 65 20 62 6c 65 6e 64 | 20 77 69 6c 6c 20 62 65 |he blend| will be|
|00002010| 20 62 65 74 77 65 65 6e | 20 74 77 6f 20 73 70 68 | between| two sph|
|00002020| 65 72 65 73 0a 63 65 6e | 74 65 72 65 64 20 61 74 |eres.cen|tered at|
|00002030| 20 74 68 65 20 6f 72 69 | 67 69 6e 2e 20 0a 0a 20 | the ori|gin. .. |
|00002040| 20 20 20 20 20 20 20 7b | 5c 65 6d 20 73 74 61 72 |       {|\em star|
|00002050| 74 7d 20 61 6e 64 20 7b | 5c 65 6d 20 73 74 6f 70 |t} and {|\em stop|
|00002060| 7d 20 64 65 66 69 6e 65 | 20 74 68 65 20 73 74 61 |} define| the sta|
|00002070| 72 74 69 6e 67 20 61 6e | 64 20 73 74 6f 70 69 6e |rting an|d stopin|
|00002080| 67 20 64 69 73 74 61 6e | 63 65 73 0a 66 6f 72 20 |g distan|ces.for |
|00002090| 74 68 65 20 67 72 61 64 | 69 65 6e 74 2e 20 20 46 |the grad|ient.  F|
|000020a0| 6f 72 20 74 68 65 20 70 | 6c 61 6e 61 72 20 64 65 |or the p|lanar de|
|000020b0| 73 69 67 6e 61 74 69 6f | 6e 2c 20 74 68 65 20 67 |signatio|n, the g|
|000020c0| 72 61 64 69 65 6e 74 20 | 77 69 6c 6c 20 62 65 20 |radient |will be |
|000020d0| 61 70 70 6c 69 65 64 0a | 66 72 6f 6d 20 74 68 65 |applied.|from the|
|000020e0| 20 24 7a 20 3d 20 73 74 | 61 72 74 24 20 74 6f 20 | $z = st|art$ to |
|000020f0| 74 68 65 20 24 7a 20 3d | 20 73 74 6f 70 24 20 70 |the $z =| stop$ p|
|00002100| 6c 61 6e 65 73 2e 20 20 | 46 6f 72 20 74 68 65 20 |lanes.  |For the |
|00002110| 72 61 64 69 61 6c 20 64 | 65 73 69 67 6e 61 74 69 |radial d|esignati|
|00002120| 6f 6e 2c 0a 74 68 65 20 | 67 72 61 64 69 65 6e 74 |on,.the |gradient|
|00002130| 20 77 69 6c 6c 20 62 65 | 20 61 70 70 6c 69 65 64 | will be| applied|
|00002140| 20 66 72 6f 6d 20 74 68 | 65 20 63 79 6c 69 6e 65 | from th|e cyline|
|00002150| 72 73 20 24 72 20 3d 20 | 73 74 61 72 74 24 20 74 |rs $r = |start$ t|
|00002160| 6f 20 24 72 20 3d 20 73 | 74 6f 70 24 20 77 68 65 |o $r = s|top$ whe|
|00002170| 72 65 0a 24 72 24 20 65 | 71 75 61 6c 73 20 74 68 |re.$r$ e|quals th|
|00002180| 65 20 64 69 73 74 61 6e | 63 65 20 66 72 6f 6d 20 |e distan|ce from |
|00002190| 74 68 65 20 24 7a 24 20 | 61 78 69 73 2e 20 20 46 |the $z$ |axis.  F|
|000021a0| 6f 72 20 74 68 65 20 73 | 70 68 65 72 69 63 61 6c |or the s|pherical|
|000021b0| 20 64 65 73 69 67 6e 61 | 74 69 6f 6e 2c 20 74 68 | designa|tion, th|
|000021c0| 65 0a 67 72 61 64 69 65 | 6e 74 20 77 69 6c 6c 20 |e.gradie|nt will |
|000021d0| 62 65 20 61 70 70 6c 69 | 65 64 20 66 6f 72 20 74 |be appli|ed for t|
|000021e0| 68 65 20 73 70 68 65 72 | 65 73 20 24 72 20 3d 20 |he spher|es $r = |
|000021f0| 73 74 61 72 74 24 20 74 | 6f 20 24 72 20 3d 20 73 |start$ t|o $r = s|
|00002200| 74 6f 70 24 20 77 68 65 | 72 65 20 24 72 24 0a 65 |top$ whe|re $r$.e|
|00002210| 71 75 61 6c 73 20 74 68 | 65 20 64 69 73 74 61 6e |quals th|e distan|
|00002220| 63 65 20 66 72 6f 6d 20 | 74 68 65 20 6f 72 69 67 |ce from |the orig|
|00002230| 69 6e 2e 0a 0a 20 20 20 | 20 20 20 20 20 7b 5c 65 |in...   |     {\e|
|00002240| 6d 20 47 72 61 64 54 79 | 70 65 7d 20 73 70 65 63 |m GradTy|pe} spec|
|00002250| 69 66 69 65 73 20 74 68 | 65 20 74 79 70 65 20 6f |ifies th|e type o|
|00002260| 66 20 62 6c 65 6e 64 2e | 20 20 7b 5c 74 74 20 70 |f blend.|  {\tt p|
|00002270| 6c 61 6e 61 72 7d 20 28 | 74 68 65 0a 64 65 66 61 |lanar} (|the.defa|
|00002280| 75 6c 74 29 20 20 77 69 | 6c 6c 20 70 72 6f 64 75 |ult)  wi|ll produ|
|00002290| 63 65 20 61 20 6c 69 6e | 65 61 72 20 62 6c 65 6e |ce a lin|ear blen|
|000022a0| 64 20 62 65 74 77 65 65 | 6e 20 74 68 65 20 74 77 |d betwee|n the tw|
|000022b0| 6f 20 73 75 72 66 61 63 | 65 73 2e 20 20 7b 5c 74 |o surfac|es.  {\t|
|000022c0| 74 20 6c 6f 67 7d 0a 77 | 69 6c 6c 20 70 72 6f 64 |t log}.w|ill prod|
|000022d0| 75 63 65 20 61 20 6c 6f | 67 61 72 69 74 68 6d 69 |uce a lo|garithmi|
|000022e0| 63 20 6c 6f 6f 6b 69 6e | 67 20 20 62 6c 65 6e 64 |c lookin|g  blend|
|000022f0| 20 77 68 65 72 65 20 74 | 68 65 20 66 69 72 73 74 | where t|he first|
|00002300| 20 73 75 72 66 61 63 65 | 20 69 73 20 71 75 69 63 | surface| is quic|
|00002310| 6b 6c 79 0a 63 68 61 6e | 67 65 64 20 74 6f 20 74 |kly.chan|ged to t|
|00002320| 68 65 20 73 65 63 6f 6e | 64 2e 20 7b 5c 74 74 20 |he secon|d. {\tt |
|00002330| 72 65 76 6c 6f 67 7d 20 | 77 69 6c 6c 20 70 72 6f |revlog} |will pro|
|00002340| 64 75 63 65 20 74 68 65 | 20 6f 70 70 6f 73 69 74 |duce the| opposit|
|00002350| 65 20 6f 66 20 60 60 6c | 6f 67 27 27 20 61 6e 64 |e of ``l|og'' and|
|00002360| 0a 77 69 6c 6c 20 73 6c | 6f 77 6c 79 20 63 68 61 |.will sl|owly cha|
|00002370| 6e 67 65 20 74 68 65 20 | 66 69 72 73 74 20 73 75 |nge the |first su|
|00002380| 72 66 61 63 65 20 74 6f | 20 74 68 65 20 73 65 63 |rface to| the sec|
|00002390| 6f 6e 64 2e 0a 0a 20 20 | 20 20 20 20 20 20 7b 5c |ond...  |      {\|
|000023a0| 74 74 20 72 61 6e 64 6f | 6d 7d 20 70 72 6f 64 75 |tt rando|m} produ|
|000023b0| 63 65 73 20 61 20 62 6c | 6f 74 63 68 79 20 67 72 |ces a bl|otchy gr|
|000023c0| 61 64 69 65 6e 74 20 74 | 65 78 74 75 72 65 20 74 |adient t|exture t|
|000023d0| 68 61 74 20 69 73 20 73 | 69 6d 69 6c 61 72 20 74 |hat is s|imilar t|
|000023e0| 6f 20 74 68 65 0a 7b 5c | 74 74 20 62 6c 6f 74 63 |o the.{\|tt blotc|
|000023f0| 68 7d 20 74 65 78 74 75 | 72 65 2e 0a 5c 65 6e 64 |h} textu|re..\end|
|00002400| 7b 64 65 66 6b 65 79 7d | 0a 0a 20 20 20 20 20 20 |{defkey}|..      |
|00002410| 20 20 41 6e 79 20 63 6f | 6d 6d 65 6e 74 73 2c 20 |  Any co|mments, |
|00002420| 62 75 67 20 72 65 70 6f | 72 74 73 2c 20 6f 72 20 |bug repo|rts, or |
|00002430| 73 75 67 67 65 73 74 69 | 6f 6e 73 20 66 6f 72 20 |suggesti|ons for |
|00002440| 63 68 61 6e 67 65 73 20 | 6f 72 20 69 6d 70 72 6f |changes |or impro|
|00002450| 76 65 6d 65 6e 74 73 0a | 77 69 6c 6c 20 62 65 20 |vements.|will be |
|00002460| 67 72 65 61 74 6c 79 20 | 61 70 70 72 65 63 69 61 |greatly |apprecia|
|00002470| 74 65 64 2e 0a 0a 54 68 | 69 73 20 61 64 64 69 74 |ted...Th|is addit|
|00002480| 69 6f 6e 20 77 61 73 20 | 70 72 6f 76 69 64 65 64 |ion was |provided|
|00002490| 20 62 79 20 4c 61 72 72 | 79 20 43 6f 66 66 69 6e | by Larr|y Coffin|
|000024a0| 20 5c 5c 0a 7b 5c 74 74 | 20 3c 6c 63 6f 66 66 69 | \\.{\tt| <lcoffi|
|000024b0| 6e 40 63 6c 63 69 72 69 | 73 2e 63 68 65 6d 2e 75 |n@clciri|s.chem.u|
|000024c0| 6d 72 2e 65 64 75 3e 7d | 0a 0a 5c 73 65 63 74 69 |mr.edu>}|..\secti|
|000024d0| 6f 6e 20 7b 49 6d 61 67 | 65 20 54 65 78 74 75 72 |on {Imag|e Textur|
|000024e0| 69 6e 67 7d 0a 0a 7b 5c | 52 61 79 73 68 61 64 65 |ing}..{\|Rayshade|
|000024f0| 7d 20 61 6c 73 6f 20 73 | 75 70 70 6f 72 74 73 20 |} also s|upports |
|00002500| 61 6e 20 7b 5c 74 74 20 | 69 6d 61 67 65 7d 20 74 |an {\tt |image} t|
|00002510| 65 78 74 75 72 65 2e 20 | 20 54 68 69 73 20 74 65 |exture. | This te|
|00002520| 78 74 75 72 65 0a 61 6c | 6c 6f 77 73 20 79 6f 75 |xture.al|lows you|
|00002530| 20 74 6f 20 75 73 65 20 | 69 6d 61 67 65 73 20 74 | to use |images t|
|00002540| 6f 20 6d 6f 64 69 66 79 | 20 74 68 65 20 63 68 61 |o modify| the cha|
|00002550| 72 61 63 74 65 72 69 73 | 74 69 63 73 20 6f 66 20 |racteris|tics of |
|00002560| 61 20 73 75 72 66 61 63 | 65 2e 20 20 59 6f 75 0a |a surfac|e.  You.|
|00002570| 63 61 6e 20 75 73 65 20 | 74 68 72 65 65 2d 63 68 |can use |three-ch|
|00002580| 61 6e 6e 65 6c 20 69 6d | 61 67 65 73 20 74 6f 20 |annel im|ages to |
|00002590| 6d 6f 64 69 66 79 20 74 | 68 65 20 61 6e 79 20 6f |modify t|he any o|
|000025a0| 72 20 61 6c 6c 20 6f 66 | 0a 74 68 65 20 61 6d 62 |r all of|.the amb|
|000025b0| 69 65 6e 74 2c 20 64 69 | 66 66 75 73 65 2c 20 61 |ient, di|ffuse, a|
|000025c0| 6e 64 20 73 70 65 63 75 | 6c 61 72 20 63 6f 6c 6f |nd specu|lar colo|
|000025d0| 72 73 20 6f 66 20 61 20 | 73 75 72 66 61 63 65 2e |rs of a |surface.|
|000025e0| 0a 49 66 20 79 6f 75 20 | 61 72 65 20 75 73 69 6e |.If you |are usin|
|000025f0| 67 20 74 68 65 20 55 74 | 61 68 20 52 61 73 74 65 |g the Ut|ah Raste|
|00002600| 72 20 54 6f 6f 6c 6b 69 | 74 2c 0a 79 6f 75 20 63 |r Toolki|t,.you c|
|00002610| 61 6e 20 61 6c 73 6f 20 | 75 73 65 20 73 69 6e 67 |an also |use sing|
|00002620| 6c 65 2d 63 68 61 6e 6e | 65 6c 20 69 6d 61 67 65 |le-chann|el image|
|00002630| 73 20 74 6f 20 6d 6f 64 | 69 66 79 20 73 75 72 66 |s to mod|ify surf|
|00002640| 61 63 65 20 72 65 66 6c | 65 63 74 61 6e 63 65 2c |ace refl|ectance,|
|00002650| 0a 74 72 61 6e 73 70 61 | 72 65 6e 63 79 2c 20 61 |.transpa|rency, a|
|00002660| 6e 64 20 74 68 65 20 73 | 70 65 63 75 6c 61 72 20 |nd the s|pecular |
|00002670| 65 78 70 6f 6e 65 6e 74 | 2e 20 20 59 6f 75 20 63 |exponent|.  You c|
|00002680| 61 6e 20 61 6c 73 6f 20 | 75 73 65 20 61 20 73 69 |an also |use a si|
|00002690| 6e 67 6c 65 2d 63 68 61 | 6e 6e 65 6c 0a 69 6d 61 |ngle-cha|nnel.ima|
|000026a0| 67 65 20 74 6f 20 61 70 | 70 6c 79 20 61 20 62 75 |ge to ap|ply a bu|
|000026b0| 6d 70 20 6d 61 70 20 74 | 6f 20 61 20 73 75 72 66 |mp map t|o a surf|
|000026c0| 61 63 65 2e 0a 0a 49 6e | 20 61 6c 6c 20 62 75 74 |ace...In| all but|
|000026d0| 20 74 68 65 20 62 75 6d | 70 2d 6d 61 70 70 69 6e | the bum|p-mappin|
|000026e0| 67 20 63 61 73 65 2c 0a | 61 20 63 6f 6d 70 6f 6e |g case,.|a compon|
|000026f0| 65 6e 74 20 69 73 20 6d | 6f 64 69 66 69 65 64 20 |ent is m|odified |
|00002700| 62 79 20 6d 75 6c 74 69 | 70 6c 79 69 6e 67 20 74 |by multi|plying t|
|00002710| 68 65 20 67 69 76 65 6e | 20 76 61 6c 75 65 20 62 |he given| value b|
|00002720| 79 20 74 68 65 20 76 61 | 6c 75 65 0a 63 6f 6d 70 |y the va|lue.comp|
|00002730| 75 74 65 64 20 62 79 20 | 74 68 65 20 74 65 78 74 |uted by |the text|
|00002740| 75 72 69 6e 67 20 66 75 | 6e 63 74 69 6f 6e 2e 20 |uring fu|nction. |
|00002750| 20 57 68 65 6e 20 75 73 | 69 6e 67 20 74 68 65 20 | When us|ing the |
|00002760| 55 74 61 68 20 52 61 73 | 74 65 72 20 54 6f 6f 6c |Utah Ras|ter Tool|
|00002770| 6b 69 74 2c 0a 73 75 72 | 66 61 63 65 20 63 68 61 |kit,.sur|face cha|
|00002780| 72 61 63 74 65 72 69 73 | 74 69 63 73 20 61 72 65 |racteris|tics are|
|00002790| 20 6d 6f 64 69 66 69 65 | 64 20 69 6e 20 70 72 6f | modifie|d in pro|
|000027a0| 70 6f 72 74 69 6f 6e 20 | 74 6f 20 74 68 65 20 76 |portion |to the v|
|000027b0| 61 6c 75 65 20 6f 66 0a | 74 68 65 20 7b 5c 65 6d |alue of.|the {\em|
|000027c0| 20 61 6c 70 68 61 7d 20 | 63 68 61 6e 6e 65 6c 20 | alpha} |channel |
|000027d0| 69 6e 20 74 68 65 20 69 | 6d 61 67 65 2e 20 20 49 |in the i|mage.  I|
|000027e0| 66 20 74 68 65 72 65 20 | 69 73 20 6e 6f 20 7b 5c |f there |is no {\|
|000027f0| 65 6d 20 61 6c 70 68 61 | 7d 20 63 68 61 6e 6e 65 |em alpha|} channe|
|00002800| 6c 2c 0a 6f 72 20 79 6f | 75 20 61 72 65 20 6e 6f |l,.or yo|u are no|
|00002810| 74 20 75 73 69 6e 67 20 | 74 68 65 20 55 74 61 68 |t using |the Utah|
|00002820| 20 52 61 73 74 65 72 20 | 54 6f 6f 6c 6b 69 74 2c | Raster |Toolkit,|
|00002830| 20 7b 5c 65 6d 20 61 6c | 70 68 61 7d 20 69 73 20 | {\em al|pha} is |
|00002840| 61 73 73 75 6d 65 64 20 | 74 6f 20 62 65 0a 65 76 |assumed |to be.ev|
|00002850| 65 72 79 77 68 65 72 65 | 0a 65 71 75 61 6c 20 74 |erywhere|.equal t|
|00002860| 6f 20 24 31 24 2e 0a 0a | 5c 62 65 67 69 6e 7b 64 |o $1$...|\begin{d|
|00002870| 65 66 6b 65 79 7d 7b 63 | 6f 6d 70 6f 6e 65 6e 74 |efkey}{c|omponent|
|00002880| 7d 7b 24 3c 24 7b 5c 65 | 6d 20 43 6f 6d 70 6f 6e |}{$<${\e|m Compon|
|00002890| 65 6e 74 7d 24 3e 24 7d | 0a 09 54 68 65 20 6e 61 |ent}$>$}|..The na|
|000028a0| 6d 65 64 20 63 6f 6d 70 | 6f 6e 65 6e 74 20 77 69 |med comp|onent wi|
|000028b0| 6c 6c 20 62 65 20 6d 6f | 64 69 66 69 65 64 2e 0a |ll be mo|dified..|
|000028c0| 5c 65 6e 64 7b 64 65 66 | 6b 65 79 7d 0a 50 6f 73 |\end{def|key}.Pos|
|000028d0| 73 69 62 6c 65 20 73 75 | 72 66 61 63 65 20 63 6f |sible su|rface co|
|000028e0| 6d 70 6f 6e 65 6e 74 73 | 20 61 72 65 3a 0a 7b 5c |mponents| are:.{\|
|000028f0| 74 74 20 61 6d 62 69 65 | 6e 74 7d 20 28 6d 6f 64 |tt ambie|nt} (mod|
|00002900| 69 66 79 20 61 6d 62 69 | 65 6e 74 20 63 6f 6c 6f |ify ambi|ent colo|
|00002910| 72 29 2c 0a 7b 5c 74 74 | 20 64 69 66 66 75 73 65 |r),.{\tt| diffuse|
|00002920| 7d 20 28 6d 6f 64 69 66 | 79 20 64 69 66 66 75 73 |} (modif|y diffus|
|00002930| 65 20 63 6f 6c 6f 72 29 | 2c 0a 7b 5c 74 74 20 73 |e color)|,.{\tt s|
|00002940| 70 65 63 75 6c 61 72 7d | 20 28 6d 6f 64 69 66 79 |pecular}| (modify|
|00002950| 20 73 70 65 63 75 6c 61 | 72 20 63 6f 6c 6f 72 29 | specula|r color)|
|00002960| 2c 0a 7b 5c 74 74 20 73 | 70 65 63 70 6f 77 7d 2c |,.{\tt s|pecpow},|
|00002970| 20 28 6d 6f 64 69 66 79 | 20 73 70 65 63 75 6c 61 | (modify| specula|
|00002980| 72 20 65 78 70 6f 6e 65 | 6e 74 29 2c 0a 7b 5c 74 |r expone|nt),.{\t|
|00002990| 74 20 72 65 66 6c 65 63 | 74 7d 2c 09 28 6d 6f 64 |t reflec|t},.(mod|
|000029a0| 69 66 79 20 72 65 66 6c | 65 63 74 69 76 69 74 79 |ify refl|ectivity|
|000029b0| 29 2c 0a 7b 5c 74 74 20 | 74 72 61 6e 73 70 7d 20 |),.{\tt |transp} |
|000029c0| 28 6d 6f 64 69 66 79 20 | 74 72 61 6e 73 70 61 72 |(modify |transpar|
|000029d0| 65 6e 63 79 29 2c 0a 7b | 5c 74 74 20 62 75 6d 70 |ency),.{|\tt bump|
|000029e0| 7d 2c 20 28 6d 6f 64 69 | 66 79 20 73 75 72 66 61 |}, (modi|fy surfa|
|000029f0| 63 65 20 6e 6f 72 6d 61 | 6c 29 2e 0a 54 68 65 20 |ce norma|l)..The |
|00002a00| 7b 5c 74 74 20 73 70 65 | 63 70 6f 77 7d 2c 20 7b |{\tt spe|cpow}, {|
|00002a10| 5c 74 74 20 72 65 66 6c | 65 63 74 7d 2c 20 7b 5c |\tt refl|ect}, {\|
|00002a20| 74 74 20 74 72 61 6e 73 | 70 7d 2c 20 61 6e 64 20 |tt trans|p}, and |
|00002a30| 7b 5c 74 74 20 62 75 6d | 70 7d 0a 63 6f 6d 70 6f |{\tt bum|p}.compo|
|00002a40| 6e 65 6e 74 73 20 72 65 | 71 75 69 72 65 20 74 68 |nents re|quire th|
|00002a50| 65 20 75 73 65 20 6f 66 | 20 61 20 73 69 6e 67 6c |e use of| a singl|
|00002a60| 65 2d 63 68 61 6e 6e 65 | 6c 20 69 6d 61 67 65 2e |e-channe|l image.|
|00002a70| 0a 0a 5c 62 65 67 69 6e | 7b 64 65 66 6b 65 79 7d |..\begin|{defkey}|
|00002a80| 7b 72 61 6e 67 65 7d 7b | 7b 5c 65 6d 20 68 69 67 |{range}{|{\em hig|
|00002a90| 68 20 6c 6f 77 7d 7d 0a | 09 53 70 65 63 69 66 79 |h low}}.|.Specify|
|00002aa0| 20 74 68 65 20 72 61 6e | 67 65 20 6f 66 20 76 61 | the ran|ge of va|
|00002ab0| 6c 75 65 73 20 74 6f 20 | 77 68 69 63 68 20 74 68 |lues to |which th|
|00002ac0| 65 20 76 61 6c 75 65 73 | 20 69 6e 20 74 68 65 0a |e values| in the.|
|00002ad0| 09 69 6d 61 67 65 20 73 | 68 6f 75 6c 64 20 62 65 |.image s|hould be|
|00002ae0| 20 6d 61 70 70 65 64 2e | 20 20 41 6e 20 76 61 6c | mapped.|  An val|
|00002af0| 75 65 20 6f 66 20 24 31 | 24 20 77 69 6c 6c 0a 09 |ue of $1|$ will..|
|00002b00| 62 65 20 6d 61 70 70 65 | 64 20 7b 5c 65 6d 20 68 |be mappe|d {\em h|
|00002b10| 69 67 68 7d 2c 20 24 30 | 24 20 74 6f 20 7b 5c 65 |igh}, $0|$ to {\e|
|00002b20| 6d 20 6c 6f 77 7d 2e 20 | 20 49 6e 74 65 72 6d 65 |m low}. | Interme|
|00002b30| 64 69 61 74 65 0a 09 76 | 61 6c 75 65 73 20 77 69 |diate..v|alues wi|
|00002b40| 6c 6c 20 62 65 20 6c 69 | 6e 65 61 72 6c 79 20 69 |ll be li|nearly i|
|00002b50| 6e 74 65 72 70 6f 6c 61 | 74 65 64 2e 0a 5c 65 6e |nterpola|ted..\en|
|00002b60| 64 7b 64 65 66 6b 65 79 | 7d 0a 0a 5c 62 65 67 69 |d{defkey|}..\begi|
|00002b70| 6e 7b 64 65 66 6b 65 79 | 7d 7b 73 6d 6f 6f 74 68 |n{defkey|}{smooth|
|00002b80| 7d 7b 7d 0a 09 57 68 65 | 6e 20 67 69 76 65 6e 2c |}{}..Whe|n given,|
|00002b90| 20 70 69 78 65 6c 20 61 | 76 65 72 61 67 69 6e 67 | pixel a|veraging|
|00002ba0| 20 77 69 6c 6c 20 62 65 | 20 70 65 72 66 6f 72 6d | will be| perform|
|00002bb0| 65 64 20 69 6e 20 6f 72 | 64 65 72 0a 09 74 6f 20 |ed in or|der..to |
|00002bc0| 73 6d 6f 6f 74 68 20 74 | 68 65 20 73 61 6d 70 6c |smooth t|he sampl|
|00002bd0| 65 64 20 69 6d 61 67 65 | 2e 20 20 49 66 20 6e 6f |ed image|.  If no|
|00002be0| 74 20 73 70 65 63 69 66 | 69 65 64 2c 20 6e 6f 20 |t specif|ied, no |
|00002bf0| 61 76 65 72 61 67 69 6e | 67 0a 09 77 69 6c 6c 20 |averagin|g..will |
|00002c00| 6f 63 63 75 72 2e 0a 5c | 65 6e 64 7b 64 65 66 6b |occur..\|end{defk|
|00002c10| 65 79 7d 0a 0a 5c 62 65 | 67 69 6e 7b 64 65 66 6b |ey}..\be|gin{defk|
|00002c20| 65 79 7d 7b 74 65 78 74 | 73 75 72 66 7d 7b 24 3c |ey}{text|surf}{$<|
|00002c30| 24 7b 5c 65 6d 20 53 75 | 72 66 61 63 65 20 53 70 |${\em Su|rface Sp|
|00002c40| 65 63 69 66 69 63 61 74 | 69 6f 6e 7d 24 3e 24 7d |ecificat|ion}$>$}|
|00002c50| 0a 09 46 6f 72 20 75 73 | 65 20 77 68 65 6e 20 6d |..For us|e when m|
|00002c60| 6f 64 69 66 79 69 6e 67 | 20 73 75 72 66 61 63 65 |odifying| surface|
|00002c70| 20 63 6f 6c 6f 72 73 2c | 20 74 68 69 73 20 6b 65 | colors,| this ke|
|00002c80| 79 77 6f 72 64 20 73 70 | 65 63 69 66 69 65 73 0a |yword sp|ecifies.|
|00002c90| 09 74 68 61 74 20 74 68 | 65 20 67 69 76 65 6e 20 |.that th|e given |
|00002ca0| 73 75 72 66 61 63 65 20 | 73 68 6f 75 6c 64 20 62 |surface |should b|
|00002cb0| 65 20 75 73 65 64 20 61 | 73 20 74 68 65 20 62 61 |e used a|s the ba|
|00002cc0| 73 65 0a 09 74 6f 20 62 | 65 20 6d 6f 64 69 66 69 |se..to b|e modifi|
|00002cd0| 65 64 20 77 68 65 6e 20 | 74 68 65 20 7b 5c 65 6d |ed when |the {\em|
|00002ce0| 20 61 6c 70 68 61 7d 20 | 76 61 6c 75 65 20 69 6e | alpha} |value in|
|00002cf0| 20 74 68 65 20 69 6d 61 | 67 65 0a 09 69 73 20 6e | the ima|ge..is n|
|00002d00| 6f 6e 2d 7a 65 72 6f 2e | 20 20 57 68 65 6e 20 7b |on-zero.|  When {|
|00002d10| 5c 65 6d 20 61 6c 70 68 | 61 7d 20 69 73 20 7a 65 |\em alph|a} is ze|
|00002d20| 72 6f 2c 20 74 68 65 20 | 6f 62 6a 65 63 74 27 73 |ro, the |object's|
|00002d30| 0a 09 75 6e 6d 6f 64 69 | 66 69 65 64 20 64 65 66 |..unmodi|fied def|
|00002d40| 61 75 6c 74 20 73 75 72 | 66 61 63 65 20 63 68 61 |ault sur|face cha|
|00002d50| 72 61 63 74 65 72 69 73 | 74 69 63 73 20 61 72 65 |racteris|tics are|
|00002d60| 20 72 65 74 61 69 6e 65 | 64 2e 0a 5c 65 6e 64 7b | retaine|d..\end{|
|00002d70| 64 65 66 6b 65 79 7d 0a | 54 68 65 20 75 73 75 61 |defkey}.|The usua|
|00002d80| 6c 20 62 65 68 61 76 69 | 6f 72 20 69 73 20 66 6f |l behavi|or is fo|
|00002d90| 72 20 74 68 65 20 6f 62 | 6a 65 63 74 27 73 20 64 |r the ob|ject's d|
|00002da0| 65 66 61 75 6c 74 20 73 | 75 72 66 61 63 65 20 70 |efault s|urface p|
|00002db0| 72 6f 70 65 72 74 69 65 | 73 20 74 6f 0a 62 65 20 |ropertie|s to.be |
|00002dc0| 75 73 65 64 2e 0a 0a 5c | 62 65 67 69 6e 7b 64 65 |used...\|begin{de|
|00002dd0| 66 6b 65 79 7d 7b 74 69 | 6c 65 7d 7b 7b 5c 65 6d |fkey}{ti|le}{{\em|
|00002de0| 20 75 6e 20 76 6e 7d 7d | 0a 09 53 70 65 63 69 66 | un vn}}|..Specif|
|00002df0| 79 20 68 6f 77 20 74 68 | 65 20 69 6d 61 67 65 20 |y how th|e image |
|00002e00| 73 68 6f 75 6c 64 20 62 | 65 20 74 69 6c 65 64 20 |should b|e tiled |
|00002e10| 28 72 65 70 65 61 74 65 | 64 29 20 61 6c 6f 6e 67 |(repeate|d) along|
|00002e20| 20 74 68 65 0a 09 24 75 | 24 20 61 6e 64 20 24 76 | the..$u|$ and $v|
|00002e30| 24 20 61 78 65 73 2e 0a | 09 49 66 20 70 6f 73 69 |$ axes..|.If posi|
|00002e40| 74 69 76 65 2c 20 74 68 | 65 20 76 61 6c 75 65 20 |tive, th|e value |
|00002e50| 6f 66 20 7b 5c 65 6d 20 | 75 6e 7d 20 67 69 76 65 |of {\em |un} give|
|00002e60| 73 20 74 68 65 20 6e 75 | 6d 62 65 72 20 6f 66 0a |s the nu|mber of.|
|00002e70| 09 74 69 6d 65 73 20 74 | 68 65 20 69 6d 61 67 65 |.times t|he image|
|00002e80| 20 73 68 6f 75 6c 64 20 | 62 65 20 72 65 70 65 61 | should |be repea|
|00002e90| 74 65 64 20 61 6c 6f 6e | 67 20 74 68 65 20 24 75 |ted alon|g the $u|
|00002ea0| 24 20 61 78 69 73 2c 20 | 73 74 61 72 74 69 6e 67 |$ axis, |starting|
|00002eb0| 0a 09 66 72 6f 6d 20 74 | 68 65 20 6f 72 69 67 69 |..from t|he origi|
|00002ec0| 6e 20 6f 66 20 74 68 65 | 20 74 65 78 74 75 72 65 |n of the| texture|
|00002ed0| 2c 20 61 6e 64 20 70 6f | 73 69 74 69 76 65 20 7b |, and po|sitive {|
|00002ee0| 5c 65 6d 20 76 6e 7d 20 | 67 69 76 65 73 20 74 68 |\em vn} |gives th|
|00002ef0| 65 0a 09 6e 75 6d 62 65 | 72 20 6f 66 20 74 69 6d |e..numbe|r of tim|
|00002f00| 65 73 20 69 74 0a 09 73 | 68 6f 75 6c 64 20 62 65 |es it..s|hould be|
|00002f10| 20 72 65 70 65 61 74 65 | 64 20 61 6c 6f 6e 67 20 | repeate|d along |
|00002f20| 74 68 65 20 24 76 24 20 | 61 78 69 73 2e 20 20 49 |the $v$ |axis.  I|
|00002f30| 66 20 65 69 74 68 65 72 | 20 76 61 6c 75 65 20 69 |f either| value i|
|00002f40| 73 20 7a 65 72 6f 2c 0a | 09 74 68 65 20 69 6d 61 |s zero,.|.the ima|
|00002f50| 67 65 20 69 73 20 72 65 | 70 65 61 74 65 64 20 69 |ge is re|peated i|
|00002f60| 6e 66 69 6e 69 74 65 6c | 79 20 61 6c 6f 6e 67 20 |nfinitel|y along |
|00002f70| 74 68 65 20 61 70 70 72 | 6f 70 72 69 61 74 65 20 |the appr|opriate |
|00002f80| 61 78 69 73 2e 0a 5c 65 | 6e 64 7b 64 65 66 6b 65 |axis..\e|nd{defke|
|00002f90| 79 7d 0a 54 69 6c 69 6e | 67 20 69 73 20 75 73 75 |y}.Tilin|g is usu|
|00002fa0| 61 6c 6c 79 20 6f 6e 6c | 79 20 61 20 63 6f 6e 63 |ally onl|y a conc|
|00002fb0| 65 72 6e 20 77 68 65 6e | 20 70 6c 61 6e 61 72 20 |ern when| planar |
|00002fc0| 6d 61 70 70 69 6e 67 20 | 69 73 20 62 65 69 6e 67 |mapping |is being|
|00002fd0| 20 75 73 65 64 2c 0a 74 | 68 6f 75 67 68 20 69 74 | used,.t|hough it|
|00002fe0| 20 6d 61 79 20 61 6c 73 | 6f 20 62 65 20 75 73 65 | may als|o be use|
|00002ff0| 64 20 69 66 20 69 6d 61 | 67 65 20 74 65 78 74 75 |d if ima|ge textu|
|00003000| 72 65 73 20 61 72 65 20 | 62 65 69 6e 67 20 73 63 |res are |being sc|
|00003010| 61 6c 65 64 2e 20 20 42 | 79 20 64 65 66 61 75 6c |aled.  B|y defaul|
|00003020| 74 0a 7b 5c 65 6d 20 75 | 6e 7d 20 61 6e 64 20 7b |t.{\em u|n} and {|
|00003030| 5c 65 6d 20 76 6e 7d 20 | 61 72 65 20 62 6f 74 68 |\em vn} |are both|
|00003040| 20 7a 65 72 6f 2e 0a 20 | 0a 0a 41 20 6d 61 70 70 | zero.. |..A mapp|
|00003050| 69 6e 67 20 66 75 6e 63 | 74 69 6f 6e 20 6d 61 79 |ing func|tion may|
|00003060| 20 61 6c 73 6f 20 62 65 | 20 61 73 73 6f 63 69 61 | also be| associa|
|00003070| 74 65 64 20 77 69 74 68 | 20 61 6e 20 69 6d 61 67 |ted with| an imag|
|00003080| 65 20 74 65 78 74 75 72 | 65 2e 0a 0a 5c 73 65 63 |e textur|e...\sec|
|00003090| 74 69 6f 6e 20 7b 4d 61 | 70 70 69 6e 67 20 46 75 |tion {Ma|pping Fu|
|000030a0| 6e 63 74 69 6f 6e 73 7d | 0a 0a 4d 61 70 70 69 6e |nctions}|..Mappin|
|000030b0| 67 20 66 75 6e 63 74 69 | 6f 6e 73 20 61 72 65 20 |g functi|ons are |
|000030c0| 75 73 65 64 20 74 6f 20 | 61 70 70 6c 79 20 74 77 |used to |apply tw|
|000030d0| 6f 2d 64 69 6d 65 6e 73 | 69 6f 6e 61 6c 20 74 65 |o-dimens|ional te|
|000030e0| 78 74 75 72 65 73 20 74 | 6f 0a 73 75 72 66 61 63 |xtures t|o.surfac|
|000030f0| 65 73 2e 20 20 45 61 63 | 68 20 6d 61 70 70 69 6e |es.  Eac|h mappin|
|00003100| 67 20 66 75 6e 63 74 69 | 6f 6e 73 20 64 65 66 69 |g functi|ons defi|
|00003110| 6e 65 73 20 61 20 64 69 | 66 66 65 72 65 6e 74 20 |nes a di|fferent |
|00003120| 6d 65 74 68 6f 64 20 6f | 66 20 74 72 61 6e 73 66 |method o|f transf|
|00003130| 6f 72 6d 69 6e 67 0a 61 | 20 74 68 72 65 65 20 64 |orming.a| three d|
|00003140| 69 6d 65 6e 73 69 6f 6e | 61 6c 20 70 6f 69 6e 74 |imension|al point|
|00003150| 20 6f 66 20 69 6e 74 65 | 72 73 65 63 74 69 6f 6e | of inte|rsection|
|00003160| 20 74 6f 20 61 20 74 77 | 6f 20 64 69 6d 65 6e 73 | to a tw|o dimens|
|00003170| 69 6f 6e 61 6c 20 24 75 | 2d 76 24 20 70 61 69 72 |ional $u|-v$ pair|
|00003180| 0a 74 65 72 6d 65 64 20 | 74 65 78 74 75 72 69 6e |.termed |texturin|
|00003190| 67 20 63 6f 6f 72 64 69 | 6e 61 74 65 73 2e 0a 54 |g coordi|nates..T|
|000031a0| 79 70 69 63 61 6c 6c 79 | 2c 20 74 68 65 20 61 72 |ypically|, the ar|
|000031b0| 67 75 6d 65 6e 74 73 20 | 74 6f 20 61 20 6d 61 70 |guments |to a map|
|000031c0| 70 69 6e 67 20 6d 65 74 | 68 6f 64 20 64 65 66 69 |ping met|hod defi|
|000031d0| 6e 65 20 61 20 63 65 6e | 74 65 72 20 6f 66 0a 61 |ne a cen|ter of.a|
|000031e0| 20 70 72 6f 6a 65 63 74 | 69 6f 6e 20 61 6e 64 20 | project|ion and |
|000031f0| 74 77 6f 20 6e 6f 6e 2d | 70 61 72 61 6c 6c 65 6c |two non-|parallel|
|00003200| 20 61 78 65 73 20 74 68 | 61 74 20 64 65 66 69 6e | axes th|at defin|
|00003210| 65 20 61 20 6c 6f 63 61 | 6c 20 63 6f 6f 72 64 69 |e a loca|l coordi|
|00003220| 6e 61 74 65 20 73 79 73 | 74 65 6d 2e 0a 0a 54 68 |nate sys|tem...Th|
|00003230| 65 20 64 65 66 61 75 6c | 74 20 6d 61 70 70 69 6e |e defaul|t mappin|
|00003240| 67 20 6d 65 74 68 6f 64 | 20 69 73 20 74 65 72 6d |g method| is term|
|00003250| 65 64 20 24 75 2d 76 24 | 20 6d 61 70 70 69 6e 67 |ed $u-v$| mapping|
|00003260| 20 6f 72 20 7b 5c 65 6d | 20 69 6e 76 65 72 73 65 | or {\em| inverse|
|00003270| 20 6d 61 70 70 69 6e 67 | 7d 2e 0a 4e 6f 72 6d 61 | mapping|}..Norma|
|00003280| 6c 6c 79 2c 20 74 68 65 | 72 65 20 69 73 20 61 20 |lly, the|re is a |
|00003290| 64 69 66 66 65 72 65 6e | 74 20 69 6e 76 65 72 73 |differen|t invers|
|000032a0| 65 20 6d 61 70 70 69 6e | 67 20 6d 65 74 68 6f 64 |e mappin|g method|
|000032b0| 20 66 6f 72 20 65 61 63 | 68 20 70 72 69 6d 69 74 | for eac|h primit|
|000032c0| 69 76 65 20 74 79 70 65 | 0a 28 73 65 65 20 63 68 |ive type|.(see ch|
|000032d0| 61 70 74 65 72 20 35 29 | 2e 0a 57 68 65 6e 20 69 |apter 5)|..When i|
|000032e0| 6e 76 65 72 73 65 20 6d | 61 70 70 69 6e 67 20 69 |nverse m|apping i|
|000032f0| 73 20 75 73 65 64 2c 20 | 74 68 65 20 70 6f 69 6e |s used, |the poin|
|00003300| 74 20 6f 66 20 69 6e 74 | 65 72 73 65 63 74 69 6f |t of int|ersectio|
|00003310| 6e 20 69 73 20 70 61 73 | 73 65 64 20 74 6f 0a 74 |n is pas|sed to.t|
|00003320| 68 65 20 24 75 76 24 20 | 6d 65 74 68 6f 64 20 66 |he $uv$ |method f|
|00003330| 6f 72 20 74 68 65 20 70 | 72 69 6d 69 74 69 76 65 |or the p|rimitive|
|00003340| 20 74 68 61 74 20 77 61 | 73 20 68 69 74 2e 0a 0a | that wa|s hit...|
|00003350| 5c 62 65 67 69 6e 7b 64 | 65 66 6b 65 79 7d 7b 6d |\begin{d|efkey}{m|
|00003360| 61 70 7d 7b 7b 5c 74 74 | 20 75 76 7d 7d 0a 09 55 |ap}{{\tt| uv}}..U|
|00003370| 73 65 20 74 68 65 20 24 | 75 76 24 20 28 69 6e 76 |se the $|uv$ (inv|
|00003380| 65 72 73 65 20 6d 61 70 | 70 69 6e 67 29 20 6d 65 |erse map|ping) me|
|00003390| 74 68 6f 64 20 61 73 73 | 6f 63 69 61 74 65 64 20 |thod ass|ociated |
|000033a0| 77 69 74 68 20 74 68 65 | 0a 09 6f 62 6a 65 63 74 |with the|..object|
|000033b0| 20 74 68 61 74 20 77 61 | 73 20 69 6e 74 65 72 73 | that wa|s inters|
|000033c0| 65 63 74 65 64 20 69 6e | 20 6f 72 64 65 72 20 74 |ected in| order t|
|000033d0| 6f 20 6d 61 70 20 66 72 | 6f 6d 20 33 44 20 74 6f |o map fr|om 3D to|
|000033e0| 20 64 65 74 65 72 6d 69 | 6e 65 0a 09 74 65 78 74 | determi|ne..text|
|000033f0| 75 72 69 6e 67 20 63 6f | 6f 72 64 69 6e 61 74 65 |uring co|ordinate|
|00003400| 73 2e 0a 5c 65 6e 64 7b | 64 65 66 6b 65 79 7d 0a |s..\end{|defkey}.|
|00003410| 54 68 65 20 69 6e 76 65 | 72 73 65 20 6d 61 70 70 |The inve|rse mapp|
|00003420| 69 6e 67 20 6d 65 74 68 | 6f 64 20 66 6f 72 20 65 |ing meth|od for e|
|00003430| 61 63 68 20 70 72 69 6d | 69 74 69 76 65 20 69 73 |ach prim|itive is|
|00003440| 20 64 65 73 63 72 69 62 | 65 64 20 69 6e 20 43 68 | describ|ed in Ch|
|00003450| 61 70 74 65 72 20 35 2e | 0a 0a 5c 62 65 67 69 6e |apter 5.|..\begin|
|00003460| 7b 64 65 66 6b 65 79 7d | 7b 6d 61 70 7d 7b 7b 5c |{defkey}|{map}{{\|
|00003470| 74 74 20 70 6c 61 6e 61 | 72 7d 20 5b 5c 65 76 65 |tt plana|r} [\eve|
|00003480| 63 7b 6f 72 69 67 69 6e | 7d 20 5c 65 76 65 63 7b |c{origin|} \evec{|
|00003490| 76 61 78 69 73 7d 20 5c | 65 76 65 63 7b 75 61 78 |vaxis} \|evec{uax|
|000034a0| 69 73 7d 5d 7d 0a 09 55 | 73 65 20 61 20 70 6c 61 |is}]}..U|se a pla|
|000034b0| 6e 61 72 20 6d 61 70 70 | 69 6e 67 20 6d 65 74 68 |nar mapp|ing meth|
|000034c0| 6f 64 2e 20 54 68 65 20 | 32 44 20 74 65 78 74 75 |od. The |2D textu|
|000034d0| 72 65 20 69 73 20 74 72 | 61 6e 73 66 6f 72 6d 65 |re is tr|ansforme|
|000034e0| 64 0a 09 73 6f 20 74 68 | 61 74 20 69 74 73 20 24 |d..so th|at its $|
|000034f0| 75 24 20 61 78 69 73 20 | 69 73 20 67 69 76 65 6e |u$ axis |is given|
|00003500| 20 62 79 20 5c 65 76 65 | 63 7b 75 61 78 69 73 7d | by \eve|c{uaxis}|
|00003510| 20 61 6e 64 20 69 74 73 | 20 24 76 24 0a 09 61 78 | and its| $v$..ax|
|00003520| 69 73 20 62 79 20 24 76 | 61 78 69 73 24 2e 20 20 |is by $v|axis$.  |
|00003530| 54 68 65 20 74 65 78 74 | 75 72 65 20 69 73 20 70 |The text|ure is p|
|00003540| 72 6f 6a 65 63 74 65 64 | 20 61 6c 6f 6e 67 20 74 |rojected| along t|
|00003550| 68 65 20 76 65 63 74 6f | 72 0a 09 64 65 66 69 6e |he vecto|r..defin|
|00003560| 65 64 20 62 79 20 74 68 | 65 20 63 72 6f 73 73 20 |ed by th|e cross |
|00003570| 70 72 6f 64 75 63 74 20 | 6f 66 20 74 68 65 20 24 |product |of the $|
|00003580| 75 24 20 61 6e 64 20 24 | 76 24 20 61 78 65 73 2c |u$ and $|v$ axes,|
|00003590| 20 77 69 74 68 0a 09 74 | 68 65 20 28 30 2c 30 29 | with..t|he (0,0)|
|000035a0| 20 69 6e 20 74 65 78 74 | 75 72 65 20 73 70 61 63 | in text|ure spac|
|000035b0| 65 20 6d 61 70 70 65 64 | 20 74 6f 20 5c 65 76 65 |e mapped| to \eve|
|000035c0| 63 7b 6f 72 69 67 69 6e | 7d 2e 0a 5c 65 6e 64 7b |c{origin|}..\end{|
|000035d0| 64 65 66 6b 65 79 7d 0a | 0a 5c 62 65 67 69 6e 7b |defkey}.|.\begin{|
|000035e0| 64 65 66 6b 65 79 7d 7b | 6d 61 70 7d 7b 7b 5c 74 |defkey}{|map}{{\t|
|000035f0| 74 20 63 79 6c 69 6e 64 | 72 69 63 61 6c 7d 20 5b |t cylind|rical} [|
|00003600| 5c 65 76 65 63 7b 6f 72 | 69 67 69 6e 7d 20 5c 65 |\evec{or|igin} \e|
|00003610| 76 65 63 7b 76 61 78 69 | 73 7d 20 5c 65 76 65 63 |vec{vaxi|s} \evec|
|00003620| 7b 75 61 78 69 73 7d 5d | 7d 0a 09 55 73 65 20 61 |{uaxis}]|}..Use a|
|00003630| 20 63 79 6c 69 6e 64 72 | 69 63 61 6c 20 6d 61 70 | cylindr|ical map|
|00003640| 70 69 6e 67 20 6d 65 74 | 68 6f 64 2e 20 20 54 68 |ping met|hod.  Th|
|00003650| 65 20 70 6f 69 6e 74 20 | 6f 66 20 69 6e 74 65 72 |e point |of inter|
|00003660| 73 65 63 74 69 6f 6e 0a | 09 69 73 20 70 72 6f 6a |section.|.is proj|
|00003670| 65 63 74 65 64 20 6f 6e | 74 6f 20 61 6e 20 69 6d |ected on|to an im|
|00003680| 61 67 69 6e 61 72 79 20 | 63 79 6c 69 6e 64 65 72 |aginary |cylinder|
|00003690| 2c 20 61 6e 64 20 74 68 | 65 20 6c 6f 63 61 74 69 |, and th|e locati|
|000036a0| 6f 6e 0a 09 6f 66 20 74 | 68 65 20 70 72 6f 6a 65 |on..of t|he proje|
|000036b0| 63 74 65 64 20 70 6f 69 | 6e 74 20 69 73 20 75 73 |cted poi|nt is us|
|000036c0| 65 64 20 74 6f 20 64 65 | 74 65 72 6d 69 6e 65 20 |ed to de|termine |
|000036d0| 74 68 65 20 74 65 78 74 | 75 72 65 20 63 6f 6f 72 |the text|ure coor|
|000036e0| 64 69 6e 61 74 65 73 2e | 0a 09 49 66 20 67 69 76 |dinates.|..If giv|
|000036f0| 65 6e 2c 20 5c 65 76 65 | 63 7b 6f 72 69 67 69 6e |en, \eve|c{origin|
|00003700| 7d 20 61 6e 64 0a 09 5c | 65 76 65 63 7b 76 61 78 |} and..\|evec{vax|
|00003710| 69 73 7d 20 64 65 66 69 | 6e 65 20 74 68 65 20 63 |is} defi|ne the c|
|00003720| 79 6c 69 6e 64 65 72 27 | 73 20 61 78 69 73 2c 20 |ylinder'|s axis, |
|00003730| 61 6e 64 20 5c 65 76 65 | 63 7b 75 61 78 69 73 7d |and \eve|c{uaxis}|
|00003740| 20 64 65 66 69 6e 65 73 | 0a 09 77 68 65 72 65 20 | defines|..where |
|00003750| 24 75 3d 30 24 20 69 73 | 20 6c 6f 63 61 74 65 64 |$u=0$ is| located|
|00003760| 2e 0a 5c 65 6e 64 7b 64 | 65 66 6b 65 79 7d 0a 53 |..\end{d|efkey}.S|
|00003770| 65 65 20 74 68 65 20 64 | 65 73 63 72 69 70 74 69 |ee the d|escripti|
|00003780| 6f 6e 20 6f 66 20 74 68 | 65 20 69 6e 76 65 72 73 |on of th|e invers|
|00003790| 65 20 6d 61 70 70 69 6e | 67 20 6d 65 74 68 6f 64 |e mappin|g method|
|000037a0| 20 66 6f 72 20 74 68 65 | 20 0a 63 79 6c 69 6e 64 | for the| .cylind|
|000037b0| 65 72 20 69 6e 20 43 68 | 61 70 74 65 72 20 35 2e |er in Ch|apter 5.|
|000037c0| 20 20 42 79 20 64 65 66 | 61 75 6c 74 2c 20 74 68 |  By def|ault, th|
|000037d0| 65 20 70 6f 69 6e 74 20 | 6f 66 20 69 6e 74 65 72 |e point |of inter|
|000037e0| 73 65 63 74 69 6f 6e 20 | 69 73 0a 70 72 6f 6a 65 |section |is.proje|
|000037f0| 63 74 65 64 20 6f 6e 74 | 6f 20 61 20 63 79 6c 69 |cted ont|o a cyli|
|00003800| 6e 64 65 72 20 74 68 61 | 74 20 72 75 6e 73 20 74 |nder tha|t runs t|
|00003810| 68 72 6f 75 67 68 20 74 | 68 65 20 6f 72 69 67 69 |hrough t|he origi|
|00003820| 6e 20 61 6c 6f 6e 67 20 | 74 68 65 20 24 7a 24 0a |n along |the $z$.|
|00003830| 61 78 69 73 2c 20 77 69 | 74 68 20 5c 65 76 65 63 |axis, wi|th \evec|
|00003840| 7b 75 61 78 69 73 7d 20 | 65 71 75 61 6c 20 74 6f |{uaxis} |equal to|
|00003850| 20 74 68 65 20 24 78 24 | 20 61 78 69 73 2e 0a 0a | the $x$| axis...|
|00003860| 5c 62 65 67 69 6e 7b 64 | 65 66 6b 65 79 7d 7b 6d |\begin{d|efkey}{m|
|00003870| 61 70 7d 7b 7b 5c 74 74 | 20 73 70 68 65 72 69 63 |ap}{{\tt| spheric|
|00003880| 61 6c 7d 20 5b 5c 65 76 | 65 63 7b 6f 72 69 67 69 |al} [\ev|ec{origi|
|00003890| 6e 7d 20 5c 65 76 65 63 | 7b 76 61 78 69 73 7d 20 |n} \evec|{vaxis} |
|000038a0| 5c 65 76 65 63 7b 75 61 | 78 69 73 7d 5d 7d 0a 09 |\evec{ua|xis}]}..|
|000038b0| 55 73 65 20 61 20 73 70 | 68 65 72 69 63 61 6c 20 |Use a sp|herical |
|000038c0| 6d 61 70 70 69 6e 67 20 | 6d 65 74 68 6f 64 2e 20 |mapping |method. |
|000038d0| 20 54 68 65 20 69 6e 74 | 65 72 73 65 63 74 69 6f | The int|ersectio|
|000038e0| 6e 20 70 6f 69 6e 74 20 | 69 73 0a 09 70 72 6f 6a |n point |is..proj|
|000038f0| 65 63 74 65 64 20 6f 6e | 74 6f 20 61 6e 20 69 6d |ected on|to an im|
|00003900| 61 67 69 6e 61 72 79 20 | 73 70 68 65 72 65 2c 20 |aginary |sphere, |
|00003910| 61 6e 64 20 74 68 65 20 | 6c 6f 63 61 74 69 6f 6e |and the |location|
|00003920| 20 6f 66 20 74 68 65 0a | 09 70 72 6f 6a 65 63 74 | of the.|.project|
|00003930| 65 64 20 70 6f 69 6e 74 | 09 20 69 73 20 75 73 65 |ed point|. is use|
|00003940| 64 20 74 6f 20 64 65 74 | 65 72 6d 69 6e 65 20 74 |d to det|ermine t|
|00003950| 68 65 20 74 65 78 74 75 | 72 69 6e 67 20 63 6f 6f |he textu|ring coo|
|00003960| 72 64 69 6e 61 74 65 73 | 0a 09 69 6e 20 61 20 6d |rdinates|..in a m|
|00003970| 61 6e 6e 65 72 20 69 64 | 65 6e 74 69 63 61 6c 20 |anner id|entical |
|00003980| 74 6f 20 74 68 61 74 20 | 75 73 65 64 20 69 6e 20 |to that |used in |
|00003990| 74 68 65 20 69 6e 76 65 | 72 73 65 20 6d 61 70 70 |the inve|rse mapp|
|000039a0| 69 6e 67 20 6d 65 74 68 | 6f 64 0a 09 66 6f 72 20 |ing meth|od..for |
|000039b0| 74 68 65 20 73 70 68 65 | 72 65 20 70 72 69 6d 69 |the sphe|re primi|
|000039c0| 74 69 76 65 2e 0a 09 49 | 66 20 67 69 76 65 6e 2c |tive...I|f given,|
|000039d0| 20 74 68 65 20 63 65 6e | 74 65 72 20 6f 66 0a 09 | the cen|ter of..|
|000039e0| 74 68 65 20 70 72 6f 6a | 65 63 74 69 6f 6e 20 69 |the proj|ection i|
|000039f0| 73 20 5c 65 76 65 63 7b | 6f 72 69 67 69 6e 7d 2c |s \evec{|origin},|
|00003a00| 20 5c 65 76 65 63 7b 76 | 61 78 69 73 7d 20 64 65 | \evec{v|axis} de|
|00003a10| 66 69 6e 65 73 0a 09 74 | 68 65 20 73 70 68 65 72 |fines..t|he spher|
|00003a20| 65 20 61 78 69 73 2c 20 | 61 6e 64 20 74 68 65 20 |e axis, |and the |
|00003a30| 70 6f 69 6e 74 20 77 68 | 65 72 65 20 74 68 65 0a |point wh|ere the.|
|00003a40| 09 6e 6f 6e 2d 70 61 72 | 61 6c 6c 65 6c 20 5c 65 |.non-par|allel \e|
|00003a50| 76 65 63 7b 75 61 78 69 | 73 7d 20 69 6e 74 65 72 |vec{uaxi|s} inter|
|00003a60| 73 65 63 74 73 20 74 68 | 65 20 73 70 68 65 72 65 |sects th|e sphere|
|00003a70| 0a 09 64 65 66 69 6e 65 | 73 20 77 68 65 72 65 20 |..define|s where |
|00003a80| 24 75 3d 30 24 20 69 73 | 20 6c 6f 63 61 74 65 64 |$u=0$ is| located|
|00003a90| 2e 0a 5c 65 6e 64 7b 64 | 65 66 6b 65 79 7d 0a 42 |..\end{d|efkey}.B|
|00003aa0| 79 20 64 65 66 61 75 6c | 74 2c 20 61 20 73 70 68 |y defaul|t, a sph|
|00003ab0| 65 72 69 63 61 6c 20 6d | 61 70 70 69 6e 67 20 70 |erical m|apping p|
|00003ac0| 72 6f 6a 65 63 74 73 20 | 70 6f 69 6e 74 73 20 74 |rojects |points t|
|00003ad0| 6f 77 61 72 64 73 20 74 | 68 65 20 6f 72 69 67 69 |owards t|he origi|
|00003ae0| 6e 2c 0a 77 69 74 68 20 | 5c 65 76 65 63 7b 76 61 |n,.with |\evec{va|
|00003af0| 78 69 73 7d 20 64 65 66 | 69 6e 65 64 20 74 6f 20 |xis} def|ined to |
|00003b00| 62 65 20 74 68 65 20 24 | 7a 24 20 61 78 69 73 20 |be the $|z$ axis |
|00003b10| 61 6e 64 0a 5c 65 76 65 | 63 7b 75 61 78 69 73 7d |and.\eve|c{uaxis}|
|00003b20| 20 64 65 66 69 6e 65 64 | 20 74 6f 20 62 65 20 74 | defined| to be t|
|00003b30| 68 65 20 24 78 24 20 61 | 78 69 73 2e 0a          |he $x$ a|xis..   |
+--------+-------------------------+-------------------------+--------+--------+