To help you get the most out of Knot, this document reveals tricks for creating complicated, unobvious effects with the program, illustrated with example documents. There’s also a look inside the program that reveals how it works, and ideas on how you can plan your drawings to make the most of the program.
How it Works
Knot draws its pictures using a ‘Z-buffer algorithm’; like the X & Y axes for width and height, Z is the name of the depth axis. Knot keeps track of how far away from the front every pixel is, and only draws new pixels when they would be closer to the front than what’s already there.
For opaque strands, the program figures out the depth and color of every point on the near surface of a single sphere. This is the same sphere that you see in the substance preview in the main editing window. The color has to be calculated once for each active light source, so that’s why it’s faster to use just one light.
During drawing, this sphere is swept all along the path of the strand and is drawn over and over at every point along the way. Because the individual spheres are so close together, they merge into a single strand.
For glowing strands, a different method is used. The program actually draws all the opaque ones first, and then adds a glowing circle of color at every point along a strand’s path. After enough dim circles are drawn on top of each other, the light adds up into a bright glow. The glowing circles are drawn only where they would be in front of the opaque strands, so they are occluded in the right places.
For ribbons, the program splits the space between two strands into a stream of triangles, and uses a simple polygon-drawing algorithm to paint that region. Every second triangle has an edge on the first ribbon, and the rest have an edge on the other ribbon. Consequently, the two sets of triangles may have totally different orientations. Drawn normally, the stream of triangles would look badly wrinkled. To improve the appearance of smooth ribbons, a moving average of the triangles' surface normals is used for to calculate shading.
For each pixel, there are 24 bits of color, 16 bits of depth, and 8 bits of binary mask and geometry flags used by the gobo lights and the ribbon-drawing engine, which you should see in version 3.6. The depth is multipled by a factor of 10 in the Z-direction in order to make the sufaces smoother.
The color model lacks gamma correction, but every attempt is made to avoid the single-hued plastic look that poisons computer graphics. Knot uses a simplified Cook illumination model, wherein the green components of reflected light are angularly compressed toward the highlight, and the blue component is compressed even more. The roughness and metallic sliders interact to set the amount of angular compression.The result is a variation of hue across the surface. The 'wet' property just dims the diffuse reflection and adds a highlight of the same color as the illmination. Iridescence rotates the hue of the color in HSL colorspace by an amount dependent on the angle between the surface normal and the illumination vector.
For normal lights, the program just uses the simple drawing technique described below, but for the gobo light, the 4 channels of color info travel with a fourth channel of angle info, which describes the angle between the surface normal and the illumination vector. After each opaque strand is drawn, the picture is scanned to find 'gobo pixels', and the angle data is combined with a stored illumination ramp and a function that provides the light intensity at any given x,y,z location; a color is calculated, it's stored, and the pixel is marked as 'completed'.
When you render an animation, you may notice that the progress box says it’s drawing more strands per frame that you think ought to be there. That’s because when the program blends two very different strands, it may need to do it by using two strands rather than a single intermediate one. For example, if you blend an opaque strand into a glowing one, the program will draw a shrinking opaque strand on top of a growing glowing one.
For morphing, the program ordinarily interpolates strand parameters in spherical coordinates, which can produce wildly flailing results at times. To avoid this, cartesian interpolation is available as an option, but is slower to render (click on a strand wtih option+command). Blending can be turned off altogether with the discontinuous interpolation mark (click with control+option).
Spheres
If you want to draw one great big sphere, make a cage with many closely spaced swoops, and make the diameter of the strand so large that the strands merge together into a single uninterrupted surface. Adjust phi so that only the top half of the cage draws, and then tip the pole down to face you. What this does is give you a hemispherical shell facing you, and saves time by not drawing the far side of the shell which you can’t see anyway. An example of this is the ocean in ‘ugly planet’.
The continents in ‘ugly planet’ are also spherical cages that have been collapsed even further into tiny caps, and made irregular by buckling. The atmosphere is a glowing half-cage that’s been broken into pearls in order to avoid a streaked appearance.
Haloes
Simplicity itself. Just make an opaque strand, copy & paste it, and make the new one larger, darker, and glowing. If your first strand was buckled, it’s important to decide upon the exact form of buckling (using option-click on the preview) before you clone the strand. That’s because there is no other way to say "I want this strand to be buckled in the same way as this other strand."
To color a halo in a sophisticated manner, layer several strands atop each other. Thin, bright ones atop thick, dim ones, all with the iridescence turned up high, produce a complex result. If you want to use just one strand, use a slightly desaturated color. That way, the brightest core of the strand will tend realistically toward white, as if the knot had been photographed on slightly overexposed film. Be sure to use very dim colors. Example: ‘layered haloes.’
Glowing strands that don’t get lost in 3-D
Glowing strands don’t look so wonderful in the stereograms as you might like because they have no features along their length for your eyes to fix on as stereo reference points. This effect can be fixed: put an opaque black strand inside your glowing strand to add some definition and helps fix it in space in a 3-D picture. ‘Hellflower’ shows this effect.
3-D Knots and Perspective
The pictures drawn by Knot have no perspective; you might wonder, then, how it manages to draw 3-D stereograms. The program settles for drawing two perspectiveless views separated by 4°, and the picture that results is enough like a real view of a real object placed far from your eyes to fool your vision. It’s fun to display a knot stereogram on your screen, put on 3-D glasses, and look at the screen from various distances away to see how the depth effect changes and fails depending on where you are.
Animation by phase change
Although you can make an animation by morphing one strand into another one with an unrelated shape, often you can get a good result by morphing from one strand to a copy of the same strand with a phase change in phi or theta. For a repeating animation that you plan to play as a loop, the phase change should be 2π, and you should not click the ‘loop’ option in the Animation window. For examples, check the documents ‘alien heart’ and ‘skewed braid’.
Explosions
One way to make explosions is to break up the knot into bright glowing pearls to make hot points of light. Buckle the strands to avoid too much regularity. Double up these strands: make large red glows superimposed on small white or bright yellow glows to create a fiery effect. In later keyframes, make the strands larger, the colors dimmer and redder, and distribute glowing points closely gathered around the south pole to create the impression of sparks falling as they burn out. Spinning the poles of the spark-strands can produce a whirling, vortical effect. An example: ‘electrosplosion.’
Segmented braids
Dividing braids into straight-line segments can make unusual strands. Open the document ‘segmented braids’ and take a look.
Keyframe 1, with green strands, has a plain braid with 26 loops, and then the same braid divided into 29, 28, 27, and 26 segments, each with a slightly different shape. In the last one, where the number of segments equals the number of loops, the braid becomes flat.
Keyframe 2 shows a flat braid with various phase offsets in phi, causing a conical effect.
Keyframe 3’s yellow strands have stronger phase offsets that collapse the braid into a straight line. The white strands are just like the yellow strands, but are curved.
Keyframe 4 has strands much like frames 2 and 3, but here there are 29 segments rather than 26, so the skewing effect looks different. Straight-line versions are at the top of the list, and equivalent curved versions below.
Unusual shapes:
There are a few effects possible only with straight-line strands. The document ‘vetruvian’ is a simple geometric shape.
‘Weird strands’ is just that: an assortment of unusually shaped strands to get ideas from.
To make a star, start with a circular hoop. Change theta to pi times some even number 4 or greater, and the number of segments to an odd number 5 or greater.
To animate a star, first make the shape more complicated so that the motion will be interesting. Next, do animation by phase change.
With ribbons, you can make polyhedra - look at ‘dodecahedron.’
To make complicated curves, try choosing for the w multipliers different numbers which have common factors (3, 9, 27) or which have no common factors at all (7, 13, 19). Experiment with the phase offsets
Design tips
The best way to make cool knots is to start with a clear idea of what you want it to look like. Use the Magic Knot window to make a few strands at a time, and then keep just the best ones, modifying them until each strand has a satisfying shape. After you’ve made several nice ones, choose a few that go together well and combine them in one knot.
You can plan the shape of whole knot this way: try to avoid a ‘spherical’ shape overall. Try to make a disk, or an ellipse, or a figure-8; anything but a sphere.
Try having groups of strands that work together as single element. For example, look at ‘wet wheel’ - there are three similar red strands, and two similar blue-grey strands.
Give a strand a neon halo. Take an opaque strand, clone it, make the clone glow and make it larger and darker. The result will be a solid strand with a halo, like in ‘energy’ or ‘disorganized tangle’.
Use different kinds of strands in a single knot - curved, straight, pearl-shaped, opaque, glowing, wet, dry, plain, iridescent, metallic, ribbon-shaped. Try the extreme color effects you can get when the iridescence, roughness, and metallic quality are at the maximum -check ‘too colorful’ for an example.
Use at least two lights all the time - one white one, and a strongly colored one off to the side. Keep the ambient light turned off.
If you want rim lighting, you may want to use two identical white lights so that the light is bright enough. Look at ‘energy’ for an example.
Always use a gobo with ribbons - otherwise the ribbons look too flat
For tiles, use soft blends as well as strands oriented diagonally to reduce the obvious ‘repeating squares’ effect that can ruin tiles. try also to keep the size of details small relative to the size of the tile.
You can get a chiaroscuro effect by turning all the lights to maximum harshness, and making them shine from the sides.
For the background, use either dark, vivid colors, or light, desaturated colors - not bright vivid or dark desaturated. Or try using grey backgrounds.
For strands, always use iridescence, and occasionally wetness. Color the strands so that they go together: pick two colors on opposite sides of the color wheel, or three equally spaced around the color wheel. Or, use several different saturation values of one hue. Make sure the background is not the same color as any strand.
Remember, even if you do all this, not every knot you make will really ‘work’. The best result happens when you have a strong design idea in your head before you start.
