Complex Domain Coloring

1 - Output Types 1.1 - domain Generic output for domain colourings. - Size/quality can be changed with bmp width and height. - Alternative generator methods: generategif, generatemapping, and generate inverse are also under this. 1.2 - graphing Graphs the function in 2D. - Red = real part, Blue = imaginary part. - start locations and end location define a line in the complex domain. - maximum parameter: - Positive → sets the maximum height of the graph. - 0 → automatically sets maximum. - Negative → directly sets the max output range (positive values take maximum value, and maximum’s minimum). 1.2.1 - polar graph Checkbox at the right makes it a polar graph. - Moving along a contour: imaginary parts set y-axis, real part sets real axis. - To make parametric: use f(x)+i*g(x). 1.3 - numerical Input the number for substituting x → get result. 1.4 - heightmap - Use capslock ON for movement: - WASDZX → move (Z/X = up/down, W/S = forward/backward). - QE, CV, RF → rotate. - Click reset camera to reset movement. 1.4.0.1 T (capital) changes heightmap from magnitude → real part → imaginary part (cycles every 3). 1.4.1 maximum height parameter: - Set to 0 → height always = 1. 1.4.2 download stl → downloads mesh as STL. ⚠ Mesh might be large or have NaN values, loading STLs may be slow. 1.4.3 download objects → downloads as .gltf (Three.js format). 1.4.4 - create gif Creates a GIF rotating around the mesh. Options: 1. default - circular path looking down (sqrt2). 2. close - smaller orbit (1). 3. ortho - camera orbits flat plane (z=0), no tilt. 4. zero - camera at origin, spins in place. 5. zeroup - camera at origin, spins while angled up. 6. topdown - rotates around Z-axis, looking down. 7. flyover - camera moves in a straight line past origin. 8. spherical - 45° up view, circle path (z=0, r=0.9). 1.4.5 Buttons at bottom left change the mesh directly (not the camera) — useful for GIFs. 2 - Image Generation 2.0 Variables: - x, z can be used. - q → nome inputs. - t → period ratio τ. - modulartransformqt = div(log(q), I, pi()) - modulartransformtq(t) = exp(mul(pi(), I, t)) - Note: t cannot be used when making GIFs. 2.01 Shortcuts for autocomplete: _: All functions, #: Total functions count, :: Random functions, ?: Show help for all prefixes, L: Long functions (>100 characters), LL: Longer functions (>200 characters), LLL: Sort by length, S: Single-parameter functions, D: Double-parameter functions 2.1 - generate Generates domain coloring on the domain canvas. 2.2 - generate inverse Colors as initial point and draws at output complex number (inverse). - Faster than newtoninv('sin(x)', x) but you cannot get values. 2.3 - generate mapping For conformal maps (e.g., Schwarz triangle function). - Similar to generate inverse, but only draws gridlines. 2.4 - generate gifs Uses variable t moving from 0 to 1. - Suggestion: use sin(pi*t) for easing. - Cannot use t for period ratio when doing this. 2.5 - mesh generation Under toggle mesh: - Heightmap generated from magnitude of output. - Use maximum height in heightmap area to limit height. 2.5.1 - create mesh Creates heightmap. 2.5.2 - create spherical mesh Uses spherical coordinates (θ, φ). - F(θ, φ) = f(θ + φ*i). - Should have periods 2π in both real and imaginary axis. 2.5.3 - create stereographic mesh For functions such as stereopcube (stereographic projection). 2.5.4 - create toroidal mesh - Function should have period 1 in both imaginary and real axes. - For functions like cm, use unwarp(x, w1, w2) where w1, w2 are periods. - Example: cm(unwarp(5*x, 5.29991625, exp(2*pi*i/3)*5.29991625)). 2.5.5 - cylindric mesh - Function should be periodic along the real axis with period 1. 3 - Color Modes 3.1 - contour Draws lines where f(x) components get close to an integer. 3.1.2 - pcontour Polar contour (uses arg & mag instead of re & im). 3.1.3 - acountour Antialias contour (slow, minimal advantages). 3.2 - HSV Value from magnitude-to-lightness function plus other methods below. 3.3 - HSL 3.4 - HCL 3.5 - LCH Cylindrical version of CIELAB color space. 3.6 - YCbCr Real part → blue, Imag part → red. 3.7 - YDbDr Same as YCbCr. 3.8 - YCoCg Real part → orange, Imag part → green. 3.9 - YPbPr Blue and red again. 3.10 - real Colors based on real part of output. 3.11 - image Colors based on imaginary part. 3.12 - 2d Technical; used to save function for use in other software. 3.13 - palette User can choose palette by loading 360×100 BMP or selecting colors. 3.14 - freeRGB [r, g, b] from magnitude-to-lightness. Example: [mag(x), sqr(x), log(1+re(x)^2)] 3.15 - freeHSV [h, s, v]. Example: [360+arg(z)/pi*180, 100, log(x+1)*100] 3.16 - freeHSL [h, s, l]. Example: [360+arg(z)/pi*180, 100, (x+1)/x*100] 3.17 - isolines Draws lines of equal magnitude. 3.18 - dynamic Changes with time (good for GIFs). 3.19 - magnitudeold (Magnitude 2) Colors by magnitude. 3.20 - magnitude (Magnitude) Colors by magnitude. 3.21 - magnitudecolour (Magnitude 3) Colors by magnitude. 3.22 - magnitude4 (Magnitude 4) Colors by magnitude. 3.23 - neon Brighter where f(x) components get close to an integer. 3.24 - differ Based on difference between means of real and imaginary parts. 3.25 - purify "Cool" effect. 3.26 - quadrant Colors depending only on angle — colors quadrants. 3.27 - upperplanebw Imag > 0 → white, else black. 3.28 - unitcirclebw Inside unit circle → white, else black. 3.29 - trisect Similar to quadrant but 3 colors. 3.30 - hexasect Similar to quadrant but 6 colors. 3.31 - octant Similar to quadrant but 8 colors. 3.32 - dodecant Similar to quadrant but 12 colors. 3.33 - griddistort Checkerboard (smooth). 3.34 - griddistorts Checkerboard (unsmooth). 3.35 - griddistortbw Checkerboard (black and white). 3.36 - realimagheat Colors based on imag^2/real and real^2/imag.