1d:{"title":"Domain Coloring for Complex Functions","cells":[{"id":1320,"value":"md`# Domain Coloring for Complex Functions`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1312,"value":"viewof plot3 = createPlot()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1698,"value":"md`**Figure**: A plot of the function ${tex`f(z) = \\displaystyle \\frac{(z^2 - 1)(z - 2 - i)^2}{z^2 + 2 + 2i}`}, where the horizontal axis is ${tex`\\mathrm{Re}(z)`} and the vertical axis is ${tex`\\mathrm{Im}(z)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1736,"value":"drawPlot3(domainColoringParameters)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1422,"value":"drawPlot3 = plot3.createDrawFunction(`vec2 f (vec2 z) {\n return cdiv(cmul(csqr(z) - vec2(1, 0), csqr(z - vec2(2, 1))), csqr(z) + vec2(2, 2));\n}`)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1148,"value":"$1f","pinned":false,"hidden":false,"mode":"ojs"},{"id":1219,"value":"viewof plot1 = createPlot()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1720,"value":"drawPlot1 = plot1.createDrawFunction(`\n vec2 f (vec2 z) {\n return z;\n }\n`)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2065,"value":"drawPlot1(domainColoringParameters)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1549,"value":"md`\nThere are a few things to note about this plot. The dark grid lines remain a uniformly spaced rectangular grid since ${tex `f(z) = z`} is the identity function. These are just inputs ${tex`z`} with integer real or imaginary parts.\n\nThe light grid lines represent the modulus and argument of the input ${tex `z`}. The eight radial lines representing the argument are spaced in equal ${tex `45^\\circ`} increments, while the modulus is displayed as concentric rings (equally spaced in the logarithmic sense).\n\nThe dark region in the center represents the zero, ${tex `f(0) = 0`}.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1238,"value":"md`\n## Visualizing poles\n\nNext, we'll plot the inverse, ${tex `f(z) = 1 / z`}. In fact, let's cut to the chase and plot ${tex `f(z) = z^n`} and start with ${tex `n = -1`} so that we can draw some additional conclusions.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1261,"value":"viewof plot2 = createPlot()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1749,"value":"tex`f(z) = z^{${n}}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1752,"value":"viewof n = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1741,"value":"drawPlot2 = plot2.createDrawFunction(`\n uniform float n;\n vec2 f (vec2 z) {\n return cpow(z, n);\n }\n`)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1744,"value":"drawPlot2(Object.assign({n}, domainColoringParameters))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1482,"value":"$20","pinned":false,"hidden":false,"mode":"ojs"},{"id":1967,"value":"md`\n## Putting the pieces together\n\nFinally, we'll plot a more complicated function and animate it. Feel free to modify the function. The GLSL code at the bottom defines a number of functions (\\`cmul\\`, \\`cdiv\\`, \\`cinv\\`, \\`cpow\\`, \\`csqrt\\`, \\`clog\\`), trig functions (\\`ccos\\`, \\`csin\\`, \\`ctan\\`, \\`cacos\\`, \\`casin\\`, \\`catan\\`), and hyperbolic trig functions (\\`csinh\\`, \\`ccosh\\`, \\`ctanh\\`). They're fun and easy to implement, but please note that they're not very robust if you're into floating point rigor.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1349,"value":"viewof plot4 = createPlot()","pinned":false,"hidden":false,"mode":"ojs"},{"id":1894,"value":"md`**Figure:** Can you locate the poles? The zeros? The zero with multiplicity 2?`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1861,"value":"viewof button = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1870,"value":"paused = {\n button;\n return !this;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1352,"value":"drawPlot4 = plot4.createDrawFunction(`\n uniform float t;\n vec2 f (vec2 z) {\n vec2 fz = C_ONE;\n vec2 a0 = cexp(vec2(0.375, -0.125 * t));\n fz = cmul(fz, z - cexp(vec2(0.0, t)));\n fz = cdiv(fz, z - cexp(vec2(0.5, 0.5 * t)));\n fz = cmul(fz, z - cexp(vec2(0.125, -0.625 * t)));\n fz = cdiv(fz, z - cexp(vec2(0.25, -0.25 * t)));\n fz = cmul(fz, csqr(z - a0));\n return fz;\n }\n`)","pinned":true,"hidden":false,"mode":"ojs"},{"id":1939,"value":"drawPlot4(Object.assign({t: 6}, domainColoringParameters))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1433,"value":"!paused && drawPlot4(Object.assign({t: (now / 1000) % 1000}, domainColoringParameters))","pinned":true,"hidden":false,"mode":"ojs"},{"id":1978,"value":"md`\n## Customizing the coloring\n\nBy dragging the sliders below, you can customize the details of the domain coloring. It's a bit difficult to make all the plots on the page look nice at the same time.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1515,"value":"viewof rectangularGridOpacity = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1518,"value":"viewof polarGridOpacity = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1570,"value":"viewof rootDarkening = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1573,"value":"viewof rootDarkeningSharpness = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1576,"value":"viewof poleLightening = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1579,"value":"viewof poleLighteningSharpness = html``","pinned":false,"hidden":false,"mode":"ojs"},{"id":1997,"value":"viewof plot5 = createPlot()","pinned":false,"hidden":false,"mode":"ojs"},{"id":2013,"value":"md`**Figure**: A plot of ${tex`f(z) = \\sin(1/z)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2000,"value":"drawPlot5 = plot5.createDrawFunction(`\n vec2 f (vec2 z) {\n z /= 16.0;\n return csin(cinv(z));\n }\n`)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2003,"value":"drawPlot5(domainColoringParameters)","pinned":true,"hidden":false,"mode":"ojs"},{"id":2028,"value":"$21","pinned":false,"hidden":false,"mode":"ojs"},{"id":2010,"value":"md`\n## The implementation details\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1490,"value":"domainColoringParameters = ({\n rootDarkening,\n poleLightening,\n rectangularGridOpacity,\n polarGridOpacity,\n rootDarkeningSharpness,\n poleLighteningSharpness\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":710,"value":"$22","pinned":false,"hidden":false,"mode":"ojs"},{"id":197,"value":"wrapREGL = require('regl')","pinned":false,"hidden":false,"mode":"ojs"},{"id":1112,"value":"$23","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W24","stdlib":"1"}