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+๏ปฟ<?xml version="1.0" encoding="utf-8" standalone="yes"?>
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+<Fractal BackgroundVisible="True" BackColor="FFFFFF00" BorderWidth="0" StripBorder="True" PixelSize="0.00548779710473845" Pixels.X="729" Pixels.Y="729" ClassicProcessing="None" OrbitTrapProcessing="ActivateControllers" ClassicMaxDwell="1" CycleDetectionActive="False" SolidGuessing="None">
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+ <FractalNotes><![CDATA[]]></FractalNotes>
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+ <FractalEquation Id="adb85431-618d-491f-b8c5-08555b119814" Title="Pixel" MaxPower="1">
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+ <Instructions><![CDATA[
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+comment:
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+
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+ This equation can be used when no equation is necessary.
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+ By setting the MinDwell/MaxDwell to 1 below, we limit
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+ the iteration to a single step for best performance.
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+
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+global:
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+
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+ FSK.OverrideValue("MinDwell", 1)
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+ FSK.OverrideValue("MaxDwell", 1)
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+
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+iterate:
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+
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+ z = c
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+]]></Instructions>
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+ <PropertyValueOverrides />
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+ </FractalEquation>
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+ <TransformationArray>
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+ <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
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+ <Instructions><![CDATA[]]></Instructions>
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+ <PropertyValueOverrides />
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+ </Transformation>
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+ </TransformationArray>
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+ <TransformationArray>
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+ <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
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+ <Instructions><![CDATA[]]></Instructions>
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+ <PropertyValueOverrides />
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+ </Transformation>
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+ </TransformationArray>
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+ <AlternateValueInfo />
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+ <AlternateValueInfo />
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+ <OrbitTrapInfo TrapValueNTrapPreProcessing="CenterReflect" TrapValueNTrapStyle="Flat" TrapValueNSmoothing="Linear" TrapValueNTrapPreProcessing1="CenterReflect" TrapValueNTrapStyle1="Flat" TrapValueNSmoothing1="Linear" TrapValueNTrapPreProcessing2="CenterReflect" TrapValueNTrapStyle2="Flat" TrapValueNSmoothing2="Linear" TrapValueNTrapPreProcessing3="CenterReflect" TrapValueNTrapStyle3="Flat" TrapValueNSmoothing3="Linear">
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+ <OrbitTrapArray TrapIndexMapping="CombinedIndex" TrapDeltaMapping="TrapDelta" BlendValues="False">
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+ <OrbitTransform />
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+ <OrbitTransform />
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+ <TransformationArray>
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+ <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
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+ <Instructions><![CDATA[]]></Instructions>
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+ <PropertyValueOverrides />
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+ </Transformation>
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+ </TransformationArray>
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+ <TransformationArray>
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+ <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
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+ <Instructions><![CDATA[]]></Instructions>
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+ <PropertyValueOverrides />
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+ </Transformation>
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+ </TransformationArray>
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+ <SymmetryTransformation Id="8301dc6c-2273-4fb9-ada8-2ded2833031f" Title="Identity">
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+ <Instructions><![CDATA[]]></Instructions>
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+ <PropertyValueOverrides />
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+ </SymmetryTransformation>
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+ <OrbitTrapItem>
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+ <OrbitTrap Id="4e4b2e73-dc52-4107-bb98-f9abb8745748" Title="Unit Circle Group">
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+ <Instructions><![CDATA[
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+comment:
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+
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+ trappedPoint.Index is the base circle index: 0 (center), 1-N (in ring)
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+ trappedPoint.Delta is the level: 0 - Steps-1
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+
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+ See the paper:
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+ "Evolution of Math into Art via Mobius Transformations"
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+ by Anne M. Burns, Department of Mathematics,
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+ Long Island University.
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+ http://myweb.cwpost.liu.edu/aburns/
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+
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+ Also, see pages 88-89 in the book:
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+ "Indra's Pearls, The Vision of Felix Klein"
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+ by David Mumford, Caroline Series, David Wright.
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+ http://klein.math.okstate.edu/IndrasPearls/
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+
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+global:
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+
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+ Complex ShowCenter[] = LC1,LC2,LC3,LC4,LC5,LC6,LC7,LC8,LC9,LC10,LC11,LC12,LC13,LC14,LC15,LC16
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+ Complex ShowRing[] = LR1,LR2,LR3,LR4,LR5,LR6,LR7,LR8,LR9,LR10,LR11,LR12,LR13,LR14,LR15,LR16
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+ AbsV = Sqrt(AbsU^2 - 1)
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+ u = AbsU * Cis(DegreeToRadian(ArgU))
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+ v = AbsV * Cis(DegreeToRadian(ArgV))
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+ Mobius UnitCircleGroup = Mobius(u, v, Conj(v), Conj(u))
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+ Mobius m[N+1]
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+ '
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+ ' Given N, find radius R such that N circles with radius R can be
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+ ' placed along the inside of the unit circle, each tangent to the
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+ ' unit circle and each of its two adjacent neighbors. The centers
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+ ' of all circles are a distance of 1-R from the origin. The centers
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+ ' of all circles form a regular polygon P with N sides and each edge
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+ ' has length = 2*(1-R)*Sin(pi/N). Since all N circles are pairwise
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+ ' tangent, the edge of P must equal 2*R, so 2*R = 2*(1-R)*Sin(pi/N).
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+ ' Solving for R yields R = 1/(1+1/Sin(pi/N)).
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+ '
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+ r = 1/(1+1/Sin(Math.PI/N))
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+
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+ step = 2*Math.PI/N
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+ ang = IIf(Shift, step/2, 0)
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+ '
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+ ' Generate the Mobius transformation that transforms the
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+ ' unit circle into the circle in the center of the ring.
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+ '
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+ m[0] = Mobius(1-2*r, 0, 0, 1)
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+ '
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+ ' Generate the set of N Mobius transformations the
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+ ' transform the unit circle into the i'th circle in the
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+ ' ring, where i = 1 to N.
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+ '
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+ for (i = 1, i <= N, i += 1) {
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+ rotate = Cis(ang)
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+
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+ m[i] = Mobius.Multiply( \
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+ Mobius(r*rotate, (1-r)*rotate, 0, 1), \
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+ UnitCircleGroup \
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+ )
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+ ang += step
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+ }
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+ '
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+ ' Assign Total = the total number of circles.
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+ '
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+ const Complex Total = 0
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+ count = N+1
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+
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+ for (i = 0, i < Steps, i += 1) {
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+ Total += count
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+ count *= N+1
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+ }
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+ const Circle c[Total]
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+ const Complex index[Total]
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+ const Complex level[Total]
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+ Circle UnitCircle = CircleC(0, 1)
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+ '
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+ ' Generate the base ring and center circles.
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+ '
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+ for (i = 0, i <= N, i += 1) {
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+ c[i] = Mobius.TransformCircle(m[i], UnitCircle)
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+ index[i] = i
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+ level[i] = 0
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+ }
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+ count = N+1
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+ max = 0
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+ '
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+ ' Generate the remaining circles.
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+ '
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+ if (Steps > 1) {
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+ for (i = 1, i < Steps, i += 1) {
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+ min = max
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+ max = count
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+
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+ for (j = min, j < max, j += 1) {
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+ for (k = 0, k < N+1, k += 1) {
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+ c[count] = Mobius.TransformCircle(m[k], c[j])
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+ '
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+ ' Skip circles that are too small.
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+ '
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+ if (c[count].Radius >= RadiusMin) {
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+ index[count] = index[j]
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+ level[count] = i
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+ count += 1
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+ }
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+ }
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+ }
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+ }
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+ }
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+ '
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+ ' Reset Total in case you skipped circles that were too small.
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+ '
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+ Total = count
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+
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+ CurveTrap.Initialize( \
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+ Center, DegreeToRadian(Angle), Scale, AlternateAngle, 6, False, LineWidth \
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+ )
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+ '
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+ ' Add the circles to the trap based on user criteria.
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+ '
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+ for (i = 0, i < Total, i += 1) {
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+ lev = level[i]
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+ idx = index[i]
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+ '
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+ ' Note: idx=N for center circles.
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+ '
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+ if (idx = 0) {
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+ if (ShowCenter[lev]) {
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+ CurveTrap.AddCircle2(c[i], Solid, IIf(Solid, lev, 0), idx, lev)
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+ }
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+ } else {
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+ if (ShowRing[lev]) {
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+ CurveTrap.AddCircle2(c[i], Solid, IIf(Solid, lev, 0), idx, lev)
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+ }
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+ }
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+ }
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+
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+trap:
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+
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+ trappedPoint = CurveTrap.Apply(z)
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+
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+properties:
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+
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+ divider {
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+ caption = "General Options"
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+ }
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+ option Center {
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+ type = Complex
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+ caption = "Center"
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+ details = "Center of trap"
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+ default = 0
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+ }
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+ option Angle {
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+ type = Float
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+ caption = "Angle"
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+ details = "Angle of rotation"
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+ default = 0
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+ range = [-360,360]
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+ }
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+ option Scale {
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+ type = Float
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+ caption = "Scale"
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+ details = "Scale factor applied to trap"
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+ range = (0,)
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+ default = 2
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+ }
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+ option Solid {
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+ type = Boolean
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+ caption = "Solid"
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+ details = "Check to create solid trap"
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+ default = False
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+ }
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+ option AlternateAngle {
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+ type = Boolean
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+ caption = "Alternate Angle"
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+ details = "Use alternate angle calculation"
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+ default = False
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+ }
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+ option LineWidth {
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+ type = Float
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+ caption = "Line Width"
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+ details = "Extent of trap on either side of curve (> 0)"
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+ range = (0,)
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+ default = 0.00411522633744855967078189300413
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+ enabled = ~Solid
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+ }
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+ divider {
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+ caption = "U/V Controls"
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+ }
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+ option AbsU {
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+ type = Float
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+ caption = "Abs(U)"
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+ details = "Magnitude of U (1-2)"
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+ default = 1.1
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+ range = [1,2]
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+ }
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+ option ArgU {
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+ type = Float
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+ caption = "Arg(U)"
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+ details = "Angle of U"
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+ default = 0
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+ range = [-360,360]
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+ }
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+ option ArgV {
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+ type = Float
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+ caption = "Arg(V)"
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+ details = "Angle of V"
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+ default = 180
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+ range = [-360,360]
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+ }
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+ divider {
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+ caption = "Circle Controls"
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+ }
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+ option N {
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+ type = IntegerEnum(3,12)
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+ caption = "N"
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+ details = "Number of base circles"
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+ default = 4
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+ }
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+ option Steps {
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+ type = IntegerEnum(1,16)
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+ caption = "Steps"
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+ details = "Number of inversion steps"
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+ default = 8
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+ }
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+ option Shift {
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+ type = Boolean
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+ caption = "Shift"
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+ details = "Check to rotate initial chain by pi/N"
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+ default = False
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+ }
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+ option RadiusMin {
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+ type = Float
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+ caption = "Radius Min"
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+ details = "Minimum acceptable circle radius"
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+ default = 0
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+ range = [0,)
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+ }
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+ #define ShowLevel(Index)
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+
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+ divider {
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+ caption = "Level #Index# Options"
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+ }
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+ option LR#Index# {
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+ type = Boolean
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+ caption = "Show Ring"
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+ details = "Show ring of circles at level #Index#"
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+ default = True
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+ enabled = Steps >= #Index#
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+ }
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+ option LC#Index# {
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+ type = Boolean
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+ caption = "Show Center"
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+ details = "Show center circle at level #Index#"
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+ default = True
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+ enabled = Steps >= #Index#
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+ }
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+ #end
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+
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+ #include ShowLevel("1")
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+ #include ShowLevel("2")
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+ #include ShowLevel("3")
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+ #include ShowLevel("4")
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+ #include ShowLevel("5")
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+ #include ShowLevel("6")
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+ #include ShowLevel("7")
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+ #include ShowLevel("8")
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+ #include ShowLevel("9")
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+ #include ShowLevel("10")
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+ #include ShowLevel("11")
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+ #include ShowLevel("12")
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+ #include ShowLevel("13")
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+ #include ShowLevel("14")
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+ #include ShowLevel("15")
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+ #include ShowLevel("16")]]></Instructions>
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+ <PropertyValueOverrides>
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+ <Option Name="Solid" Type="BooleanOption" Value="True" />
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+ <Option Name="LineWidth" Type="FloatOption" Value="0.00411522633744855" />
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+ <Option Name="ArgU" Type="FloatOption" Value="180" />
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+ <Option Name="ArgV" Type="FloatOption" Value="0" />
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+ <Option Name="LR1" Type="BooleanOption" Value="False" />
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+ <Option Name="LR2" Type="BooleanOption" Value="False" />
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+ <Option Name="LR3" Type="BooleanOption" Value="False" />
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+ <Option Name="LR4" Type="BooleanOption" Value="False" />
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+ <Option Name="LR5" Type="BooleanOption" Value="False" />
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+ <Option Name="LR6" Type="BooleanOption" Value="False" />
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+ <Option Name="LR7" Type="BooleanOption" Value="False" />
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+ <Option Name="LR8" Type="BooleanOption" Value="False" />
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+ <Option Name="LR9" Type="BooleanOption" Value="False" />
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+ <Option Name="LR10" Type="BooleanOption" Value="False" />
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+ <Option Name="LR11" Type="BooleanOption" Value="False" />
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+ <Option Name="LR12" Type="BooleanOption" Value="False" />
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+ <Option Name="LR13" Type="BooleanOption" Value="False" />
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349
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+ <Option Name="LR14" Type="BooleanOption" Value="False" />
|
|
350
|
+ <Option Name="LR15" Type="BooleanOption" Value="False" />
|
|
351
|
+ <Option Name="LR16" Type="BooleanOption" Value="False" />
|
|
352
|
+ </PropertyValueOverrides>
|
|
353
|
+ </OrbitTrap>
|
|
354
|
+ </OrbitTrapItem>
|
|
355
|
+ </OrbitTrapArray>
|
|
356
|
+ <OrbitTrapMasterController Id="02699986-4214-4098-b68b-48285874bc4e" Title="โ ">
|
|
357
|
+ <Instructions><![CDATA[
|
|
358
|
+comment:
|
|
359
|
+
|
|
360
|
+ Return color computed by 1st controller.
|
|
361
|
+
|
|
362
|
+color:
|
|
363
|
+
|
|
364
|
+ color = Controller.Color(0)
|
|
365
|
+]]></Instructions>
|
|
366
|
+ <PropertyValueOverrides />
|
|
367
|
+ </OrbitTrapMasterController>
|
|
368
|
+ <OrbitTrapControllerArray>
|
|
369
|
+ <OrbitTrapController Id="29fbd0db-b470-4d0d-ad93-e4d5d937ef86" Title="โตยทโต" ApplyDepth="False" Apply3D="False">
|
|
370
|
+ <GradientArray />
|
|
371
|
+ <TextureArray />
|
|
372
|
+ <Instructions><![CDATA[
|
|
373
|
+comment:
|
|
374
|
+
|
|
375
|
+ This controller colors each trap based on the
|
|
376
|
+ property 'Index Map' and the point's 'Trap Dwell'
|
|
377
|
+ 'Trap Index' or 'Trap Delta'. You can define up to
|
|
378
|
+ 16 colors and these are mapped to the index map
|
|
379
|
+ values. Offset can be used to shift the set of
|
|
380
|
+ colors with respect to value 0.
|
|
381
|
+
|
|
382
|
+ If Cutouts are active, the trap is divided into
|
|
383
|
+ N sectors and a user defined percent of each
|
|
384
|
+ sector is inverted with respect to the 3D shading
|
|
385
|
+ effect. This gives the appearance that part of
|
|
386
|
+ the trap surface has been cutout or folded over.
|
|
387
|
+
|
|
388
|
+ The Cutout effect only looks good if the trap
|
|
389
|
+ is a solid shape. For example, the 'Shape',
|
|
390
|
+ 'Tangent Circles', 'Shapes' and 'Squares'
|
|
391
|
+ traps support a 'Solid' property that is used to
|
|
392
|
+ produce a solid figure. You should check this
|
|
393
|
+ property to display Cutouts.
|
|
394
|
+
|
|
395
|
+ Also note that the Cutout effect only works if
|
|
396
|
+ the trap's angle is given relative to the center
|
|
397
|
+ of the solid shape. This is always the case when
|
|
398
|
+ the trap is composed of a single shape. However,
|
|
399
|
+ when the trap is composed of several shapes arranged
|
|
400
|
+ in a pattern like the 'Tangent Circles', 'Shapes' and
|
|
401
|
+ 'Squares' traps, the angle is given relative to the
|
|
402
|
+ center of the pattern not the individual shapes.
|
|
403
|
+ In those cases, you will need to check the
|
|
404
|
+ 'Alternate Angle' property on the properties page
|
|
405
|
+ for the specific trap so the angle is given relative
|
|
406
|
+ to the individual shapes.
|
|
407
|
+
|
|
408
|
+global:
|
|
409
|
+
|
|
410
|
+ const Complex angleShift = SectorAngle / 360
|
|
411
|
+ const Complex percentSolid = 1 - PercentCutout
|
|
412
|
+
|
|
413
|
+color:
|
|
414
|
+
|
|
415
|
+ switch (IndexMap) {
|
|
416
|
+ case IndexMapTypes.TrapDwell: index = Sample.TrapDwellRaw
|
|
417
|
+ case IndexMapTypes.TrapIndex: index = Sample.TrapIndexRaw
|
|
418
|
+ case IndexMapTypes.TrapDelta: index = Sample.TrapDeltaRaw
|
|
419
|
+ case IndexMapTypes.Alternate1Index: index = Sample.Alternate1IndexRaw
|
|
420
|
+ }
|
|
421
|
+ color = Colors[WrapIndex(Offset + index, Count)]
|
|
422
|
+
|
|
423
|
+ if (ApplyCutouts) {
|
|
424
|
+ r = Sectors * Wrap(Sample.TrapAngle - angleShift)
|
|
425
|
+ r = r - Int(r)
|
|
426
|
+ v = Abs(Sample.TrapValue)
|
|
427
|
+
|
|
428
|
+ compositeHeight = IIf(r < percentSolid, v, Blend(0.75, 0.25, v))
|
|
429
|
+ } else {
|
|
430
|
+ compositeHeight = Sample.TrapValue
|
|
431
|
+ }
|
|
432
|
+
|
|
433
|
+properties:
|
|
434
|
+
|
|
435
|
+ divider {
|
|
436
|
+ caption = "Index Map"
|
|
437
|
+ }
|
|
438
|
+ enum IndexMapTypes {
|
|
439
|
+ TrapDwell, "Trap Dwell"
|
|
440
|
+ TrapIndex, "Trap Index"
|
|
441
|
+ TrapDelta, "Trap Delta"
|
|
442
|
+ Alternate1Index , "Alternate 1 Index"
|
|
443
|
+ }
|
|
444
|
+ option IndexMap {
|
|
445
|
+ type = IndexMapTypes
|
|
446
|
+ caption = "Index Map"
|
|
447
|
+ details = "Color index map"
|
|
448
|
+ default = IndexMapTypes.TrapDwell
|
|
449
|
+ }
|
|
450
|
+ divider {
|
|
451
|
+ caption = "Color Map"
|
|
452
|
+ }
|
|
453
|
+ option Count {
|
|
454
|
+ type = IntegerEnum(1,32)
|
|
455
|
+ caption = "Count"
|
|
456
|
+ details = "Number of colors to use"
|
|
457
|
+ default = 8
|
|
458
|
+ }
|
|
459
|
+ option Colors {
|
|
460
|
+ type = ColorSet[Count]
|
|
461
|
+ caption = "Colors"
|
|
462
|
+ default = FF00AE, FF2020, FF8200, FFFF00, 00FF6C, 00FFFF, 4040FF, AE60FF
|
|
463
|
+ }
|
|
464
|
+ option Offset {
|
|
465
|
+ type = IntegerEnum(0,31)
|
|
466
|
+ caption = "Offset"
|
|
467
|
+ details = "Offset into set of colors"
|
|
468
|
+ default = 0
|
|
469
|
+ }
|
|
470
|
+ divider {
|
|
471
|
+ caption = "Cutouts (Solid traps only)"
|
|
472
|
+ }
|
|
473
|
+ option ApplyCutouts {
|
|
474
|
+ type = Boolean
|
|
475
|
+ caption = "Active"
|
|
476
|
+ details = "Activate cutout effects"
|
|
477
|
+ default = False
|
|
478
|
+ }
|
|
479
|
+ option Sectors {
|
|
480
|
+ type = IntegerEnum(2,16)
|
|
481
|
+ caption = "Sectors"
|
|
482
|
+ details = "Number of sectors"
|
|
483
|
+ default = 2
|
|
484
|
+ enabled = ApplyCutouts
|
|
485
|
+ }
|
|
486
|
+ option SectorAngle {
|
|
487
|
+ type = Float
|
|
488
|
+ caption = "Angle"
|
|
489
|
+ details = "Starting angle of 1st sector"
|
|
490
|
+ default = 0
|
|
491
|
+ range = [-360,360]
|
|
492
|
+ enabled = ApplyCutouts
|
|
493
|
+ }
|
|
494
|
+ option PercentCutout {
|
|
495
|
+ type = Float
|
|
496
|
+ caption = "Percent Cutout"
|
|
497
|
+ details = "Percent of sector that is cutout"
|
|
498
|
+ range = [0,1]
|
|
499
|
+ default = 0.5
|
|
500
|
+ enabled = ApplyCutouts
|
|
501
|
+ }
|
|
502
|
+]]></Instructions>
|
|
503
|
+ <PropertyValueOverrides>
|
|
504
|
+ <Option Name="IndexMap" Type="EnumOption" Value="IndexMapTypes.TrapDelta" />
|
|
505
|
+ <Option Name="Count" Type="IntegerEnumOption" Value="10" />
|
|
506
|
+ <Option Name="Colors" Type="ColorSetOption" Value="A9A9A9,B2B2B2,BABABA,C3C3C3,CBCBCB,D4D4D4,DCDCDC,E5E5E5,EDEDED,F6F6F6" />
|
|
507
|
+ </PropertyValueOverrides>
|
|
508
|
+ </OrbitTrapController>
|
|
509
|
+ </OrbitTrapControllerArray>
|
|
510
|
+ <SampleDataNormalizer>
|
|
511
|
+ <NormalizerDescriptor />
|
|
512
|
+ </SampleDataNormalizer>
|
|
513
|
+ <SampleDataNormalizer>
|
|
514
|
+ <NormalizerDescriptor />
|
|
515
|
+ </SampleDataNormalizer>
|
|
516
|
+ <SampleDataNormalizer>
|
|
517
|
+ <NormalizerDescriptor />
|
|
518
|
+ </SampleDataNormalizer>
|
|
519
|
+ </OrbitTrapInfo>
|
|
520
|
+ <TransformationArray>
|
|
521
|
+ <Transformation Id="7fc4bc2c-b961-4431-b074-277d7b29842b" Title="Identity">
|
|
522
|
+ <Instructions><![CDATA[]]></Instructions>
|
|
523
|
+ <PropertyValueOverrides />
|
|
524
|
+ </Transformation>
|
|
525
|
+ </TransformationArray>
|
|
526
|
+ <TriangleMetricSet>
|
|
527
|
+ <Instructions><![CDATA[metric = p1]]></Instructions>
|
|
528
|
+ <TriangleMetric />
|
|
529
|
+ <TriangleMetric />
|
|
530
|
+ <TriangleMetric />
|
|
531
|
+ </TriangleMetricSet>
|
|
532
|
+</Fractal>
|