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Added bezier_points new improved fast code. (Did not remove old code.)
This commit is contained in:
Adrian Mariano
parent
1988031e43
commit
52ef8089bb
1 changed files with 112 additions and 1 deletions
113
beziers.scad
113
beziers.scad
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@ -38,6 +38,117 @@ include <skin.scad>
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// Section: Segment Functions
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// Section: Segment Functions
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// Function: bezier_points()
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// Usage:
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// bezier_points(curve, u)
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// Description:
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// Computes bezier points for bezier with control points specified by `curve` at parameter values specified by `u`, which can be a scalar or a list.
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// This function uses an optimized method which is best when `u` is a long list and the bezier degree is 10 or less.
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// Ugly but speed optimized code for computing bezier curves using the matrix representation
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// See https://pomax.github.io/bezierinfo/#matrix for explanation.
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//
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// All of the loop unrolling makes and the use of the matrix lookup table make a big difference
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// in the speed of execution. For orders 10 and below this code is 10-20 times faster than
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// the recursive code using the de Casteljau method depending on the bezier order and the
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// number of points evaluated in one call (more points is faster). For orders 11 and above without the
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// lookup table or hard coded powers list the code is about twice as fast as the recursive method.
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// Note that everything I tried to simplify or tidy this code made is slower, sometimes a lot slower.
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function bezier_points(curve, u) =
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is_num(u) ? bezier_points(pts,[u])[0] :
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let(
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N = len(curve)-1,
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M = _bezier_matrix(N)*curve
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)
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N==0 ? [for(uval=u)[1]*M] :
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N==1 ? [for(uval=u)[1, uval]*M] :
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N==2 ? [for(uval=u)[1, uval, uval*uval]*M] :
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N==3 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval]*M] : // It appears that pow() is as fast or faster for powers 5 or above
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N==4 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval]*M] :
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N==5 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5)]*M] :
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N==6 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5),pow(uval,6)]*M] :
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N==7 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5),pow(uval,6), pow(uval,7)]*M] :
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N==8 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5),pow(uval,6), pow(uval,7), pow(uval,8)]*M] :
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N==9 ? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5),pow(uval,6), pow(uval,7), pow(uval,8), pow(uval,9)]*M] :
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N==10? [for(uval=u)[1, uval, uval*uval, uval*uval*uval, uval*uval*uval*uval, pow(uval,5),pow(uval,6), pow(uval,7), pow(uval,8), pow(uval,9), pow(uval,10)]*M] :
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/* N>=11 */ [for(uval=u)[for (i=[0:1:N]) pow(uval,i)]*M];
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function _signed_pascals_triangle(N,tri=[[-1]]) =
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len(tri)==N+1 ? tri :
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let(last=tri[len(tri)-1])
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_signed_pascals_triangle(N,concat(tri,[[-1, for(i=[0:1:len(tri)-2]) (i%2==1?-1:1)*(abs(last[i])+abs(last[i+1])),len(last)%2==0? -1:1]]));
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function _compute_bez_matrix(N) =
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let(tri = _signed_pascals_triangle(N))
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[for(i=[0:N]) concat(tri[N][i]*tri[i], repeat(0,N-i))];
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// The bezier matrix, which is related to Pascal's triangle, enables nonrecursive computation
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// of bezier points. This method is much faster than the recursive de Casteljau method
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// in OpenScad, but we have to precompute the matrices to reap the full benefit.
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_bezier_matrix_table = [
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[[1]],
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[[ 1, 0],
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[-1, 1]],
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[[1, 0, 0],
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[-2, 2, 0],
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[1, -2, 1]],
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[[ 1, 0, 0, 0],
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[-3, 3, 0, 0],
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[ 3,-6, 3, 0],
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[-1, 3,-3, 1]],
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[[ 1, 0, 0, 0, 0],
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[-4, 4, 0, 0, 0],
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[ 6,-12, 6, 0, 0],
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[-4, 12,-12, 4, 0],
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[ 1, -4, 6,-4, 1]],
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[[ 1, 0, 0, 0, 0, 0],
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[ -5, 5, 0, 0, 0, 0],
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[ 10,-20, 10, 0, 0, 0],
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[-10, 30,-30, 10, 0, 0],
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[ 5,-20, 30,-20, 5, 0],
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[ -1, 5,-10, 10,-5, 1]],
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[[ 1, 0, 0, 0, 0, 0, 0],
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[ -6, 6, 0, 0, 0, 0, 0],
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[ 15,-30, 15, 0, 0, 0, 0],
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[-20, 60,-60, 20, 0, 0, 0],
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[ 15,-60, 90,-60, 15, 0, 0],
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[ -6, 30,-60, 60,-30, 6, 0],
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[ 1, -6, 15,-20, 15,-6, 1]],
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[[ 1, 0, 0, 0, 0, 0, 0, 0],
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[ -7, 7, 0, 0, 0, 0, 0, 0],
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[ 21, -42, 21, 0, 0, 0, 0, 0],
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[-35, 105,-105, 35, 0, 0, 0, 0],
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[ 35,-140, 210,-140, 35, 0, 0, 0],
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[-21, 105,-210, 210,-105, 21, 0, 0],
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[ 7, -42, 105,-140, 105,-42, 7, 0],
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[ -1, 7, -21, 35, -35, 21,-7, 1]],
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[[ 1, 0, 0, 0, 0, 0, 0, 0, 0],
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[ -8, 8, 0, 0, 0, 0, 0, 0, 0],
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[ 28, -56, 28, 0, 0, 0, 0, 0, 0],
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[-56, 168,-168, 56, 0, 0, 0, 0, 0],
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[ 70,-280, 420,-280, 70, 0, 0, 0, 0],
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[-56, 280,-560, 560,-280, 56, 0, 0, 0],
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[ 28,-168, 420,-560, 420,-168, 28, 0, 0],
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[ -8, 56,-168, 280,-280, 168,-56, 8, 0],
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[ 1, -8, 28, -56, 70, -56, 28,-8, 1]],
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[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 9, 0, 0, 0, 0, 0, 0, 0, 0], [36, -72, 36, 0, 0, 0, 0, 0, 0, 0], [-84, 252, -252, 84, 0, 0, 0, 0, 0, 0],
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[126, -504, 756, -504, 126, 0, 0, 0, 0, 0], [-126, 630, -1260, 1260, -630, 126, 0, 0, 0, 0], [84, -504, 1260, -1680, 1260, -504, 84, 0, 0, 0],
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[-36, 252, -756, 1260, -1260, 756, -252, 36, 0, 0], [9, -72, 252, -504, 630, -504, 252, -72, 9, 0], [-1, 9, -36, 84, -126, 126, -84, 36, -9, 1]],
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[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [45, -90, 45, 0, 0, 0, 0, 0, 0, 0, 0], [-120, 360, -360, 120, 0, 0, 0, 0, 0, 0, 0],
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[210, -840, 1260, -840, 210, 0, 0, 0, 0, 0, 0], [-252, 1260, -2520, 2520, -1260, 252, 0, 0, 0, 0, 0],
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[210, -1260, 3150, -4200, 3150, -1260, 210, 0, 0, 0, 0], [-120, 840, -2520, 4200, -4200, 2520, -840, 120, 0, 0, 0],
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[45, -360, 1260, -2520, 3150, -2520, 1260, -360, 45, 0, 0], [-10, 90, -360, 840, -1260, 1260, -840, 360, -90, 10, 0],
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[1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1]]
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];
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function _bezier_matrix(N) =
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N>10 ? _compute_bez_matrix(N)
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: _bezier_matrix_table[N];
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// Function: bez_point()
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// Function: bez_point()
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// Usage:
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// Usage:
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// bez_point(curve, u)
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// bez_point(curve, u)
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@ -943,7 +1054,7 @@ function bezier_patch(patch, splinesteps=16, vnf=EMPTY_VNF, style="default") =
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],
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],
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pts = [
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pts = [
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for(step=[0:1:splinesteps.y]) [
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for(step=[0:1:splinesteps.y]) [
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for(bezparm=bpatch)
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for(bezparm=bpatch)
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bez_point(bezparm, step/splinesteps.y)
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bez_point(bezparm, step/splinesteps.y)
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]
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]
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],
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],
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