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 diff --git a/third_party/spiro/curves/poly3.py b/third_party/spiro/curves/poly3.pynew file mode 100644index 0000000..daf6f87--- /dev/null+++ b/third_party/spiro/curves/poly3.py@@ -0,0 +1,148 @@+# Numerical techniques for solving 3rd order polynomial spline systems++# The standard representation is the vector of derivatives at s=0,+# with -.5 <= s <= 5.+#+# Thus, \kappa(s) = k0 + k1 s + 1/2 k2 s^2 + 1/6 k3 s^3++from math import *++def eval_cubic(a, b, c, d, x):+ return ((d * x + c) * x + b) * x + a++# integrate over s = [0, 1]+def int_3spiro_poly(ks, n):+ x, y = 0, 0+ th = 0+ ds = 1.0 / n+ th1, th2, th3, th4 = ks[0], .5 * ks[1], (1./6) * ks[2], (1./24) * ks[3]+ k0, k1, k2, k3 = ks[0] * ds, ks[1] * ds, ks[2] * ds, ks[3] * ds+ s = 0+ result = [(x, y)]+ for i in range(n):+ sm = s + 0.5 * ds+ th = sm * eval_cubic(th1, th2, th3, th4, sm)+ cth = cos(th)+ sth = sin(th)++ km0 = ((1./6 * k3 * sm + .5 * k2) * sm + k1) * sm + k0+ km1 = ((.5 * k3 * sm + k2) * sm + k1) * ds+ km2 = (k3 * sm + k2) * ds * ds+ km3 = k3 * ds * ds * ds+ #print km0, km1, km2, km3+ u = 1 - km0 * km0 / 24+ v = km1 / 24++ u = 1 - km0 * km0 / 24 + (km0 ** 4 - 4 * km0 * km2 - 3 * km1 * km1) / 1920+ v = km1 / 24 + (km3 - 6 * km0 * km0 * km1) / 1920++ x += cth * u - sth * v+ y += cth * v + sth * u+ result.append((ds * x, ds * y))++ s += ds++ return result++def integ_chord(k, n = 64):+ ks = (k[0] * .5, k[1] * .25, k[2] * .125, k[3] * .0625)+ xp, yp = int_3spiro_poly(ks, n)[-1]+ ks = (k[0] * -.5, k[1] * .25, k[2] * -.125, k[3] * .0625)+ xm, ym = int_3spiro_poly(ks, n)[-1]+ dx, dy = .5 * (xp + xm), .5 * (yp + ym)+ return hypot(dx, dy), atan2(dy, dx)++# Return th0, th1, k0, k1 for given params+def calc_thk(ks):+ chord, ch_th = integ_chord(ks)+ th0 = ch_th - (-.5 * ks[0] + .125 * ks[1] - 1./48 * ks[2] + 1./384 * ks[3])+ th1 = (.5 * ks[0] + .125 * ks[1] + 1./48 * ks[2] + 1./384 * ks[3]) - ch_th+ k0 = chord * (ks[0] - .5 * ks[1] + .125 * ks[2] - 1./48 * ks[3])+ k1 = chord * (ks[0] + .5 * ks[1] + .125 * ks[2] + 1./48 * ks[3])+ #print '%', (-.5 * ks[0] + .125 * ks[1] - 1./48 * ks[2] + 1./384 * ks[3]), (.5 * ks[0] + .125 * ks[1] + 1./48 * ks[2] + 1./384 * ks[3]), ch_th+ return th0, th1, k0, k1++def calc_k1k2(ks):+ chord, ch_th = integ_chord(ks)+ k1l = chord * chord * (ks[1] - .5 * ks[2] + .125 * ks[3])+ k1r = chord * chord * (ks[1] + .5 * ks[2] + .125 * ks[3])+ k2l = chord * chord * chord * (ks[2] - .5 * ks[3])+ k2r = chord * chord * chord * (ks[2] + .5 * ks[3])+ return k1l, k1r, k2l, k2r+ +def plot(ks):+ ksp = (ks[0] * .5, ks[1] * .25, ks[2] * .125, ks[3] * .0625)+ pside = int_3spiro_poly(ksp, 64)+ ksm = (ks[0] * -.5, ks[1] * .25, ks[2] * -.125, ks[3] * .0625)+ mside = int_3spiro_poly(ksm, 64)+ mside.reverse()+ for i in range(len(mside)):+ mside[i] = (-mside[i][0], -mside[i][1])+ pts = mside + pside[1:]+ cmd = "moveto"+ for j in range(len(pts)):+ x, y = pts[j]+ print 306 + 300 * x, 400 + 300 * y, cmd+ cmd = "lineto"+ print "stroke"+ x, y = pts[0]+ print 306 + 300 * x, 400 + 300 * y, "moveto"+ x, y = pts[-1]+ print 306 + 300 * x, 400 + 300 * y, "lineto .5 setlinewidth stroke"+ print "showpage"++def solve_3spiro(th0, th1, k0, k1):+ ks = [0, 0, 0, 0]+ for i in range(5):+ th0_a, th1_a, k0_a, k1_a = calc_thk(ks)+ dth0 = th0 - th0_a+ dth1 = th1 - th1_a+ dk0 = k0 - k0_a+ dk1 = k1 - k1_a+ ks[0] += (dth0 + dth1) * 1.5 + (dk0 + dk1) * -.25+ ks[1] += (dth1 - dth0) * 15 + (dk0 - dk1) * 1.5+ ks[2] += (dth0 + dth1) * -12 + (dk0 + dk1) * 6+ ks[3] += (dth0 - dth1) * 360 + (dk1 - dk0) * 60+ #print '% ks =', ks+ return ks++def iter_spline(pts, ths, ks):+ pass++def solve_vee():+ kss = []+ for i in range(10):+ kss.append([0, 0, 0, 0])+ thl = [0] * len(kss)+ thr = [0] * len(kss)+ k0l = [0] * len(kss)+ k0r = [0] * len(kss)+ k1l = [0] * len(kss)+ k1r = [0] * len(kss)+ k2l = [0] * len(kss)+ k2r = [0] * len(kss)+ for i in range(10):+ for j in range(len(kss)):+ thl[j], thr[j], k0l[j], k0r[j] = calc_thk(kss[j])+ k0l[j], k1r[j], k2l[j], k2r[j] = calc_k1k2(kss[j])+ for j in range(len(kss) - 1):+ dth = thl[j + 1] + thr[j]+ if j == 5: dth += .1+ dk0 = k0l[j + 1] - k0r[j]+ dk1 = k1l[j + 1] - k1r[j]+ dk2 = k2l[j + 1] - k2r[j]+++if __name__ == '__main__':+ k0 = pi * 3+ ks = [0, k0, -2 * k0, 0]+ ks = [0, 0, 0, 0.01]+ #plot(ks)+ thk = calc_thk(ks)+ print '%', thk+ + ks = solve_3spiro(0, 0, 0, 0.001)+ print '% thk =', calc_thk(ks)+ #plot(ks)+ print '%', ks+ print calc_k1k2(ks)