1 files changed, 148 insertions, 0 deletions
diff --git a/third_party/spiro/curves/poly3.py b/third_party/spiro/curves/poly3.py
new file mode 100644
index 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)