2 from __future__ import print_function
5 from numpy import cos, sin
9 def augment(v): return np.append(v, 1)
10 def augment0(v): return np.append(v, 0)
11 def unaugment(v): return v[0:3]
13 class HelixishCurve():
17 dp = unit_v(cp[1]-cp[0])
18 dq = unit_v(cp[3]-cp[2])
21 # - solve in the plane containing dP and dQ
22 # - total distance normal to that plane gives mu
23 # - now resulting curve is not parallel to dP at P
24 # nor dQ at Q, so tilt it
25 # - [[ pick as the hinge point the half of the curve
26 # with the larger s or t ]] not yet implemented
27 # - increase the other distance {t,s} by a bodge factor
28 # approx distance between {Q,P} and {Q,P}' due to hinging
29 # but minimum is 10% of (wlog) {s,t} [[ not quite like this ]]
31 dPQplane_normal = np.cross(dp, dq)
32 if (np.norm(dPQplane_normal) < 1E6):
33 dPQplane_normal += [0, 0, 1E5]
34 dPQplane_normal = unit_v(dPQplane_normal)
36 dPQplane_basis = np.column_stack(np.cross(dp, dPQplane_normal),
40 dPQplane_basis = np.vstack(dPQplane_basis, [0,0,0,1])
41 dPQplane_into = np.linalg.inv(dPQplane_basis)
43 dp_plane = unaugment(dPQplane_into * augment0(dp))
44 dq_plane = unaugment(dPQplane_into * augment0(dq))
45 q_plane = unaugment(dPQplane_into * augment(q))
46 dist_pq_plane = np.linalg.norm(q_plane)
48 # two circular arcs of equal maximum possible radius
49 # algorithm courtesy of Simon Tatham (`Railway problem',
50 # pers.comm. to ijackson@chiark 23.1.2004)
51 railway_angleoffset = atan2(*q_plane[0:1])
52 railway_theta = tau/4 - railway_angleoffset
53 railway_phi = atan2(*dq_plane[0:1]) - railway_angleoffset
54 railway_cos_theta = cos(railway_theta)
55 railway_cos_phi = cos(railway_phi)
56 if railway_cos_theta**2 + railway_cos_phi**2 > 1E6:
57 railway_roots = np.roots([
58 2 * (1 + cos(railway_theta - railway_phi)),
59 2 * (railway_cos_theta - railway_cos_phi),
62 for railway_r in railway_roots:
63 def railway_CPQ(pq, dpq):
65 return pq + railway_r * [-dpq[1], dpq[0]]
67 railway_CP = railway_CPQ([0,0,0], dp_plane)
68 railway_QP = railway_CPQ(q_plane[0:2], -dq_plane)
69 railway_midpt = 0.5 * (railway_CP + railway_QP)
72 def railway_ST(C, start, end):
74 delta = atan2(*(end - C)[0:2]) - atan2(start - C)[0:2]
77 try_s = railway_ST(railway_CP, [0,0], midpt)
78 try_t = railway_ST(railway_CP, midpt, q_plane)
79 try_st = try_s + try_t
80 if best_st is None or try_st < best_st:
85 start_mu = q_plane[2] / (start_s + start_t)
87 else: # twoarcs algorithm is not well defined
89 start_s = dist_pq_plane * .65
90 start_t = dist_pq_plane * .35
93 bodge = max( q_plane[2] * mu,
94 (start_s + start_t) * 0.1 )
95 start_s += 0.5 * bodge
96 start_t += 0.5 * bodge
101 tilt_basis = np.array([
103 0, cos(tilt), -sin(tilt), 0,
104 0, sin(tilt), cos(tilt), 0,
107 findcurve_basis = dPQplane_basis * tilt_basis
108 findcurve_into = np.linalg.inv(findcurve_basis)
110 q_findcurve = unaugment(findcurve_into, augment(q))
111 dq_findcurve = unaugment(findcurve_into, augment0(dq))
113 findcurve_target = np.concatenate(q_findcurve, dq_findcurve)
114 findcurve_start = (sqrt(start_s), sqrt(start_t), start_la,
115 start_mu, start_gamma, start_kappa)
120 # we work in two additional coordinate systems:
125 # dP is the +ve y axis
126 # Q lies in the x/y plane
127 # for calculating the initial attempt: