2 from __future__ import print_function
5 from numpy import cos, sin
7 from bezier import BezierSegment
13 origin = np.array((0,0,0))
14 unit_x = np.array((1,0,0))
15 unit_y = np.array((0,1,0))
16 unit_z = np.array((0,0,1))
19 return v / np.linalg.norm(v)
21 class DoubleCubicBezier():
23 single = BezierSegment(cp)
24 midpoint = np.array(single.point_at_t(0.5))
25 mid_dirn = single.point_at_t(0.5 + 0.001) - midpoint
26 mid_dirn /= np.linalg.norm(mid_dirn)
28 mid_scale = ocp_factor * 0.5 * (np.linalg.norm(cp[1] - cp[0]) +
29 np.linalg.norm(cp[3] - cp[2]))
30 db.b0 = BezierSegment([ cp[0],
31 cp[1] * ocp_factor + cp[0] * (1-ocp_factor),
32 midpoint - mid_dirn * mid_scale,
34 db.b1 = BezierSegment([ midpoint,
35 midpoint + mid_dirn * mid_scale,
36 cp[2] * ocp_factor + cp[3] * (1-ocp_factor),
38 def point_at_t(db, t):
40 return db.b0.point_at_t(t*2)
42 return db.b1.point_at_t(t*2 - 1)
44 class ParametricCircle:
45 def __init__(pc, c, r0, r1):
46 ''' circle centred on c
47 with theta=0 point at c+r0
48 and with theta=tau/4 point at c+r1 '''
52 def radius(pc, theta):
53 return pc._r0 * cos(theta) + pc._r1 * sin(theta)
55 return pc._c + pc.radius(theta)
57 class Twirler(ParametricCircle):
58 def __init__(tw, c, r0, r1, cycles, begin_zeta):
59 ''' circle centred on c, etc.
60 but with an orientation at each point, orthogonal to
62 the orientation circles round cycles times during the
63 whole cycle (if cycles<1, to get a whole circling of
64 the dirn around the circle, must pass some theta>tau)
65 begin_zeta is the angle from outwards at theta==0
66 positive meaning in the direction of r0 x r1 from r0
68 ParametricCircle.__init__(tw, c, r0, r1)
70 tw._begin_zeta = begin_zeta
71 tw._axis = np.cross(r0, r1)
72 def dirn(tw, theta, extra_zeta=0):
73 ''' unit vector for dirn at theta,
74 but rotated extra_theta more around the circle
76 zeta = tw._begin_zeta + theta * tw._cycles + extra_zeta
78 return cos(zeta) * r + sin(zeta) * tw._axis
83 MoebiusHalf().edge is a Twirler for the edge,
84 expecting theta = u * tau (see MoebiusHalf().point)
85 with dirn pointing into the surface
87 m.edge = Twirler(origin, unit_z, unit_x, -2, tau/2)
88 m.midline = Twirler(-unit_z, unit_z, unit_y, -0.5, 0)
90 m._thetas = [ u * tau for u in np.linspace(0, 1, nu+1) ]
91 m._cp2b = BezierSegment([ (c,) for c in [0.33,0.33, 1.50]])
92 m._beziers = [ m._bezier(theta) for theta in m._thetas ]
93 def _bezier(m, theta):
95 cp[0] = m.edge .point(theta)
96 cp[3] = m.midline.point(theta*2)
97 ncp3 = np.linalg.norm(cp[3])
99 cp2scale = m._cp2b.point_at_t(ncp3/2)
100 cp1scale = 1.0 * cpt + 0.33 * (1-cpt)
101 #print('u=%d ncp3=%f cp2scale=%f' % (u, ncp3, cp2scale),
103 cp[1] = cp[0] + cp1scale * m.edge .dirn (theta)
104 cp[2] = cp[3] + cp2scale * m.midline.dirn (theta*2)
105 return DoubleCubicBezier(cp)
108 0 <= iu <= nu meaning 0 <= u <= 1
109 along the extent (well, along the edge)
110 0 and 1 are both the top half of the flat traverse
111 0.5 is the bottom half of the flat traverse
112 0 <= t <= 1 across the half-traverse
113 0 is the edge, 1 is the midline
115 return np.array(m._beziers[iu].point_at_t(t))
117 def details(m, iu, t):
119 returns tuple of 4 vectors:
121 - normal (+ve is in the +ve y direction at iu=t=0) unit vector
122 - along extent (towrds +ve iu) unit vector
123 - along traverse (towards +ve t) unit vector
126 return m.details(0, t)
128 vec_t = unit_v( m.point(iu, t + 0.01) - p )
130 normal = m.edge.dirn(m._thetas[iu], extra_zeta=-tau/4)
131 vec_u = np.cross(vec_t, normal)
133 vec_u = unit_v( m.point(iu+1, t) - p )
134 normal = np.cross(vec_u, vec_t)
135 return p, normal, vec_u, vec_t
137 def point_offset(m, iu, t, offset):
139 offset by offset perpendicular to the surface
140 at the top (iu=t=0), +ve offset is in the +ve y direction
142 p, normal, dummy, dummy = m.details(iu, t)
143 return p + offset * normal
146 def __init__(m, nv, nw):
148 0 <= v <= nw along the extent, v=0 is the flat traverse
149 0 <= w <= nv across the traverse nw must be even
150 the top is both v=0, w=0 v=nv, w=nw
156 m._t_vals = np.linspace(0, 1, m.nt+1)
157 m.h = MoebiusHalf(nu=nv*2)
159 def _vw2tiu_kw(m, v, w):
166 #print('v,w=%d,%d => it,iu=%d,%d' % (v,w,it,iu),
168 return { 't': m._t_vals[it], 'iu': iu }
171 return m.h.point(**m._vw2tiu_kw(v,w))
173 def point_offset(m, v, w, offset):
174 return m.h.point_offset(offset=
175 offset if w <= m.nt else -offset,
178 def details(m, v, w):
180 returns tuple of 4 vectors:
182 - normal (+ve is in the +ve y direction at iu=t=0) unit vector
183 - along extent (towrds +ve v) unit vector
184 - along traverse (towards +ve w) unit vector
186 p, normal, vec_v, vec_w = m.h.details(**m._vw2tiu_kw(v,w))
190 return p, normal, vec_v, vec_w