-
Notifications
You must be signed in to change notification settings - Fork 0
/
interpolations.py
192 lines (146 loc) · 5.49 KB
/
interpolations.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
# interpolations - routines for numerical interpolations between control values
'''
Routines for numerical interpolation between various control values.
SYNOPSIS
>>> import interpolations ; from interpolations import *
>>> import vectors ; from vectors import *
Linear:
>>> linear(0.5, 10, 20)
15.0
>>> linear(1.0, 2.0, 1.5, 10, 20)
15.0
>>> linear(1.0, 2.0, 1.5, V(10,100,1000), V(20,200,2000))
V(15.0, 150.0, 1500.0)
Bezier:
>>> bezier(0.5, 10, 20, 10)
15.0
>>> bezier(0.5, 10, 20, 20, 10)
175.0
>>> bezier(0.5, V(10,100), V(20,200), V(20,200), V(10,100))
V(17.5, 175.0)
AUTHOR
Ed Halley (ed@halley.cc) 25 October 2007
REFERENCES
Bezier code adapted and ported to python from a C++ v3 implementation:
http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/bezier/index2.html
'''
__all__ = [ 'linear', 'bezier' ]
#----------------------------------------------------------------------------
import math
#----------------------------------------------------------------------------
def linear(*args):
'''Find a point along a linear path of two controls.
Controls may be scalars, or vectors if arithmetic operators defined.
May take three arguments, linear(i, A, B).
Interpolated values equal control A at i==0.0, and control B at i==1.0.
May take five arguments, linear(x, y, i, A, B).
Interpolated values equal control A at i==x, and control B at i==y.
'''
if len(args) == 3:
(xmin, xmax) = (0., 1.)
(x, hmin, hmax) = args
elif len(args) == 5:
(xmin, xmax, x, hmin, hmax) = args
else:
raise ValueError, 'linterp() takes 3 or 5 arguments'
return ( ((x)-(xmin)) * ((hmax)-(hmin)) / ((xmax)-(xmin)) + (hmin) )
#----------------------------------------------------------------------------
# ref:
# http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/bezier/index2.html
def bezier3(i, A, B, C):
'''Find a point along a bezier curve of three controls.
Controls may be scalars, or vectors if arithmetic operators defined.
Takes four arguments, bezier3(i, A, B, C).
Interpolated values touch A at i==0.0, and C at i==1.0.
Interpolated values may not touch B value.
'''
ii = 1.-i
i2 = i*i
ii2 = ii*ii
j = A*ii2 + 2*B*ii*i + C*i2
return j
def bezier4(i, A, B, C, D):
'''Find a point along a bezier curve of four controls.
Controls may be scalars, or vectors if arithmetic operators defined.
Takes five arguments, bezier4(i, A, B, C, D).
Interpolated values touch A at i==0.0, and D at i==1.0.
Interpolated values may not touch B or C values.
'''
ii = 1.-i
i3 = i*i*i
ii3 = ii*ii*ii
j = A*ii3 + 3*B*i*ii*ii + 3*C*i*i*ii + D*i3
return j
def bezier(i, *A):
'''Find a point along a bezier curve of arbitrary number of controls.
Controls may be scalars, or vectors if arithmetic operators defined.
Takes at least two arguments, bezier(i, A, ...).
Interpolated values touch A at i==0.0, and last control at i==1.0.
Interpolated values may not touch any intervening control value.
Uses linear interpolation for two controls (A, B),
or constant if only given one control (A).
'''
n = len(A)-1
if n < 4:
if n < 0: raise ValueError, 'need at least one control value'
if n == 0: return A[0]
if n == 1: return linear(i, *A)
if n == 2: return bezier3(i, *A)
if n == 3: return bezier4(i, *A)
ik = 1
ii = 1-i
ink = math.pow(ii, float(n))
if i == 1.0:
return A[-1]*1.0
j = A[0]*0.0
for k in range(n+1):
nn = n
kn = k
nkn = n - k
blend = ik*ink
ik *= i
ink /= ii
while (nn >= 1):
blend *= nn
if (kn > 1):
blend /= kn
kn -= 1
if (nkn > 1):
blend /= nkn
nkn -= 1
nn -= 1
j += A[k]*blend
return j
#----------------------------------------------------------------------------
def __test__():
from testing import __ok__, __report__
import vectors ; from vectors import V,equal,zero
print 'Testing interpolations...'
__ok__( linear(0.5, 50, 60), 55 )
__ok__( linear(1, 2, 1.5, 50, 60), 55 )
__ok__( linear(1, 2, 1.5, V(50,500), V(60,600)), V(55,550) )
__ok__( bezier3(0.0, 0.0, 1.0, 0.0), 0.0 )
__ok__( bezier3(0.5, 0.0, 1.0, 0.0), 0.5 )
__ok__( equal( bezier3(0.2, 0.0, 1.0, 0.0),
bezier3(0.8, 0.0, 1.0, 0.0) ) )
__ok__( bezier3(1.0, 0.0, 1.0, 0.0), 0.0 )
__ok__( bezier4(0.0, 0.0, 1.0, 1.0, 0.0), 0.0 )
__ok__( bezier4(0.5, 0.0, 1.0, 1.0, 0.0), 0.75 )
__ok__( equal( bezier4(0.2, 0.0, 1.0, 1.0, 0.0),
bezier4(0.8, 0.0, 1.0, 1.0, 0.0) ) )
__ok__( bezier4(1.0, 0.0, 1.0, 1.0, 0.0), 0.0 )
__ok__( bezier(0.0, 0.0, 1.0, 1.0, 1.0, 0.0), 0.0 )
__ok__( bezier(0.5, 0.0, 1.0, 1.0, 1.0, 0.0), 0.875 )
__ok__( equal( bezier(0.2, 0.0, 1.0, 1.0, 1.0, 0.0),
bezier(0.8, 0.0, 1.0, 1.0, 1.0, 0.0) ) )
__ok__( bezier(1.0, 0.0, 1.0, 1.0, 1.0, 0.0), 0.0 )
__ok__( bezier(1.0, 0.0, 1.0, 1.0, 1.0, 0.0), 0.0 )
__report__()
def __table__():
from testing import __ok__, __report__
for x in range(40+1):
i = x/40.0
print i, ',', bezier(i, 1.0,2.0,0.0,2.0,1.0)
if __name__ == '__main__':
raise Exception, \
'This module is not a stand-alone script. Import it in a program.'