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
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
|
"""Pen calculating 0th, 1st, and 2nd moments of area of glyph shapes.
This is low-level, autogenerated pen. Use statisticsPen instead."""
from fontTools.misc.py23 import *
from fontTools.pens.basePen import BasePen
__all__ = ["MomentsPen"]
class MomentsPen(BasePen):
def __init__(self, glyphset=None):
BasePen.__init__(self, glyphset)
self.area = 0
self.momentX = 0
self.momentY = 0
self.momentXX = 0
self.momentXY = 0
self.momentYY = 0
def _moveTo(self, p0):
self.__startPoint = p0
def _closePath(self):
p0 = self._getCurrentPoint()
if p0 != self.__startPoint:
self._lineTo(self.__startPoint)
def _endPath(self):
p0 = self._getCurrentPoint()
if p0 != self.__startPoint:
# Green theorem is not defined on open contours.
raise NotImplementedError
def _lineTo(self, p1):
x0,y0 = self._getCurrentPoint()
x1,y1 = p1
r0 = x1*y0
r1 = x1*y1
r2 = x1**2
r3 = x0**2
r4 = 2*y0
r5 = y0 - y1
r6 = r5*x0
r7 = y0**2
r8 = y1**2
r9 = x1**3
r10 = r4*y1
r11 = y0**3
r12 = y1**3
self.area += -r0/2 - r1/2 + x0*(y0 + y1)/2
self.momentX += -r2*y0/6 - r2*y1/3 + r3*(r4 + y1)/6 - r6*x1/6
self.momentY += -r0*y1/6 - r7*x1/6 - r8*x1/6 + x0*(r7 + r8 + y0*y1)/6
self.momentXX += -r2*r6/12 - r3*r5*x1/12 - r9*y0/12 - r9*y1/4 + x0**3*(3*y0 + y1)/12
self.momentXY += -r10*r2/24 - r2*r7/24 - r2*r8/8 + r3*(r10 + 3*r7 + r8)/24 - x0*x1*(r7 - r8)/12
self.momentYY += -r0*r8/12 - r1*r7/12 - r11*x1/12 - r12*x1/12 + x0*(r11 + r12 + r7*y1 + r8*y0)/12
def _qCurveToOne(self, p1, p2):
x0,y0 = self._getCurrentPoint()
x1,y1 = p1
x2,y2 = p2
r0 = 2*x1
r1 = r0*y2
r2 = 2*y1
r3 = r2*x2
r4 = 3*y2
r5 = r4*x2
r6 = 3*y0
r7 = x1**2
r8 = 2*y2
r9 = x2**2
r10 = 4*y1
r11 = 10*y2
r12 = r0*x2
r13 = x0**2
r14 = 10*y0
r15 = x2*y2
r16 = r0*y1 + r15
r17 = 4*x1
r18 = x2*y0
r19 = r10*r15
r20 = y1**2
r21 = 2*r20
r22 = y2**2
r23 = r22*x2
r24 = 5*r23
r25 = y0**2
r26 = y0*y2
r27 = 5*r25
r28 = 8*x1**3
r29 = x2**3
r30 = 30*y1
r31 = 6*y1
r32 = 10*r9*x1
r33 = 4*r7
r34 = 5*y2
r35 = 12*r7
r36 = r5 + 20*x1*y1
r37 = 30*x1
r38 = 12*x1
r39 = 20*r7
r40 = 8*r7*y1
r41 = r34*r9
r42 = 60*y1
r43 = 20*r20
r44 = 4*r20
r45 = 15*r22
r46 = r38*x2
r47 = y1*y2
r48 = 8*r20*x1 + r24
r49 = 6*x1
r50 = 8*y1**3
r51 = y2**3
r52 = y0**3
r53 = 10*y1
r54 = 12*y1
r55 = 12*r20
self.area += r1/6 - r3/6 - r5/6 + x0*(r2 + r6 + y2)/6 - y0*(r0 + x2)/6
self.momentX += -r10*r9/30 - r11*r9/30 - r12*(-r8 + y1)/30 + r13*(r10 + r14 + y2)/30 + r7*r8/30 + x0*(r1 + r16 - r17*y0 - r18)/30 - y0*(r12 + 2*r7 + r9)/30
self.momentY += r1*(r8 + y1)/30 - r19/30 - r21*x2/30 - r24/30 - r25*(r17 + x2)/30 + x0*(r10*y0 + r2*y2 + r21 + r22 + r26 + r27)/30 - y0*(r16 + r3)/30
self.momentXX += r13*(r11*x1 - 5*r18 + r3 + r36 - r37*y0)/420 + r28*y2/420 - r29*r30/420 - r29*y2/4 - r32*(r2 - r4)/420 - r33*x2*(r2 - r34)/420 + x0**3*(r31 + 21*y0 + y2)/84 - x0*(-r15*r38 + r18*r38 + r2*r9 - r35*y2 + r39*y0 - r40 - r41 + r6*r9)/420 - y0*(r28 + 5*r29 + r32 + r35*x2)/420
self.momentXY += r13*(r14*y2 + 3*r22 + 105*r25 + r42*y0 + r43 + 12*r47)/840 - r17*x2*(r44 - r45)/840 - r22*r9/8 - r25*(r39 + r46 + 3*r9)/840 + r33*y2*(r10 + r34)/840 - r42*r9*y2/840 - r43*r9/840 + x0*(-r10*r18 + r17*r26 + r19 + r22*r49 - r25*r37 - r27*x2 + r38*r47 + r48)/420 - y0*(r15*r17 + r31*r9 + r40 + r41 + r46*y1)/420
self.momentYY += r1*(r11*y1 + r44 + r45)/420 - r15*r43/420 - r23*r30/420 - r25*(r1 + r36 + r53*x2)/420 - r50*x2/420 - r51*x2/12 - r52*(r49 + x2)/84 + x0*(r22*r53 + r22*r6 + r25*r30 + r25*r34 + r26*r54 + r43*y0 + r50 + 5*r51 + 35*r52 + r55*y2)/420 - y0*(-r0*r22 + r15*r54 + r48 + r55*x2)/420
def _curveToOne(self, p1, p2, p3):
x0,y0 = self._getCurrentPoint()
x1,y1 = p1
x2,y2 = p2
x3,y3 = p3
r0 = 6*x2
r1 = r0*y3
r2 = 6*y2
r3 = 10*y3
r4 = r3*x3
r5 = 3*x1
r6 = 3*y1
r7 = 6*x1
r8 = 3*x2
r9 = 6*y1
r10 = 3*y2
r11 = x2**2
r12 = r11*y3
r13 = 45*r12
r14 = x3**2
r15 = r14*y2
r16 = r14*y3
r17 = x2*x3
r18 = 15*r17
r19 = 7*y3
r20 = x1**2
r21 = 9*r20
r22 = x0**2
r23 = 21*y1
r24 = 9*r11
r25 = 9*x2
r26 = x2*y3
r27 = 15*r26
r28 = -r25*y1 + r27
r29 = r25*y2
r30 = r9*x3
r31 = 45*x1
r32 = x1*x3
r33 = 45*r20
r34 = 5*r14
r35 = x2*y2
r36 = 18*r35
r37 = 5*x3
r38 = r37*y3
r39 = r31*y1 + r36 + r38
r40 = x1*y0
r41 = x1*y3
r42 = x2*y0
r43 = x3*y1
r44 = r10*x3
r45 = x3*y2*y3
r46 = y2**2
r47 = 45*r46
r48 = r47*x3
r49 = y3**2
r50 = r49*x3
r51 = y1**2
r52 = 9*r51
r53 = y0**2
r54 = 21*x1
r55 = x3*y2
r56 = 15*r55
r57 = 9*y2
r58 = y2*y3
r59 = 15*r58
r60 = 9*r46
r61 = 3*y3
r62 = 45*y1
r63 = r8*y3
r64 = y0*y1
r65 = y0*y2
r66 = 30*r65
r67 = 5*y3
r68 = y1*y3
r69 = 45*r51
r70 = 5*r49
r71 = x2**3
r72 = x3**3
r73 = 126*x3
r74 = x1**3
r75 = r14*x2
r76 = 63*r11
r77 = r76*x3
r78 = 15*r35
r79 = r19*x3
r80 = x1*y1
r81 = 63*r35
r82 = r38 + 378*r80 + r81
r83 = x1*y2
r84 = x2*y1
r85 = x3*y0
r86 = x2*x3*y1
r87 = x2*x3*y3
r88 = r11*y2
r89 = 27*r88
r90 = 42*y3
r91 = r14*r90
r92 = 90*x1*x2
r93 = 189*x2
r94 = 30*x1*x3
r95 = 14*r16 + 126*r20*y1 + 45*r88 + r94*y2
r96 = x1*x2
r97 = 252*r96
r98 = x1*x2*y2
r99 = 42*r32
r100 = x1*x3*y1
r101 = 30*r17
r102 = 18*r17
r103 = 378*r20
r104 = 189*y2
r105 = r20*y3
r106 = r11*y1
r107 = r14*y1
r108 = 378*r46
r109 = 252*y2
r110 = y1*y2
r111 = x2*x3*y2
r112 = y0*y3
r113 = 378*r51
r114 = 63*r46
r115 = 27*x2
r116 = r115*r46 + 42*r50
r117 = x2*y1*y3
r118 = x3*y1*y2
r119 = r49*x2
r120 = r51*x3
r121 = x3*y3
r122 = 14*x3
r123 = 30*r117 + r122*r49 + r47*x2 + 126*r51*x1
r124 = x1*y1*y3
r125 = x1*y2*y3
r126 = x2*y1*y2
r127 = 54*y3
r128 = 21*r55
r129 = 630*r53
r130 = r46*x1
r131 = r49*x1
r132 = 126*r53
r133 = y2**3
r134 = y3**3
r135 = 630*r49
r136 = y1**3
r137 = y0**3
r138 = r114*y3 + r23*r49
r139 = r49*y2
self.area += r1/20 - r2*x3/20 - r4/20 + r5*(y2 + y3)/20 - r6*(x2 + x3)/20 + x0*(r10 + r9 + 10*y0 + y3)/20 - y0*(r7 + r8 + x3)/20
self.momentX += r13/840 - r15/8 - r16/3 - r18*(r10 - r19)/840 + r21*(r10 + 2*y3)/840 + r22*(r2 + r23 + 56*y0 + y3)/168 + r5*(r28 + r29 - r30 + r4)/840 - r6*(10*r14 + r18 + r24)/840 + x0*(12*r26 + r31*y2 - r37*y0 + r39 - 105*r40 + 15*r41 - 30*r42 - 3*r43 + r44)/840 - y0*(18*r11 + r18 + r31*x2 + 12*r32 + r33 + r34)/840
self.momentY += r27*(r10 + r19)/840 - r45/8 - r48/840 + r5*(10*r49 + r57*y1 + r59 + r60 + r9*y3)/840 - r50/6 - r52*(r8 + 2*x3)/840 - r53*(r0 + r54 + x3)/168 - r6*(r29 + r4 + r56)/840 + x0*(18*r46 + 140*r53 + r59 + r62*y2 + 105*r64 + r66 + r67*y0 + 12*r68 + r69 + r70)/840 - y0*(r39 + 15*r43 + 12*r55 - r61*x1 + r62*x2 + r63)/840
self.momentXX += -r11*r73*(-r61 + y2)/9240 + r21*(r28 - r37*y1 + r44 + r78 + r79)/9240 + r22*(21*r26 - 630*r40 + 42*r41 - 126*r42 + r57*x3 + r82 + 210*r83 + 42*r84 - 14*r85)/9240 - r5*(r11*r62 + r14*r23 + 14*r15 - r76*y3 + 54*r86 - 84*r87 - r89 - r91)/9240 - r6*(27*r71 + 42*r72 + 70*r75 + r77)/9240 + 3*r71*y3/220 - 3*r72*y2/44 - r72*y3/4 + 3*r74*(r57 + r67)/3080 - r75*(378*y2 - 630*y3)/9240 + x0**3*(r57 + r62 + 165*y0 + y3)/660 + x0*(-18*r100 - r101*y0 - r101*y1 + r102*y2 - r103*y0 + r104*r20 + 63*r105 - 27*r106 - 9*r107 + r13 - r34*y0 - r76*y0 + 42*r87 + r92*y3 + r94*y3 + r95 - r97*y0 + 162*r98 - r99*y0)/9240 - y0*(135*r11*x1 + r14*r54 + r20*r93 + r33*x3 + 45*r71 + 14*r72 + 126*r74 + 42*r75 + r77 + r92*x3)/9240
self.momentXY += -r108*r14/18480 + r12*(r109 + 378*y3)/18480 - r14*r49/8 - 3*r14*r58/44 - r17*(252*r46 - 1260*r49)/18480 + r21*(18*r110 + r3*y1 + 15*r46 + 7*r49 + 18*r58)/18480 + r22*(252*r110 + 28*r112 + r113 + r114 + 2310*r53 + 30*r58 + 1260*r64 + 252*r65 + 42*r68 + r70)/18480 - r52*(r102 + 15*r11 + 7*r14)/18480 - r53*(r101 + r103 + r34 + r76 + r97 + r99)/18480 + r7*(-r115*r51 + r116 + 18*r117 - 18*r118 + 42*r119 - 15*r120 + 28*r45 + r81*y3)/18480 - r9*(63*r111 + 42*r15 + 28*r87 + r89 + r91)/18480 + x0*(r1*y0 + r104*r80 + r112*r54 + 21*r119 - 9*r120 - r122*r53 + r123 + 54*r124 + 60*r125 + 54*r126 + r127*r35 + r128*y3 - r129*x1 + 81*r130 + 15*r131 - r132*x2 - r2*r85 - r23*r85 + r30*y3 + 84*r40*y2 - 84*r42*y1 + r60*x3)/9240 - y0*(54*r100 - 9*r105 + 81*r106 + 15*r107 + 54*r111 + r121*r7 + 21*r15 + r24*y3 + 60*r86 + 21*r87 + r95 + 189*r96*y1 + 54*r98)/9240
self.momentYY += -r108*r121/9240 - r133*r73/9240 - r134*x3/12 - r135*r55/9240 - 3*r136*(r25 + r37)/3080 - r137*(r25 + r31 + x3)/660 + r26*(r135 + 126*r46 + 378*y2*y3)/9240 + r5*(r110*r127 + 27*r133 + 42*r134 + r138 + 70*r139 + r46*r62 + 27*r51*y2 + 15*r51*y3)/9240 - r52*(r56 + r63 + r78 + r79)/9240 - r53*(r128 + r25*y3 + 42*r43 + r82 + 42*r83 + 210*r84)/9240 - r6*(r114*x3 + r116 - 14*r119 + 84*r45)/9240 + x0*(r104*r51 + r109*r64 + 90*r110*y3 + r113*y0 + r114*y0 + r129*y1 + r132*y2 + 45*r133 + 14*r134 + 126*r136 + 770*r137 + r138 + 42*r139 + 135*r46*y1 + 14*r53*y3 + r64*r90 + r66*y3 + r69*y3 + r70*y0)/9240 - y0*(90*r118 + 63*r120 + r123 - 18*r124 - 30*r125 + 162*r126 - 27*r130 - 9*r131 + r36*y3 + 30*r43*y3 + 42*r45 + r48 + r51*r93)/9240
if __name__ == '__main__':
from fontTools.misc.symfont import x, y, printGreenPen
printGreenPen('MomentsPen', [
('area', 1),
('momentX', x),
('momentY', y),
('momentXX', x**2),
('momentXY', x*y),
('momentYY', y**2),
])
|
