/usr/lib/python2.7/dist-packages/FontTools/fontTools/misc/bezierTools.py is in fonttools 3.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 | """fontTools.misc.bezierTools.py -- tools for working with bezier path segments.
"""
from __future__ import print_function, division, absolute_import
from fontTools.misc.py23 import *
__all__ = [
"calcQuadraticBounds",
"calcCubicBounds",
"splitLine",
"splitQuadratic",
"splitCubic",
"splitQuadraticAtT",
"splitCubicAtT",
"solveQuadratic",
"solveCubic",
]
from fontTools.misc.arrayTools import calcBounds
epsilon = 1e-12
def calcQuadraticBounds(pt1, pt2, pt3):
"""Return the bounding rectangle for a qudratic bezier segment.
pt1 and pt3 are the "anchor" points, pt2 is the "handle".
>>> calcQuadraticBounds((0, 0), (50, 100), (100, 0))
(0, 0, 100, 50.0)
>>> calcQuadraticBounds((0, 0), (100, 0), (100, 100))
(0.0, 0.0, 100, 100)
"""
(ax, ay), (bx, by), (cx, cy) = calcQuadraticParameters(pt1, pt2, pt3)
ax2 = ax*2.0
ay2 = ay*2.0
roots = []
if ax2 != 0:
roots.append(-bx/ax2)
if ay2 != 0:
roots.append(-by/ay2)
points = [(ax*t*t + bx*t + cx, ay*t*t + by*t + cy) for t in roots if 0 <= t < 1] + [pt1, pt3]
return calcBounds(points)
def calcCubicBounds(pt1, pt2, pt3, pt4):
"""Return the bounding rectangle for a cubic bezier segment.
pt1 and pt4 are the "anchor" points, pt2 and pt3 are the "handles".
>>> calcCubicBounds((0, 0), (25, 100), (75, 100), (100, 0))
(0, 0, 100, 75.0)
>>> calcCubicBounds((0, 0), (50, 0), (100, 50), (100, 100))
(0.0, 0.0, 100, 100)
>>> print("%f %f %f %f" % calcCubicBounds((50, 0), (0, 100), (100, 100), (50, 0)))
35.566243 0.000000 64.433757 75.000000
"""
(ax, ay), (bx, by), (cx, cy), (dx, dy) = calcCubicParameters(pt1, pt2, pt3, pt4)
# calc first derivative
ax3 = ax * 3.0
ay3 = ay * 3.0
bx2 = bx * 2.0
by2 = by * 2.0
xRoots = [t for t in solveQuadratic(ax3, bx2, cx) if 0 <= t < 1]
yRoots = [t for t in solveQuadratic(ay3, by2, cy) if 0 <= t < 1]
roots = xRoots + yRoots
points = [(ax*t*t*t + bx*t*t + cx * t + dx, ay*t*t*t + by*t*t + cy * t + dy) for t in roots] + [pt1, pt4]
return calcBounds(points)
def splitLine(pt1, pt2, where, isHorizontal):
"""Split the line between pt1 and pt2 at position 'where', which
is an x coordinate if isHorizontal is False, a y coordinate if
isHorizontal is True. Return a list of two line segments if the
line was successfully split, or a list containing the original
line.
>>> printSegments(splitLine((0, 0), (100, 100), 50, True))
((0, 0), (50, 50))
((50, 50), (100, 100))
>>> printSegments(splitLine((0, 0), (100, 100), 100, True))
((0, 0), (100, 100))
>>> printSegments(splitLine((0, 0), (100, 100), 0, True))
((0, 0), (0, 0))
((0, 0), (100, 100))
>>> printSegments(splitLine((0, 0), (100, 100), 0, False))
((0, 0), (0, 0))
((0, 0), (100, 100))
>>> printSegments(splitLine((100, 0), (0, 0), 50, False))
((100, 0), (50, 0))
((50, 0), (0, 0))
>>> printSegments(splitLine((0, 100), (0, 0), 50, True))
((0, 100), (0, 50))
((0, 50), (0, 0))
"""
pt1x, pt1y = pt1
pt2x, pt2y = pt2
ax = (pt2x - pt1x)
ay = (pt2y - pt1y)
bx = pt1x
by = pt1y
a = (ax, ay)[isHorizontal]
if a == 0:
return [(pt1, pt2)]
t = (where - (bx, by)[isHorizontal]) / a
if 0 <= t < 1:
midPt = ax * t + bx, ay * t + by
return [(pt1, midPt), (midPt, pt2)]
else:
return [(pt1, pt2)]
def splitQuadratic(pt1, pt2, pt3, where, isHorizontal):
"""Split the quadratic curve between pt1, pt2 and pt3 at position 'where',
which is an x coordinate if isHorizontal is False, a y coordinate if
isHorizontal is True. Return a list of curve segments.
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 150, False))
((0, 0), (50, 100), (100, 0))
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, False))
((0, 0), (25, 50), (50, 50))
((50, 50), (75, 50), (100, 0))
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, False))
((0, 0), (12.5, 25), (25, 37.5))
((25, 37.5), (62.5, 75), (100, 0))
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, True))
((0, 0), (7.32233, 14.6447), (14.6447, 25))
((14.6447, 25), (50, 75), (85.3553, 25))
((85.3553, 25), (92.6777, 14.6447), (100, -7.10543e-15))
>>> # XXX I'm not at all sure if the following behavior is desirable:
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, True))
((0, 0), (25, 50), (50, 50))
((50, 50), (50, 50), (50, 50))
((50, 50), (75, 50), (100, 0))
"""
a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
solutions = solveQuadratic(a[isHorizontal], b[isHorizontal],
c[isHorizontal] - where)
solutions = sorted([t for t in solutions if 0 <= t < 1])
if not solutions:
return [(pt1, pt2, pt3)]
return _splitQuadraticAtT(a, b, c, *solutions)
def splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal):
"""Split the cubic curve between pt1, pt2, pt3 and pt4 at position 'where',
which is an x coordinate if isHorizontal is False, a y coordinate if
isHorizontal is True. Return a list of curve segments.
>>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 150, False))
((0, 0), (25, 100), (75, 100), (100, 0))
>>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 50, False))
((0, 0), (12.5, 50), (31.25, 75), (50, 75))
((50, 75), (68.75, 75), (87.5, 50), (100, 0))
>>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 25, True))
((0, 0), (2.29379, 9.17517), (4.79804, 17.5085), (7.47414, 25))
((7.47414, 25), (31.2886, 91.6667), (68.7114, 91.6667), (92.5259, 25))
((92.5259, 25), (95.202, 17.5085), (97.7062, 9.17517), (100, 1.77636e-15))
"""
a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
solutions = solveCubic(a[isHorizontal], b[isHorizontal], c[isHorizontal],
d[isHorizontal] - where)
solutions = sorted([t for t in solutions if 0 <= t < 1])
if not solutions:
return [(pt1, pt2, pt3, pt4)]
return _splitCubicAtT(a, b, c, d, *solutions)
def splitQuadraticAtT(pt1, pt2, pt3, *ts):
"""Split the quadratic curve between pt1, pt2 and pt3 at one or more
values of t. Return a list of curve segments.
>>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5))
((0, 0), (25, 50), (50, 50))
((50, 50), (75, 50), (100, 0))
>>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5, 0.75))
((0, 0), (25, 50), (50, 50))
((50, 50), (62.5, 50), (75, 37.5))
((75, 37.5), (87.5, 25), (100, 0))
"""
a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
return _splitQuadraticAtT(a, b, c, *ts)
def splitCubicAtT(pt1, pt2, pt3, pt4, *ts):
"""Split the cubic curve between pt1, pt2, pt3 and pt4 at one or more
values of t. Return a list of curve segments.
>>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5))
((0, 0), (12.5, 50), (31.25, 75), (50, 75))
((50, 75), (68.75, 75), (87.5, 50), (100, 0))
>>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5, 0.75))
((0, 0), (12.5, 50), (31.25, 75), (50, 75))
((50, 75), (59.375, 75), (68.75, 68.75), (77.3438, 56.25))
((77.3438, 56.25), (85.9375, 43.75), (93.75, 25), (100, 0))
"""
a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
return _splitCubicAtT(a, b, c, d, *ts)
def _splitQuadraticAtT(a, b, c, *ts):
ts = list(ts)
segments = []
ts.insert(0, 0.0)
ts.append(1.0)
ax, ay = a
bx, by = b
cx, cy = c
for i in range(len(ts) - 1):
t1 = ts[i]
t2 = ts[i+1]
delta = (t2 - t1)
# calc new a, b and c
a1x = ax * delta**2
a1y = ay * delta**2
b1x = (2*ax*t1 + bx) * delta
b1y = (2*ay*t1 + by) * delta
c1x = ax*t1**2 + bx*t1 + cx
c1y = ay*t1**2 + by*t1 + cy
pt1, pt2, pt3 = calcQuadraticPoints((a1x, a1y), (b1x, b1y), (c1x, c1y))
segments.append((pt1, pt2, pt3))
return segments
def _splitCubicAtT(a, b, c, d, *ts):
ts = list(ts)
ts.insert(0, 0.0)
ts.append(1.0)
segments = []
ax, ay = a
bx, by = b
cx, cy = c
dx, dy = d
for i in range(len(ts) - 1):
t1 = ts[i]
t2 = ts[i+1]
delta = (t2 - t1)
# calc new a, b, c and d
a1x = ax * delta**3
a1y = ay * delta**3
b1x = (3*ax*t1 + bx) * delta**2
b1y = (3*ay*t1 + by) * delta**2
c1x = (2*bx*t1 + cx + 3*ax*t1**2) * delta
c1y = (2*by*t1 + cy + 3*ay*t1**2) * delta
d1x = ax*t1**3 + bx*t1**2 + cx*t1 + dx
d1y = ay*t1**3 + by*t1**2 + cy*t1 + dy
pt1, pt2, pt3, pt4 = calcCubicPoints((a1x, a1y), (b1x, b1y), (c1x, c1y), (d1x, d1y))
segments.append((pt1, pt2, pt3, pt4))
return segments
#
# Equation solvers.
#
from math import sqrt, acos, cos, pi
def solveQuadratic(a, b, c,
sqrt=sqrt):
"""Solve a quadratic equation where a, b and c are real.
a*x*x + b*x + c = 0
This function returns a list of roots. Note that the returned list
is neither guaranteed to be sorted nor to contain unique values!
"""
if abs(a) < epsilon:
if abs(b) < epsilon:
# We have a non-equation; therefore, we have no valid solution
roots = []
else:
# We have a linear equation with 1 root.
roots = [-c/b]
else:
# We have a true quadratic equation. Apply the quadratic formula to find two roots.
DD = b*b - 4.0*a*c
if DD >= 0.0:
rDD = sqrt(DD)
roots = [(-b+rDD)/2.0/a, (-b-rDD)/2.0/a]
else:
# complex roots, ignore
roots = []
return roots
def solveCubic(a, b, c, d):
"""Solve a cubic equation where a, b, c and d are real.
a*x*x*x + b*x*x + c*x + d = 0
This function returns a list of roots. Note that the returned list
is neither guaranteed to be sorted nor to contain unique values!
"""
#
# adapted from:
# CUBIC.C - Solve a cubic polynomial
# public domain by Ross Cottrell
# found at: http://www.strangecreations.com/library/snippets/Cubic.C
#
if abs(a) < epsilon:
# don't just test for zero; for very small values of 'a' solveCubic()
# returns unreliable results, so we fall back to quad.
return solveQuadratic(b, c, d)
a = float(a)
a1 = b/a
a2 = c/a
a3 = d/a
Q = (a1*a1 - 3.0*a2)/9.0
R = (2.0*a1*a1*a1 - 9.0*a1*a2 + 27.0*a3)/54.0
R2_Q3 = R*R - Q*Q*Q
if R2_Q3 < 0:
theta = acos(R/sqrt(Q*Q*Q))
rQ2 = -2.0*sqrt(Q)
x0 = rQ2*cos(theta/3.0) - a1/3.0
x1 = rQ2*cos((theta+2.0*pi)/3.0) - a1/3.0
x2 = rQ2*cos((theta+4.0*pi)/3.0) - a1/3.0
return [x0, x1, x2]
else:
if Q == 0 and R == 0:
x = 0
else:
x = pow(sqrt(R2_Q3)+abs(R), 1/3.0)
x = x + Q/x
if R >= 0.0:
x = -x
x = x - a1/3.0
return [x]
#
# Conversion routines for points to parameters and vice versa
#
def calcQuadraticParameters(pt1, pt2, pt3):
x2, y2 = pt2
x3, y3 = pt3
cx, cy = pt1
bx = (x2 - cx) * 2.0
by = (y2 - cy) * 2.0
ax = x3 - cx - bx
ay = y3 - cy - by
return (ax, ay), (bx, by), (cx, cy)
def calcCubicParameters(pt1, pt2, pt3, pt4):
x2, y2 = pt2
x3, y3 = pt3
x4, y4 = pt4
dx, dy = pt1
cx = (x2 -dx) * 3.0
cy = (y2 -dy) * 3.0
bx = (x3 - x2) * 3.0 - cx
by = (y3 - y2) * 3.0 - cy
ax = x4 - dx - cx - bx
ay = y4 - dy - cy - by
return (ax, ay), (bx, by), (cx, cy), (dx, dy)
def calcQuadraticPoints(a, b, c):
ax, ay = a
bx, by = b
cx, cy = c
x1 = cx
y1 = cy
x2 = (bx * 0.5) + cx
y2 = (by * 0.5) + cy
x3 = ax + bx + cx
y3 = ay + by + cy
return (x1, y1), (x2, y2), (x3, y3)
def calcCubicPoints(a, b, c, d):
ax, ay = a
bx, by = b
cx, cy = c
dx, dy = d
x1 = dx
y1 = dy
x2 = (cx / 3.0) + dx
y2 = (cy / 3.0) + dy
x3 = (bx + cx) / 3.0 + x2
y3 = (by + cy) / 3.0 + y2
x4 = ax + dx + cx + bx
y4 = ay + dy + cy + by
return (x1, y1), (x2, y2), (x3, y3), (x4, y4)
def _segmentrepr(obj):
"""
>>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], [0.1, 2.2]]]])
'(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))'
"""
try:
it = iter(obj)
except TypeError:
return "%g" % obj
else:
return "(%s)" % ", ".join([_segmentrepr(x) for x in it])
def printSegments(segments):
"""Helper for the doctests, displaying each segment in a list of
segments on a single line as a tuple.
"""
for segment in segments:
print(_segmentrepr(segment))
if __name__ == "__main__":
import sys
import doctest
sys.exit(doctest.testmod().failed)
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