"""fontTools.misc.bezierTools.py -- tools for working with bezier path segments."""
__all__ = [
"calcQuadraticBounds",
"calcCubicBounds",
"splitLine",
"splitQuadratic",
"splitCubic",
"splitQuadraticAtT",
"splitCubicAtT",
"solveQuadratic",
"solveCubic",
]
from fontTools.misc.arrayTools import calcBounds
import numpy
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, 0.0, 100.0, 50.0)
>>> calcQuadraticBounds((0, 0), (100, 0), (100, 100))
(0.0, 0.0, 100.0, 100.0)
"""
a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
# calc first derivative
ax, ay = a * 2
bx, by = b
roots = []
if ax != 0:
roots.append(-bx/ax)
if ay != 0:
roots.append(-by/ay)
points = [a*t*t + b*t + c 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, 0.0, 100.0, 75.0)
>>> calcCubicBounds((0, 0), (50, 0), (100, 50), (100, 100))
(0.0, 0.0, 100.0, 100.0)
>>> calcCubicBounds((50, 0), (0, 100), (100, 100), (50, 0))
(35.5662432703, 0.0, 64.4337567297, 75.0)
"""
a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
# calc first derivative
ax, ay = a * 3.0
bx, by = b * 2.0
cx, cy = c
xRoots = [t for t in solveQuadratic(ax, bx, cx) if 0 <= t < 1]
yRoots = [t for t in solveQuadratic(ay, by, cy) if 0 <= t < 1]
roots = xRoots + yRoots
points = [(a*t*t*t + b*t*t + c * t + d) 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), (50.0, 50.0))
((50.0, 50.0), (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))
((0.0, 0.0), (100, 100))
>>> printSegments(splitLine((0, 0), (100, 100), 0, False))
((0, 0), (0.0, 0.0))
((0.0, 0.0), (100, 100))
"""
pt1, pt2 = numpy.array((pt1, pt2))
a = (pt2 - pt1)
b = pt1
ax = a[isHorizontal]
if ax == 0:
return [(pt1, pt2)]
t = float(where - b[isHorizontal]) / ax
if 0 <= t < 1:
midPt = a * t + b
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, 0.0), (25.0, 50.0), (50.0, 50.0))
((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, False))
((0.0, 0.0), (12.5, 25.0), (25.0, 37.5))
((25.0, 37.5), (62.5, 75.0), (100.0, 0.0))
>>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, True))
((0.0, 0.0), (7.32233047034, 14.6446609407), (14.6446609407, 25.0))
((14.6446609407, 25.0), (50.0, 75.0), (85.3553390593, 25.0))
((85.3553390593, 25.0), (92.6776695297, 14.6446609407), (100.0, -7.1054273576e-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, 0.0), (25.0, 50.0), (50.0, 50.0))
((50.0, 50.0), (50.0, 50.0), (50.0, 50.0))
((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
"""
a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
solutions = solveQuadratic(a[isHorizontal], b[isHorizontal],
c[isHorizontal] - where)
solutions = [t for t in solutions if 0 <= t < 1]
solutions.sort()
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, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
((50.0, 75.0), (68.75, 75.0), (87.5, 50.0), (100.0, 0.0))
>>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 25, True))
((0.0, 0.0), (2.2937927384, 9.17517095361), (4.79804488188, 17.5085042869), (7.47413641001, 25.0))
((7.47413641001, 25.0), (31.2886200204, 91.6666666667), (68.7113799796, 91.6666666667), (92.52586359, 25.0))
((92.52586359, 25.0), (95.2019551181, 17.5085042869), (97.7062072616, 9.17517095361), (100.0, 1.7763568394e-15))
"""
a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
solutions = solveCubic(a[isHorizontal], b[isHorizontal], c[isHorizontal],
d[isHorizontal] - where)
solutions = [t for t in solutions if 0 <= t < 1]
solutions.sort()
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, 0.0), (25.0, 50.0), (50.0, 50.0))
((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
>>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5, 0.75))
((0.0, 0.0), (25.0, 50.0), (50.0, 50.0))
((50.0, 50.0), (62.5, 50.0), (75.0, 37.5))
((75.0, 37.5), (87.5, 25.0), (100.0, 0.0))
"""
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, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
((50.0, 75.0), (68.75, 75.0), (87.5, 50.0), (100.0, 0.0))
>>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5, 0.75))
((0.0, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
((50.0, 75.0), (59.375, 75.0), (68.75, 68.75), (77.34375, 56.25))
((77.34375, 56.25), (85.9375, 43.75), (93.75, 25.0), (100.0, 0.0))
"""
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)
for i in range(len(ts) - 1):
t1 = ts[i]
t2 = ts[i+1]
delta = (t2 - t1)
# calc new a, b and c
a1 = a * delta**2
b1 = (2*a*t1 + b) * delta
c1 = a*t1**2 + b*t1 + c
pt1, pt2, pt3 = calcQuadraticPoints(a1, b1, c1)
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 = []
for i in range(len(ts) - 1):
t1 = ts[i]
t2 = ts[i+1]
delta = (t2 - t1)
# calc new a, b, c and d
a1 = a * delta**3
b1 = (3*a*t1 + b) * delta**2
c1 = (2*b*t1 + c + 3*a*t1**2) * delta
d1 = a*t1**3 + b*t1**2 + c*t1 + d
pt1, pt2, pt3, pt4 = calcCubicPoints(a1, b1, c1, d1)
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,
abs=abs, pow=pow, sqrt=sqrt, cos=cos, acos=acos, pi=pi):
"""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):
pt1, pt2, pt3 = numpy.array((pt1, pt2, pt3))
c = pt1
b = (pt2 - c) * 2.0
a = pt3 - c - b
return a, b, c
def calcCubicParameters(pt1, pt2, pt3, pt4):
pt1, pt2, pt3, pt4 = numpy.array((pt1, pt2, pt3, pt4))
d = pt1
c = (pt2 - d) * 3.0
b = (pt3 - pt2) * 3.0 - c
a = pt4 - d - c - b
return a, b, c, d
def calcQuadraticPoints(a, b, c):
pt1 = c
pt2 = (b * 0.5) + c
pt3 = a + b + c
return pt1, pt2, pt3
def calcCubicPoints(a, b, c, d):
pt1 = d
pt2 = (c / 3.0) + d
pt3 = (b + c) / 3.0 + pt2
pt4 = a + d + c + b
return pt1, pt2, pt3, pt4
def _segmentrepr(obj):
"""
>>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], numpy.array([0.1, 2.2])]]])
'(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))'
"""
try:
it = iter(obj)
except TypeError:
return str(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 doctest
doctest.testmod()
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