"""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), (50.0, 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))
"""
pt1x, pt1y = pt1
pt2x, pt2y = pt2
ax = (pt2x - pt1x)
ay = (pt2y - pt1y)
bx = pt1x
by = pt1y
if ax == 0:
return [(pt1, pt2)]
t = (where - (bx, by)[isHorizontal]) / ax
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, 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 = 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, 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 = 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, 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)
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 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()