# -*- coding: utf-8 -*-
"""fontTools.misc.bezierTools.py -- tools for working with Bezier path segments.
"""
from fontTools.misc.arrayTools import calcBounds, sectRect, rectArea
from fontTools.misc.transform import Identity
import math
from collections import namedtuple
try:
import cython
COMPILED = cython.compiled
except (AttributeError, ImportError):
# if cython not installed, use mock module with no-op decorators and types
from fontTools.misc import cython
COMPILED = False
Intersection = namedtuple("Intersection", ["pt", "t1", "t2"])
__all__ = [
"approximateCubicArcLength",
"approximateCubicArcLengthC",
"approximateQuadraticArcLength",
"approximateQuadraticArcLengthC",
"calcCubicArcLength",
"calcCubicArcLengthC",
"calcQuadraticArcLength",
"calcQuadraticArcLengthC",
"calcCubicBounds",
"calcQuadraticBounds",
"splitLine",
"splitQuadratic",
"splitCubic",
"splitQuadraticAtT",
"splitCubicAtT",
"splitCubicAtTC",
"splitCubicIntoTwoAtTC",
"solveQuadratic",
"solveCubic",
"quadraticPointAtT",
"cubicPointAtT",
"cubicPointAtTC",
"linePointAtT",
"segmentPointAtT",
"lineLineIntersections",
"curveLineIntersections",
"curveCurveIntersections",
"segmentSegmentIntersections",
]
[docs]
def calcCubicArcLength(pt1, pt2, pt3, pt4, tolerance=0.005):
"""Calculates the arc length for a cubic Bezier segment.
Whereas :func:`approximateCubicArcLength` approximates the length, this
function calculates it by "measuring", recursively dividing the curve
until the divided segments are shorter than ``tolerance``.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples.
tolerance: Controls the precision of the calcuation.
Returns:
Arc length value.
"""
return calcCubicArcLengthC(
complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4), tolerance
)
def _split_cubic_into_two(p0, p1, p2, p3):
mid = (p0 + 3 * (p1 + p2) + p3) * 0.125
deriv3 = (p3 + p2 - p1 - p0) * 0.125
return (
(p0, (p0 + p1) * 0.5, mid - deriv3, mid),
(mid, mid + deriv3, (p2 + p3) * 0.5, p3),
)
@cython.returns(cython.double)
@cython.locals(
p0=cython.complex,
p1=cython.complex,
p2=cython.complex,
p3=cython.complex,
)
@cython.locals(mult=cython.double, arch=cython.double, box=cython.double)
def _calcCubicArcLengthCRecurse(mult, p0, p1, p2, p3):
arch = abs(p0 - p3)
box = abs(p0 - p1) + abs(p1 - p2) + abs(p2 - p3)
if arch * mult >= box:
return (arch + box) * 0.5
else:
one, two = _split_cubic_into_two(p0, p1, p2, p3)
return _calcCubicArcLengthCRecurse(mult, *one) + _calcCubicArcLengthCRecurse(
mult, *two
)
[docs]
@cython.returns(cython.double)
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
)
@cython.locals(
tolerance=cython.double,
mult=cython.double,
)
def calcCubicArcLengthC(pt1, pt2, pt3, pt4, tolerance=0.005):
"""Calculates the arc length for a cubic Bezier segment.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers.
tolerance: Controls the precision of the calcuation.
Returns:
Arc length value.
"""
mult = 1.0 + 1.5 * tolerance # The 1.5 is a empirical hack; no math
return _calcCubicArcLengthCRecurse(mult, pt1, pt2, pt3, pt4)
epsilonDigits = 6
epsilon = 1e-10
@cython.cfunc
@cython.inline
@cython.returns(cython.double)
@cython.locals(v1=cython.complex, v2=cython.complex)
def _dot(v1, v2):
return (v1 * v2.conjugate()).real
@cython.cfunc
@cython.inline
@cython.returns(cython.double)
@cython.locals(x=cython.double)
def _intSecAtan(x):
# In : sympy.integrate(sp.sec(sp.atan(x)))
# Out: x*sqrt(x**2 + 1)/2 + asinh(x)/2
return x * math.sqrt(x**2 + 1) / 2 + math.asinh(x) / 2
[docs]
def calcQuadraticArcLength(pt1, pt2, pt3):
"""Calculates the arc length for a quadratic Bezier segment.
Args:
pt1: Start point of the Bezier as 2D tuple.
pt2: Handle point of the Bezier as 2D tuple.
pt3: End point of the Bezier as 2D tuple.
Returns:
Arc length value.
Example::
>>> calcQuadraticArcLength((0, 0), (0, 0), (0, 0)) # empty segment
0.0
>>> calcQuadraticArcLength((0, 0), (50, 0), (80, 0)) # collinear points
80.0
>>> calcQuadraticArcLength((0, 0), (0, 50), (0, 80)) # collinear points vertical
80.0
>>> calcQuadraticArcLength((0, 0), (50, 20), (100, 40)) # collinear points
107.70329614269008
>>> calcQuadraticArcLength((0, 0), (0, 100), (100, 0))
154.02976155645263
>>> calcQuadraticArcLength((0, 0), (0, 50), (100, 0))
120.21581243984076
>>> calcQuadraticArcLength((0, 0), (50, -10), (80, 50))
102.53273816445825
>>> calcQuadraticArcLength((0, 0), (40, 0), (-40, 0)) # collinear points, control point outside
66.66666666666667
>>> calcQuadraticArcLength((0, 0), (40, 0), (0, 0)) # collinear points, looping back
40.0
"""
return calcQuadraticArcLengthC(complex(*pt1), complex(*pt2), complex(*pt3))
[docs]
@cython.returns(cython.double)
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
d0=cython.complex,
d1=cython.complex,
d=cython.complex,
n=cython.complex,
)
@cython.locals(
scale=cython.double,
origDist=cython.double,
a=cython.double,
b=cython.double,
x0=cython.double,
x1=cython.double,
Len=cython.double,
)
def calcQuadraticArcLengthC(pt1, pt2, pt3):
"""Calculates the arc length for a quadratic Bezier segment.
Args:
pt1: Start point of the Bezier as a complex number.
pt2: Handle point of the Bezier as a complex number.
pt3: End point of the Bezier as a complex number.
Returns:
Arc length value.
"""
# Analytical solution to the length of a quadratic bezier.
# Documentation: https://github.com/fonttools/fonttools/issues/3055
d0 = pt2 - pt1
d1 = pt3 - pt2
d = d1 - d0
n = d * 1j
scale = abs(n)
if scale == 0.0:
return abs(pt3 - pt1)
origDist = _dot(n, d0)
if abs(origDist) < epsilon:
if _dot(d0, d1) >= 0:
return abs(pt3 - pt1)
a, b = abs(d0), abs(d1)
return (a * a + b * b) / (a + b)
x0 = _dot(d, d0) / origDist
x1 = _dot(d, d1) / origDist
Len = abs(2 * (_intSecAtan(x1) - _intSecAtan(x0)) * origDist / (scale * (x1 - x0)))
return Len
[docs]
def approximateQuadraticArcLength(pt1, pt2, pt3):
"""Calculates the arc length for a quadratic Bezier segment.
Uses Gauss-Legendre quadrature for a branch-free approximation.
See :func:`calcQuadraticArcLength` for a slower but more accurate result.
Args:
pt1: Start point of the Bezier as 2D tuple.
pt2: Handle point of the Bezier as 2D tuple.
pt3: End point of the Bezier as 2D tuple.
Returns:
Approximate arc length value.
"""
return approximateQuadraticArcLengthC(complex(*pt1), complex(*pt2), complex(*pt3))
[docs]
@cython.returns(cython.double)
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
)
@cython.locals(
v0=cython.double,
v1=cython.double,
v2=cython.double,
)
def approximateQuadraticArcLengthC(pt1, pt2, pt3):
"""Calculates the arc length for a quadratic Bezier segment.
Uses Gauss-Legendre quadrature for a branch-free approximation.
See :func:`calcQuadraticArcLength` for a slower but more accurate result.
Args:
pt1: Start point of the Bezier as a complex number.
pt2: Handle point of the Bezier as a complex number.
pt3: End point of the Bezier as a complex number.
Returns:
Approximate arc length value.
"""
# This, essentially, approximates the length-of-derivative function
# to be integrated with the best-matching fifth-degree polynomial
# approximation of it.
#
# https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Legendre_quadrature
# abs(BezierCurveC[2].diff(t).subs({t:T})) for T in sorted(.5, .5±sqrt(3/5)/2),
# weighted 5/18, 8/18, 5/18 respectively.
v0 = abs(
-0.492943519233745 * pt1 + 0.430331482911935 * pt2 + 0.0626120363218102 * pt3
)
v1 = abs(pt3 - pt1) * 0.4444444444444444
v2 = abs(
-0.0626120363218102 * pt1 - 0.430331482911935 * pt2 + 0.492943519233745 * pt3
)
return v0 + v1 + v2
[docs]
def calcQuadraticBounds(pt1, pt2, pt3):
"""Calculates the bounding rectangle for a quadratic Bezier segment.
Args:
pt1: Start point of the Bezier as a 2D tuple.
pt2: Handle point of the Bezier as a 2D tuple.
pt3: End point of the Bezier as a 2D tuple.
Returns:
A four-item tuple representing the bounding rectangle ``(xMin, yMin, xMax, yMax)``.
Example::
>>> 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)
[docs]
def approximateCubicArcLength(pt1, pt2, pt3, pt4):
"""Approximates the arc length for a cubic Bezier segment.
Uses Gauss-Lobatto quadrature with n=5 points to approximate arc length.
See :func:`calcCubicArcLength` for a slower but more accurate result.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples.
Returns:
Arc length value.
Example::
>>> approximateCubicArcLength((0, 0), (25, 100), (75, 100), (100, 0))
190.04332968932817
>>> approximateCubicArcLength((0, 0), (50, 0), (100, 50), (100, 100))
154.8852074945903
>>> approximateCubicArcLength((0, 0), (50, 0), (100, 0), (150, 0)) # line; exact result should be 150.
149.99999999999991
>>> approximateCubicArcLength((0, 0), (50, 0), (100, 0), (-50, 0)) # cusp; exact result should be 150.
136.9267662156362
>>> approximateCubicArcLength((0, 0), (50, 0), (100, -50), (-50, 0)) # cusp
154.80848416537057
"""
return approximateCubicArcLengthC(
complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4)
)
[docs]
@cython.returns(cython.double)
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
)
@cython.locals(
v0=cython.double,
v1=cython.double,
v2=cython.double,
v3=cython.double,
v4=cython.double,
)
def approximateCubicArcLengthC(pt1, pt2, pt3, pt4):
"""Approximates the arc length for a cubic Bezier segment.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers.
Returns:
Arc length value.
"""
# This, essentially, approximates the length-of-derivative function
# to be integrated with the best-matching seventh-degree polynomial
# approximation of it.
#
# https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules
# abs(BezierCurveC[3].diff(t).subs({t:T})) for T in sorted(0, .5±(3/7)**.5/2, .5, 1),
# weighted 1/20, 49/180, 32/90, 49/180, 1/20 respectively.
v0 = abs(pt2 - pt1) * 0.15
v1 = abs(
-0.558983582205757 * pt1
+ 0.325650248872424 * pt2
+ 0.208983582205757 * pt3
+ 0.024349751127576 * pt4
)
v2 = abs(pt4 - pt1 + pt3 - pt2) * 0.26666666666666666
v3 = abs(
-0.024349751127576 * pt1
- 0.208983582205757 * pt2
- 0.325650248872424 * pt3
+ 0.558983582205757 * pt4
)
v4 = abs(pt4 - pt3) * 0.15
return v0 + v1 + v2 + v3 + v4
[docs]
def calcCubicBounds(pt1, pt2, pt3, pt4):
"""Calculates the bounding rectangle for a quadratic Bezier segment.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples.
Returns:
A four-item tuple representing the bounding rectangle ``(xMin, yMin, xMax, yMax)``.
Example::
>>> 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)
[docs]
def splitLine(pt1, pt2, where, isHorizontal):
"""Split a line at a given coordinate.
Args:
pt1: Start point of line as 2D tuple.
pt2: End point of line as 2D tuple.
where: Position at which to split the line.
isHorizontal: Direction of the ray splitting the line. If true,
``where`` is interpreted as a Y coordinate; if false, then
``where`` is interpreted as an X coordinate.
Returns:
A list of two line segments (each line segment being two 2D tuples)
if the line was successfully split, or a list containing the original
line.
Example::
>>> 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)]
[docs]
def splitQuadratic(pt1, pt2, pt3, where, isHorizontal):
"""Split a quadratic Bezier curve at a given coordinate.
Args:
pt1,pt2,pt3: Control points of the Bezier as 2D tuples.
where: Position at which to split the curve.
isHorizontal: Direction of the ray splitting the curve. If true,
``where`` is interpreted as a Y coordinate; if false, then
``where`` is interpreted as an X coordinate.
Returns:
A list of two curve segments (each curve segment being three 2D tuples)
if the curve was successfully split, or a list containing the original
curve.
Example::
>>> 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)
[docs]
def splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal):
"""Split a cubic Bezier curve at a given coordinate.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples.
where: Position at which to split the curve.
isHorizontal: Direction of the ray splitting the curve. If true,
``where`` is interpreted as a Y coordinate; if false, then
``where`` is interpreted as an X coordinate.
Returns:
A list of two curve segments (each curve segment being four 2D tuples)
if the curve was successfully split, or a list containing the original
curve.
Example::
>>> 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)
[docs]
def splitQuadraticAtT(pt1, pt2, pt3, *ts):
"""Split a quadratic Bezier curve at one or more values of t.
Args:
pt1,pt2,pt3: Control points of the Bezier as 2D tuples.
*ts: Positions at which to split the curve.
Returns:
A list of curve segments (each curve segment being three 2D tuples).
Examples::
>>> 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)
[docs]
def splitCubicAtT(pt1, pt2, pt3, pt4, *ts):
"""Split a cubic Bezier curve at one or more values of t.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as 2D tuples.
*ts: Positions at which to split the curve.
Returns:
A list of curve segments (each curve segment being four 2D tuples).
Examples::
>>> 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)
[docs]
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
a=cython.complex,
b=cython.complex,
c=cython.complex,
d=cython.complex,
)
def splitCubicAtTC(pt1, pt2, pt3, pt4, *ts):
"""Split a cubic Bezier curve at one or more values of t.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers..
*ts: Positions at which to split the curve.
Yields:
Curve segments (each curve segment being four complex numbers).
"""
a, b, c, d = calcCubicParametersC(pt1, pt2, pt3, pt4)
yield from _splitCubicAtTC(a, b, c, d, *ts)
[docs]
@cython.returns(cython.complex)
@cython.locals(
t=cython.double,
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
pointAtT=cython.complex,
off1=cython.complex,
off2=cython.complex,
)
@cython.locals(
t2=cython.double, _1_t=cython.double, _1_t_2=cython.double, _2_t_1_t=cython.double
)
def splitCubicIntoTwoAtTC(pt1, pt2, pt3, pt4, t):
"""Split a cubic Bezier curve at t.
Args:
pt1,pt2,pt3,pt4: Control points of the Bezier as complex numbers.
t: Position at which to split the curve.
Returns:
A tuple of two curve segments (each curve segment being four complex numbers).
"""
t2 = t * t
_1_t = 1 - t
_1_t_2 = _1_t * _1_t
_2_t_1_t = 2 * t * _1_t
pointAtT = (
_1_t_2 * _1_t * pt1 + 3 * (_1_t_2 * t * pt2 + _1_t * t2 * pt3) + t2 * t * pt4
)
off1 = _1_t_2 * pt1 + _2_t_1_t * pt2 + t2 * pt3
off2 = _1_t_2 * pt2 + _2_t_1_t * pt3 + t2 * pt4
pt2 = pt1 + (pt2 - pt1) * t
pt3 = pt4 + (pt3 - pt4) * _1_t
return ((pt1, pt2, off1, pointAtT), (pointAtT, off2, pt3, pt4))
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
delta_2 = delta * delta
a1x = ax * delta_2
a1y = ay * delta_2
b1x = (2 * ax * t1 + bx) * delta
b1y = (2 * ay * t1 + by) * delta
t1_2 = t1 * t1
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
delta_2 = delta * delta
delta_3 = delta * delta_2
t1_2 = t1 * t1
t1_3 = t1 * t1_2
# 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
@cython.locals(
a=cython.complex,
b=cython.complex,
c=cython.complex,
d=cython.complex,
t1=cython.double,
t2=cython.double,
delta=cython.double,
delta_2=cython.double,
delta_3=cython.double,
a1=cython.complex,
b1=cython.complex,
c1=cython.complex,
d1=cython.complex,
)
def _splitCubicAtTC(a, b, c, d, *ts):
ts = list(ts)
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
delta_2 = delta * delta
delta_3 = delta * delta_2
t1_2 = t1 * t1
t1_3 = t1 * t1_2
# 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 = calcCubicPointsC(a1, b1, c1, d1)
yield (pt1, pt2, pt3, pt4)
#
# Equation solvers.
#
from math import sqrt, acos, cos, pi
[docs]
def solveQuadratic(a, b, c, sqrt=sqrt):
"""Solve a quadratic equation.
Solves *a*x*x + b*x + c = 0* where a, b and c are real.
Args:
a: coefficient of *x²*
b: coefficient of *x*
c: constant term
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
[docs]
def solveCubic(a, b, c, d):
"""Solve a cubic equation.
Solves *a*x*x*x + b*x*x + c*x + d = 0* where a, b, c and d are real.
Args:
a: coefficient of *x³*
b: coefficient of *x²*
c: coefficient of *x*
d: constant term
Returns:
A list of roots. Note that the returned list is neither guaranteed to
be sorted nor to contain unique values!
Examples::
>>> solveCubic(1, 1, -6, 0)
[-3.0, -0.0, 2.0]
>>> solveCubic(-10.0, -9.0, 48.0, -29.0)
[-2.9, 1.0, 1.0]
>>> solveCubic(-9.875, -9.0, 47.625, -28.75)
[-2.911392, 1.0, 1.0]
>>> solveCubic(1.0, -4.5, 6.75, -3.375)
[1.5, 1.5, 1.5]
>>> solveCubic(-12.0, 18.0, -9.0, 1.50023651123)
[0.5, 0.5, 0.5]
>>> solveCubic(
... 9.0, 0.0, 0.0, -7.62939453125e-05
... ) == [-0.0, -0.0, -0.0]
True
"""
#
# 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 = R * R
Q3 = Q * Q * Q
R2 = 0 if R2 < epsilon else R2
Q3 = 0 if abs(Q3) < epsilon else Q3
R2_Q3 = R2 - Q3
if R2 == 0.0 and Q3 == 0.0:
x = round(-a1 / 3.0, epsilonDigits)
return [x, x, x]
elif R2_Q3 <= epsilon * 0.5:
# The epsilon * .5 above ensures that Q3 is not zero.
theta = acos(max(min(R / sqrt(Q3), 1.0), -1.0))
rQ2 = -2.0 * sqrt(Q)
a1_3 = a1 / 3.0
x0 = rQ2 * cos(theta / 3.0) - a1_3
x1 = rQ2 * cos((theta + 2.0 * pi) / 3.0) - a1_3
x2 = rQ2 * cos((theta + 4.0 * pi) / 3.0) - a1_3
x0, x1, x2 = sorted([x0, x1, x2])
# Merge roots that are close-enough
if x1 - x0 < epsilon and x2 - x1 < epsilon:
x0 = x1 = x2 = round((x0 + x1 + x2) / 3.0, epsilonDigits)
elif x1 - x0 < epsilon:
x0 = x1 = round((x0 + x1) / 2.0, epsilonDigits)
x2 = round(x2, epsilonDigits)
elif x2 - x1 < epsilon:
x0 = round(x0, epsilonDigits)
x1 = x2 = round((x1 + x2) / 2.0, epsilonDigits)
else:
x0 = round(x0, epsilonDigits)
x1 = round(x1, epsilonDigits)
x2 = round(x2, epsilonDigits)
return [x0, x1, x2]
else:
x = pow(sqrt(R2_Q3) + abs(R), 1 / 3.0)
x = x + Q / x
if R >= 0.0:
x = -x
x = round(x - a1 / 3.0, epsilonDigits)
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)
@cython.cfunc
@cython.inline
@cython.locals(
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
a=cython.complex,
b=cython.complex,
c=cython.complex,
)
def calcCubicParametersC(pt1, pt2, pt3, pt4):
c = (pt2 - pt1) * 3.0
b = (pt3 - pt2) * 3.0 - c
a = pt4 - pt1 - c - b
return (a, b, c, pt1)
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)
@cython.cfunc
@cython.inline
@cython.locals(
a=cython.complex,
b=cython.complex,
c=cython.complex,
d=cython.complex,
p2=cython.complex,
p3=cython.complex,
p4=cython.complex,
)
def calcCubicPointsC(a, b, c, d):
p2 = c * (1 / 3) + d
p3 = (b + c) * (1 / 3) + p2
p4 = a + b + c + d
return (d, p2, p3, p4)
#
# Point at time
#
[docs]
def linePointAtT(pt1, pt2, t):
"""Finds the point at time `t` on a line.
Args:
pt1, pt2: Coordinates of the line as 2D tuples.
t: The time along the line.
Returns:
A 2D tuple with the coordinates of the point.
"""
return ((pt1[0] * (1 - t) + pt2[0] * t), (pt1[1] * (1 - t) + pt2[1] * t))
[docs]
def quadraticPointAtT(pt1, pt2, pt3, t):
"""Finds the point at time `t` on a quadratic curve.
Args:
pt1, pt2, pt3: Coordinates of the curve as 2D tuples.
t: The time along the curve.
Returns:
A 2D tuple with the coordinates of the point.
"""
x = (1 - t) * (1 - t) * pt1[0] + 2 * (1 - t) * t * pt2[0] + t * t * pt3[0]
y = (1 - t) * (1 - t) * pt1[1] + 2 * (1 - t) * t * pt2[1] + t * t * pt3[1]
return (x, y)
[docs]
def cubicPointAtT(pt1, pt2, pt3, pt4, t):
"""Finds the point at time `t` on a cubic curve.
Args:
pt1, pt2, pt3, pt4: Coordinates of the curve as 2D tuples.
t: The time along the curve.
Returns:
A 2D tuple with the coordinates of the point.
"""
t2 = t * t
_1_t = 1 - t
_1_t_2 = _1_t * _1_t
x = (
_1_t_2 * _1_t * pt1[0]
+ 3 * (_1_t_2 * t * pt2[0] + _1_t * t2 * pt3[0])
+ t2 * t * pt4[0]
)
y = (
_1_t_2 * _1_t * pt1[1]
+ 3 * (_1_t_2 * t * pt2[1] + _1_t * t2 * pt3[1])
+ t2 * t * pt4[1]
)
return (x, y)
[docs]
@cython.returns(cython.complex)
@cython.locals(
t=cython.double,
pt1=cython.complex,
pt2=cython.complex,
pt3=cython.complex,
pt4=cython.complex,
)
@cython.locals(t2=cython.double, _1_t=cython.double, _1_t_2=cython.double)
def cubicPointAtTC(pt1, pt2, pt3, pt4, t):
"""Finds the point at time `t` on a cubic curve.
Args:
pt1, pt2, pt3, pt4: Coordinates of the curve as complex numbers.
t: The time along the curve.
Returns:
A complex number with the coordinates of the point.
"""
t2 = t * t
_1_t = 1 - t
_1_t_2 = _1_t * _1_t
return _1_t_2 * _1_t * pt1 + 3 * (_1_t_2 * t * pt2 + _1_t * t2 * pt3) + t2 * t * pt4
[docs]
def segmentPointAtT(seg, t):
if len(seg) == 2:
return linePointAtT(*seg, t)
elif len(seg) == 3:
return quadraticPointAtT(*seg, t)
elif len(seg) == 4:
return cubicPointAtT(*seg, t)
raise ValueError("Unknown curve degree")
#
# Intersection finders
#
def _line_t_of_pt(s, e, pt):
sx, sy = s
ex, ey = e
px, py = pt
if abs(sx - ex) < epsilon and abs(sy - ey) < epsilon:
# Line is a point!
return -1
# Use the largest
if abs(sx - ex) > abs(sy - ey):
return (px - sx) / (ex - sx)
else:
return (py - sy) / (ey - sy)
def _both_points_are_on_same_side_of_origin(a, b, origin):
xDiff = (a[0] - origin[0]) * (b[0] - origin[0])
yDiff = (a[1] - origin[1]) * (b[1] - origin[1])
return not (xDiff <= 0.0 and yDiff <= 0.0)
[docs]
def lineLineIntersections(s1, e1, s2, e2):
"""Finds intersections between two line segments.
Args:
s1, e1: Coordinates of the first line as 2D tuples.
s2, e2: Coordinates of the second line as 2D tuples.
Returns:
A list of ``Intersection`` objects, each object having ``pt``, ``t1``
and ``t2`` attributes containing the intersection point, time on first
segment and time on second segment respectively.
Examples::
>>> a = lineLineIntersections( (310,389), (453, 222), (289, 251), (447, 367))
>>> len(a)
1
>>> intersection = a[0]
>>> intersection.pt
(374.44882952482897, 313.73458370177315)
>>> (intersection.t1, intersection.t2)
(0.45069111555824465, 0.5408153767394238)
"""
s1x, s1y = s1
e1x, e1y = e1
s2x, s2y = s2
e2x, e2y = e2
if (
math.isclose(s2x, e2x) and math.isclose(s1x, e1x) and not math.isclose(s1x, s2x)
): # Parallel vertical
return []
if (
math.isclose(s2y, e2y) and math.isclose(s1y, e1y) and not math.isclose(s1y, s2y)
): # Parallel horizontal
return []
if math.isclose(s2x, e2x) and math.isclose(s2y, e2y): # Line segment is tiny
return []
if math.isclose(s1x, e1x) and math.isclose(s1y, e1y): # Line segment is tiny
return []
if math.isclose(e1x, s1x):
x = s1x
slope34 = (e2y - s2y) / (e2x - s2x)
y = slope34 * (x - s2x) + s2y
pt = (x, y)
return [
Intersection(
pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt)
)
]
if math.isclose(s2x, e2x):
x = s2x
slope12 = (e1y - s1y) / (e1x - s1x)
y = slope12 * (x - s1x) + s1y
pt = (x, y)
return [
Intersection(
pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt)
)
]
slope12 = (e1y - s1y) / (e1x - s1x)
slope34 = (e2y - s2y) / (e2x - s2x)
if math.isclose(slope12, slope34):
return []
x = (slope12 * s1x - s1y - slope34 * s2x + s2y) / (slope12 - slope34)
y = slope12 * (x - s1x) + s1y
pt = (x, y)
if _both_points_are_on_same_side_of_origin(
pt, e1, s1
) and _both_points_are_on_same_side_of_origin(pt, s2, e2):
return [
Intersection(
pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt)
)
]
return []
def _alignment_transformation(segment):
# Returns a transformation which aligns a segment horizontally at the
# origin. Apply this transformation to curves and root-find to find
# intersections with the segment.
start = segment[0]
end = segment[-1]
angle = math.atan2(end[1] - start[1], end[0] - start[0])
return Identity.rotate(-angle).translate(-start[0], -start[1])
def _curve_line_intersections_t(curve, line):
aligned_curve = _alignment_transformation(line).transformPoints(curve)
if len(curve) == 3:
a, b, c = calcQuadraticParameters(*aligned_curve)
intersections = solveQuadratic(a[1], b[1], c[1])
elif len(curve) == 4:
a, b, c, d = calcCubicParameters(*aligned_curve)
intersections = solveCubic(a[1], b[1], c[1], d[1])
else:
raise ValueError("Unknown curve degree")
return sorted(i for i in intersections if 0.0 <= i <= 1)
[docs]
def curveLineIntersections(curve, line):
"""Finds intersections between a curve and a line.
Args:
curve: List of coordinates of the curve segment as 2D tuples.
line: List of coordinates of the line segment as 2D tuples.
Returns:
A list of ``Intersection`` objects, each object having ``pt``, ``t1``
and ``t2`` attributes containing the intersection point, time on first
segment and time on second segment respectively.
Examples::
>>> curve = [ (100, 240), (30, 60), (210, 230), (160, 30) ]
>>> line = [ (25, 260), (230, 20) ]
>>> intersections = curveLineIntersections(curve, line)
>>> len(intersections)
3
>>> intersections[0].pt
(84.9000930760723, 189.87306176459828)
"""
if len(curve) == 3:
pointFinder = quadraticPointAtT
elif len(curve) == 4:
pointFinder = cubicPointAtT
else:
raise ValueError("Unknown curve degree")
intersections = []
for t in _curve_line_intersections_t(curve, line):
pt = pointFinder(*curve, t)
# Back-project the point onto the line, to avoid problems with
# numerical accuracy in the case of vertical and horizontal lines
line_t = _line_t_of_pt(*line, pt)
pt = linePointAtT(*line, line_t)
intersections.append(Intersection(pt=pt, t1=t, t2=line_t))
return intersections
def _curve_bounds(c):
if len(c) == 3:
return calcQuadraticBounds(*c)
elif len(c) == 4:
return calcCubicBounds(*c)
raise ValueError("Unknown curve degree")
def _split_segment_at_t(c, t):
if len(c) == 2:
s, e = c
midpoint = linePointAtT(s, e, t)
return [(s, midpoint), (midpoint, e)]
if len(c) == 3:
return splitQuadraticAtT(*c, t)
elif len(c) == 4:
return splitCubicAtT(*c, t)
raise ValueError("Unknown curve degree")
def _curve_curve_intersections_t(
curve1, curve2, precision=1e-3, range1=None, range2=None
):
bounds1 = _curve_bounds(curve1)
bounds2 = _curve_bounds(curve2)
if not range1:
range1 = (0.0, 1.0)
if not range2:
range2 = (0.0, 1.0)
# If bounds don't intersect, go home
intersects, _ = sectRect(bounds1, bounds2)
if not intersects:
return []
def midpoint(r):
return 0.5 * (r[0] + r[1])
# If they do overlap but they're tiny, approximate
if rectArea(bounds1) < precision and rectArea(bounds2) < precision:
return [(midpoint(range1), midpoint(range2))]
c11, c12 = _split_segment_at_t(curve1, 0.5)
c11_range = (range1[0], midpoint(range1))
c12_range = (midpoint(range1), range1[1])
c21, c22 = _split_segment_at_t(curve2, 0.5)
c21_range = (range2[0], midpoint(range2))
c22_range = (midpoint(range2), range2[1])
found = []
found.extend(
_curve_curve_intersections_t(
c11, c21, precision, range1=c11_range, range2=c21_range
)
)
found.extend(
_curve_curve_intersections_t(
c12, c21, precision, range1=c12_range, range2=c21_range
)
)
found.extend(
_curve_curve_intersections_t(
c11, c22, precision, range1=c11_range, range2=c22_range
)
)
found.extend(
_curve_curve_intersections_t(
c12, c22, precision, range1=c12_range, range2=c22_range
)
)
unique_key = lambda ts: (int(ts[0] / precision), int(ts[1] / precision))
seen = set()
unique_values = []
for ts in found:
key = unique_key(ts)
if key in seen:
continue
seen.add(key)
unique_values.append(ts)
return unique_values
def _is_linelike(segment):
maybeline = _alignment_transformation(segment).transformPoints(segment)
return all(math.isclose(p[1], 0.0) for p in maybeline)
[docs]
def curveCurveIntersections(curve1, curve2):
"""Finds intersections between a curve and a curve.
Args:
curve1: List of coordinates of the first curve segment as 2D tuples.
curve2: List of coordinates of the second curve segment as 2D tuples.
Returns:
A list of ``Intersection`` objects, each object having ``pt``, ``t1``
and ``t2`` attributes containing the intersection point, time on first
segment and time on second segment respectively.
Examples::
>>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ]
>>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ]
>>> intersections = curveCurveIntersections(curve1, curve2)
>>> len(intersections)
3
>>> intersections[0].pt
(81.7831487395506, 109.88904552375288)
"""
if _is_linelike(curve1):
line1 = curve1[0], curve1[-1]
if _is_linelike(curve2):
line2 = curve2[0], curve2[-1]
return lineLineIntersections(*line1, *line2)
else:
return curveLineIntersections(curve2, line1)
elif _is_linelike(curve2):
line2 = curve2[0], curve2[-1]
return curveLineIntersections(curve1, line2)
intersection_ts = _curve_curve_intersections_t(curve1, curve2)
return [
Intersection(pt=segmentPointAtT(curve1, ts[0]), t1=ts[0], t2=ts[1])
for ts in intersection_ts
]
[docs]
def segmentSegmentIntersections(seg1, seg2):
"""Finds intersections between two segments.
Args:
seg1: List of coordinates of the first segment as 2D tuples.
seg2: List of coordinates of the second segment as 2D tuples.
Returns:
A list of ``Intersection`` objects, each object having ``pt``, ``t1``
and ``t2`` attributes containing the intersection point, time on first
segment and time on second segment respectively.
Examples::
>>> curve1 = [ (10,100), (90,30), (40,140), (220,220) ]
>>> curve2 = [ (5,150), (180,20), (80,250), (210,190) ]
>>> intersections = segmentSegmentIntersections(curve1, curve2)
>>> len(intersections)
3
>>> intersections[0].pt
(81.7831487395506, 109.88904552375288)
>>> curve3 = [ (100, 240), (30, 60), (210, 230), (160, 30) ]
>>> line = [ (25, 260), (230, 20) ]
>>> intersections = segmentSegmentIntersections(curve3, line)
>>> len(intersections)
3
>>> intersections[0].pt
(84.9000930760723, 189.87306176459828)
"""
# Arrange by degree
swapped = False
if len(seg2) > len(seg1):
seg2, seg1 = seg1, seg2
swapped = True
if len(seg1) > 2:
if len(seg2) > 2:
intersections = curveCurveIntersections(seg1, seg2)
else:
intersections = curveLineIntersections(seg1, seg2)
elif len(seg1) == 2 and len(seg2) == 2:
intersections = lineLineIntersections(*seg1, *seg2)
else:
raise ValueError("Couldn't work out which intersection function to use")
if not swapped:
return intersections
return [Intersection(pt=i.pt, t1=i.t2, t2=i.t1) for i in intersections]
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)