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# -*- 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 | |
EPSILON = 1e-9 | |
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", | |
] | |
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), | |
) | |
def _calcCubicArcLengthCRecurse(mult, p0, p1, p2, p3): | |
arch = abs(p0 - p3) | |
box = abs(p0 - p1) + abs(p1 - p2) + abs(p2 - p3) | |
if arch * mult + EPSILON >= box: | |
return (arch + box) * 0.5 | |
else: | |
one, two = _split_cubic_into_two(p0, p1, p2, p3) | |
return _calcCubicArcLengthCRecurse(mult, *one) + _calcCubicArcLengthCRecurse( | |
mult, *two | |
) | |
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 | |
def _dot(v1, v2): | |
return (v1 * v2.conjugate()).real | |
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 | |
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)) | |
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 | |
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)) | |
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 | |
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) | |
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) | |
) | |
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 | |
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) | |
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)] | |
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) | |
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) | |
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) | |
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) | |
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) | |
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 | |
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 | |
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 | |
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) | |
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) | |
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 | |
# | |
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)) | |
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) | |
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) | |
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 | |
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) | |
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) | |
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) | |
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 | |
] | |
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) | |