bezierTools: Routines for working with Bezier curves
fontTools.misc.bezierTools.py – tools for working with Bezier path segments.
- fontTools.misc.bezierTools.approximateCubicArcLength(pt1, pt2, pt3, pt4)[source]
Approximates the arc length for a cubic Bezier segment.
Uses Gauss-Lobatto quadrature with n=5 points to approximate arc length. See
calcCubicArcLength()
for a slower but more accurate result.- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
pt3 – Control points of the Bezier as 2D tuples.
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
- fontTools.misc.bezierTools.approximateCubicArcLengthC(pt1, pt2, pt3, pt4)[source]
Approximates the arc length for a cubic Bezier segment.
- Parameters:
pt1 – Control points of the Bezier as complex numbers.
pt2 – Control points of the Bezier as complex numbers.
pt3 – Control points of the Bezier as complex numbers.
pt4 – Control points of the Bezier as complex numbers.
- Returns:
Arc length value.
- fontTools.misc.bezierTools.approximateQuadraticArcLength(pt1, pt2, pt3)[source]
Calculates the arc length for a quadratic Bezier segment.
Uses Gauss-Legendre quadrature for a branch-free approximation. See
calcQuadraticArcLength()
for a slower but more accurate result.- Parameters:
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.
- fontTools.misc.bezierTools.approximateQuadraticArcLengthC(pt1, pt2, pt3)[source]
Calculates the arc length for a quadratic Bezier segment.
Uses Gauss-Legendre quadrature for a branch-free approximation. See
calcQuadraticArcLength()
for a slower but more accurate result.- Parameters:
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.
- fontTools.misc.bezierTools.calcCubicArcLength(pt1, pt2, pt3, pt4, tolerance=0.005)[source]
Calculates the arc length for a cubic Bezier segment.
Whereas
approximateCubicArcLength()
approximates the length, this function calculates it by “measuring”, recursively dividing the curve until the divided segments are shorter thantolerance
.- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
pt3 – Control points of the Bezier as 2D tuples.
pt4 – Control points of the Bezier as 2D tuples.
tolerance – Controls the precision of the calcuation.
- Returns:
Arc length value.
- fontTools.misc.bezierTools.calcCubicArcLengthC(pt1, pt2, pt3, pt4, tolerance=0.005)[source]
Calculates the arc length for a cubic Bezier segment.
- Parameters:
pt1 – Control points of the Bezier as complex numbers.
pt2 – Control points of the Bezier as complex numbers.
pt3 – Control points of the Bezier as complex numbers.
pt4 – Control points of the Bezier as complex numbers.
tolerance – Controls the precision of the calcuation.
- Returns:
Arc length value.
- fontTools.misc.bezierTools.calcQuadraticArcLength(pt1, pt2, pt3)[source]
Calculates the arc length for a quadratic Bezier segment.
- Parameters:
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
- fontTools.misc.bezierTools.calcQuadraticArcLengthC(pt1, pt2, pt3)[source]
Calculates the arc length for a quadratic Bezier segment.
- Parameters:
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.
- fontTools.misc.bezierTools.calcCubicBounds(pt1, pt2, pt3, pt4)[source]
Calculates the bounding rectangle for a quadratic Bezier segment.
- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
pt3 – Control points of the Bezier as 2D tuples.
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
- fontTools.misc.bezierTools.calcQuadraticBounds(pt1, pt2, pt3)[source]
Calculates the bounding rectangle for a quadratic Bezier segment.
- Parameters:
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)
- fontTools.misc.bezierTools.splitLine(pt1, pt2, where, isHorizontal)[source]
Split a line at a given coordinate.
- Parameters:
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, thenwhere
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))
- fontTools.misc.bezierTools.splitQuadratic(pt1, pt2, pt3, where, isHorizontal)[source]
Split a quadratic Bezier curve at a given coordinate.
- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
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, thenwhere
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))
- fontTools.misc.bezierTools.splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal)[source]
Split a cubic Bezier curve at a given coordinate.
- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
pt3 – Control points of the Bezier as 2D tuples.
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, thenwhere
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))
- fontTools.misc.bezierTools.splitQuadraticAtT(pt1, pt2, pt3, *ts)[source]
Split a quadratic Bezier curve at one or more values of t.
- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
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))
- fontTools.misc.bezierTools.splitCubicAtT(pt1, pt2, pt3, pt4, *ts)[source]
Split a cubic Bezier curve at one or more values of t.
- Parameters:
pt1 – Control points of the Bezier as 2D tuples.
pt2 – Control points of the Bezier as 2D tuples.
pt3 – Control points of the Bezier as 2D tuples.
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))
- fontTools.misc.bezierTools.splitCubicAtTC(pt1, pt2, pt3, pt4, *ts)[source]
Split a cubic Bezier curve at one or more values of t.
- Parameters:
pt1 – Control points of the Bezier as complex numbers..
pt2 – Control points of the Bezier as complex numbers..
pt3 – Control points of the Bezier as complex numbers..
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).
- fontTools.misc.bezierTools.splitCubicIntoTwoAtTC(pt1, pt2, pt3, pt4, t)[source]
Split a cubic Bezier curve at t.
- Parameters:
pt1 – Control points of the Bezier as complex numbers.
pt2 – Control points of the Bezier as complex numbers.
pt3 – Control points of the Bezier as complex numbers.
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).
- fontTools.misc.bezierTools.solveQuadratic(a, b, c, sqrt=<built-in function sqrt>)[source]
Solve a quadratic equation.
Solves a*x*x + b*x + c = 0 where a, b and c are real.
- Parameters:
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!
- fontTools.misc.bezierTools.solveCubic(a, b, c, d)[source]
Solve a cubic equation.
Solves a*x*x*x + b*x*x + c*x + d = 0 where a, b, c and d are real.
- Parameters:
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
- fontTools.misc.bezierTools.quadraticPointAtT(pt1, pt2, pt3, t)[source]
Finds the point at time t on a quadratic curve.
- Parameters:
pt1 – Coordinates of the curve as 2D tuples.
pt2 – Coordinates of the curve as 2D tuples.
pt3 – Coordinates of the curve as 2D tuples.
t – The time along the curve.
- Returns:
A 2D tuple with the coordinates of the point.
- fontTools.misc.bezierTools.cubicPointAtT(pt1, pt2, pt3, pt4, t)[source]
Finds the point at time t on a cubic curve.
- Parameters:
pt1 – Coordinates of the curve as 2D tuples.
pt2 – Coordinates of the curve as 2D tuples.
pt3 – Coordinates of the curve as 2D tuples.
pt4 – Coordinates of the curve as 2D tuples.
t – The time along the curve.
- Returns:
A 2D tuple with the coordinates of the point.
- fontTools.misc.bezierTools.cubicPointAtTC(pt1, pt2, pt3, pt4, t)[source]
Finds the point at time t on a cubic curve.
- Parameters:
pt1 – Coordinates of the curve as complex numbers.
pt2 – Coordinates of the curve as complex numbers.
pt3 – Coordinates of the curve as complex numbers.
pt4 – Coordinates of the curve as complex numbers.
t – The time along the curve.
- Returns:
A complex number with the coordinates of the point.
- fontTools.misc.bezierTools.linePointAtT(pt1, pt2, t)[source]
Finds the point at time t on a line.
- Parameters:
pt1 – Coordinates of the line as 2D tuples.
pt2 – Coordinates of the line as 2D tuples.
t – The time along the line.
- Returns:
A 2D tuple with the coordinates of the point.
- fontTools.misc.bezierTools.lineLineIntersections(s1, e1, s2, e2)[source]
Finds intersections between two line segments.
- Parameters:
s1 – Coordinates of the first line as 2D tuples.
e1 – Coordinates of the first line as 2D tuples.
s2 – Coordinates of the second line as 2D tuples.
e2 – Coordinates of the second line as 2D tuples.
- Returns:
A list of
Intersection
objects, each object havingpt
,t1
andt2
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)
- fontTools.misc.bezierTools.curveLineIntersections(curve, line)[source]
Finds intersections between a curve and a line.
- Parameters:
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 havingpt
,t1
andt2
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)
- fontTools.misc.bezierTools.curveCurveIntersections(curve1, curve2)[source]
Finds intersections between a curve and a curve.
- Parameters:
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 havingpt
,t1
andt2
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)
- fontTools.misc.bezierTools.segmentSegmentIntersections(seg1, seg2)[source]
Finds intersections between two segments.
- Parameters:
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 havingpt
,t1
andt2
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)