# 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.

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.

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 than `tolerance`.

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.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
```

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
```

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.

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.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.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 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.pt
(81.7831487395506, 109.88904552375288)
```
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 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.pt
(84.9000930760723, 189.87306176459828)
```
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 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
>>> intersection.pt
(374.44882952482897, 313.73458370177315)
>>> (intersection.t1, intersection.t2)
(0.45069111555824465, 0.5408153767394238)
```
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.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.segmentPointAtT(seg, t)[source]
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 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.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.pt
(84.9000930760723, 189.87306176459828)
```
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

• b – coefficient of

• 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.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

• 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.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, 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))
```
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.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, 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))
```
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, 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))
```
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))
```