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-"""fontTools.misc.bezierTools.py -- tools for working with bezier path segments.
-Rewritten to elimate the numpy dependency
-"""
-
-
-__all__ = [
- "calcQuadraticBounds",
- "calcCubicBounds",
- "splitLine",
- "splitQuadratic",
- "splitCubic",
- "splitQuadraticAtT",
- "splitCubicAtT",
- "solveQuadratic",
- "solveCubic",
-]
-
-from robofab.misc.arrayTools import calcBounds
-
-epsilon = 1e-12
-
-
-def calcQuadraticBounds(pt1, pt2, pt3):
- """Return the bounding rectangle for a qudratic bezier segment.
- pt1 and pt3 are the "anchor" points, pt2 is the "handle".
-
- >>> calcQuadraticBounds((0, 0), (50, 100), (100, 0))
- (0, 0, 100, 50.0)
- >>> calcQuadraticBounds((0, 0), (100, 0), (100, 100))
- (0.0, 0.0, 100, 100)
- """
- (ax, ay), (bx, by), (cx, cy) = calcQuadraticParameters(pt1, pt2, pt3)
- ax2 = ax*2.0
- ay2 = ay*2.0
- roots = []
- if ax2 != 0:
- roots.append(-bx/ax2)
- if ay2 != 0:
- roots.append(-by/ay2)
- points = [(ax*t*t + bx*t + cx, ay*t*t + by*t + cy) for t in roots if 0 <= t < 1] + [pt1, pt3]
- return calcBounds(points)
-
-
-def calcCubicBounds(pt1, pt2, pt3, pt4):
- """Return the bounding rectangle for a cubic bezier segment.
- pt1 and pt4 are the "anchor" points, pt2 and pt3 are the "handles".
-
- >>> calcCubicBounds((0, 0), (25, 100), (75, 100), (100, 0))
- (0, 0, 100, 75.0)
- >>> calcCubicBounds((0, 0), (50, 0), (100, 50), (100, 100))
- (0.0, 0.0, 100, 100)
- >>> calcCubicBounds((50, 0), (0, 100), (100, 100), (50, 0))
- (35.566243270259356, 0, 64.43375672974068, 75.0)
- """
- (ax, ay), (bx, by), (cx, cy), (dx, dy) = calcCubicParameters(pt1, pt2, pt3, pt4)
- # calc first derivative
- ax3 = ax * 3.0
- ay3 = ay * 3.0
- bx2 = bx * 2.0
- by2 = by * 2.0
- xRoots = [t for t in solveQuadratic(ax3, bx2, cx) if 0 <= t < 1]
- yRoots = [t for t in solveQuadratic(ay3, by2, cy) if 0 <= t < 1]
- roots = xRoots + yRoots
-
- points = [(ax*t*t*t + bx*t*t + cx * t + dx, ay*t*t*t + by*t*t + cy * t + dy) for t in roots] + [pt1, pt4]
- return calcBounds(points)
-
-
-def splitLine(pt1, pt2, where, isHorizontal):
- """Split the line between pt1 and pt2 at position 'where', which
- is an x coordinate if isHorizontal is False, a y coordinate if
- isHorizontal is True. Return a list of two line segments if the
- line was successfully split, or a list containing the original
- line.
-
- >>> printSegments(splitLine((0, 0), (100, 200), 50, False))
- ((0, 0), (50.0, 100.0))
- ((50.0, 100.0), (100, 200))
- >>> printSegments(splitLine((0, 0), (100, 200), 50, True))
- ((0, 0), (25.0, 50.0))
- ((25.0, 50.0), (100, 200))
- >>> printSegments(splitLine((0, 0), (100, 100), 50, True))
- ((0, 0), (50.0, 50.0))
- ((50.0, 50.0), (100, 100))
- >>> printSegments(splitLine((0, 0), (100, 100), 100, True))
- ((0, 0), (100, 100))
- >>> printSegments(splitLine((0, 0), (100, 100), 0, True))
- ((0, 0), (0.0, 0.0))
- ((0.0, 0.0), (100, 100))
- >>> printSegments(splitLine((0, 0), (100, 100), 0, False))
- ((0, 0), (0.0, 0.0))
- ((0.0, 0.0), (100, 100))
- """
- pt1x, pt1y = pt1
- pt2x, pt2y = pt2
-
- ax = (pt2x - pt1x)
- ay = (pt2y - pt1y)
-
- bx = pt1x
- by = pt1y
-
- ax1 = (ax, ay)[isHorizontal]
-
- if ax1 == 0:
- return [(pt1, pt2)]
-
- t = float(where - (bx, by)[isHorizontal]) / ax1
- if 0 <= t < 1:
- midPt = ax * t + bx, ay * t + by
- return [(pt1, midPt), (midPt, pt2)]
- else:
- return [(pt1, pt2)]
-
-
-def splitQuadratic(pt1, pt2, pt3, where, isHorizontal):
- """Split the quadratic curve between pt1, pt2 and pt3 at position 'where',
- which is an x coordinate if isHorizontal is False, a y coordinate if
- isHorizontal is True. Return a list of curve segments.
-
- >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 150, False))
- ((0, 0), (50, 100), (100, 0))
- >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, False))
- ((0.0, 0.0), (25.0, 50.0), (50.0, 50.0))
- ((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
- >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, False))
- ((0.0, 0.0), (12.5, 25.0), (25.0, 37.5))
- ((25.0, 37.5), (62.5, 75.0), (100.0, 0.0))
- >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 25, True))
- ((0.0, 0.0), (7.32233047034, 14.6446609407), (14.6446609407, 25.0))
- ((14.6446609407, 25.0), (50.0, 75.0), (85.3553390593, 25.0))
- ((85.3553390593, 25.0), (92.6776695297, 14.6446609407), (100.0, -7.1054273576e-15))
- >>> # XXX I'm not at all sure if the following behavior is desirable:
- >>> printSegments(splitQuadratic((0, 0), (50, 100), (100, 0), 50, True))
- ((0.0, 0.0), (25.0, 50.0), (50.0, 50.0))
- ((50.0, 50.0), (50.0, 50.0), (50.0, 50.0))
- ((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
- """
- a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
- solutions = solveQuadratic(a[isHorizontal], b[isHorizontal],
- c[isHorizontal] - where)
- solutions = [t for t in solutions if 0 <= t < 1]
- solutions.sort()
- if not solutions:
- return [(pt1, pt2, pt3)]
- return _splitQuadraticAtT(a, b, c, *solutions)
-
-
-def splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal):
- """Split the cubic curve between pt1, pt2, pt3 and pt4 at position 'where',
- which is an x coordinate if isHorizontal is False, a y coordinate if
- isHorizontal is True. Return a list of curve segments.
-
- >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 150, False))
- ((0, 0), (25, 100), (75, 100), (100, 0))
- >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 50, False))
- ((0.0, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
- ((50.0, 75.0), (68.75, 75.0), (87.5, 50.0), (100.0, 0.0))
- >>> printSegments(splitCubic((0, 0), (25, 100), (75, 100), (100, 0), 25, True))
- ((0.0, 0.0), (2.2937927384, 9.17517095361), (4.79804488188, 17.5085042869), (7.47413641001, 25.0))
- ((7.47413641001, 25.0), (31.2886200204, 91.6666666667), (68.7113799796, 91.6666666667), (92.52586359, 25.0))
- ((92.52586359, 25.0), (95.2019551181, 17.5085042869), (97.7062072616, 9.17517095361), (100.0, 1.7763568394e-15))
- """
- a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
- solutions = solveCubic(a[isHorizontal], b[isHorizontal], c[isHorizontal],
- d[isHorizontal] - where)
- solutions = [t for t in solutions if 0 <= t < 1]
- solutions.sort()
- if not solutions:
- return [(pt1, pt2, pt3, pt4)]
- return _splitCubicAtT(a, b, c, d, *solutions)
-
-
-def splitQuadraticAtT(pt1, pt2, pt3, *ts):
- """Split the quadratic curve between pt1, pt2 and pt3 at one or more
- values of t. Return a list of curve segments.
-
- >>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5))
- ((0.0, 0.0), (25.0, 50.0), (50.0, 50.0))
- ((50.0, 50.0), (75.0, 50.0), (100.0, 0.0))
- >>> printSegments(splitQuadraticAtT((0, 0), (50, 100), (100, 0), 0.5, 0.75))
- ((0.0, 0.0), (25.0, 50.0), (50.0, 50.0))
- ((50.0, 50.0), (62.5, 50.0), (75.0, 37.5))
- ((75.0, 37.5), (87.5, 25.0), (100.0, 0.0))
- """
- a, b, c = calcQuadraticParameters(pt1, pt2, pt3)
- return _splitQuadraticAtT(a, b, c, *ts)
-
-
-def splitCubicAtT(pt1, pt2, pt3, pt4, *ts):
- """Split the cubic curve between pt1, pt2, pt3 and pt4 at one or more
- values of t. Return a list of curve segments.
-
- >>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5))
- ((0.0, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
- ((50.0, 75.0), (68.75, 75.0), (87.5, 50.0), (100.0, 0.0))
- >>> printSegments(splitCubicAtT((0, 0), (25, 100), (75, 100), (100, 0), 0.5, 0.75))
- ((0.0, 0.0), (12.5, 50.0), (31.25, 75.0), (50.0, 75.0))
- ((50.0, 75.0), (59.375, 75.0), (68.75, 68.75), (77.34375, 56.25))
- ((77.34375, 56.25), (85.9375, 43.75), (93.75, 25.0), (100.0, 0.0))
- """
- a, b, c, d = calcCubicParameters(pt1, pt2, pt3, pt4)
- return _splitCubicAtT(a, b, c, d, *ts)
-
-
-def _splitQuadraticAtT(a, b, c, *ts):
- ts = list(ts)
- segments = []
- ts.insert(0, 0.0)
- ts.append(1.0)
- ax, ay = a
- bx, by = b
- cx, cy = c
- for i in range(len(ts) - 1):
- t1 = ts[i]
- t2 = ts[i+1]
- delta = (t2 - t1)
- # calc new a, b and c
- a1x = ax * delta**2
- a1y = ay * delta**2
- b1x = (2*ax*t1 + bx) * delta
- b1y = (2*ay*t1 + by) * delta
- c1x = ax*t1**2 + bx*t1 + cx
- c1y = ay*t1**2 + by*t1 + cy
-
- pt1, pt2, pt3 = calcQuadraticPoints((a1x, a1y), (b1x, b1y), (c1x, c1y))
- segments.append((pt1, pt2, pt3))
- return segments
-
-
-def _splitCubicAtT(a, b, c, d, *ts):
- ts = list(ts)
- ts.insert(0, 0.0)
- ts.append(1.0)
- segments = []
- ax, ay = a
- bx, by = b
- cx, cy = c
- dx, dy = d
- for i in range(len(ts) - 1):
- t1 = ts[i]
- t2 = ts[i+1]
- delta = (t2 - t1)
- # calc new a, b, c and d
- a1x = ax * delta**3
- a1y = ay * delta**3
- b1x = (3*ax*t1 + bx) * delta**2
- b1y = (3*ay*t1 + by) * delta**2
- c1x = (2*bx*t1 + cx + 3*ax*t1**2) * delta
- c1y = (2*by*t1 + cy + 3*ay*t1**2) * delta
- d1x = ax*t1**3 + bx*t1**2 + cx*t1 + dx
- d1y = ay*t1**3 + by*t1**2 + cy*t1 + dy
- pt1, pt2, pt3, pt4 = calcCubicPoints((a1x, a1y), (b1x, b1y), (c1x, c1y), (d1x, d1y))
- segments.append((pt1, pt2, pt3, pt4))
- return segments
-
-
-#
-# Equation solvers.
-#
-
-from math import sqrt, acos, cos, pi
-
-
-def solveQuadratic(a, b, c,
- sqrt=sqrt):
- """Solve a quadratic equation where a, b and c are real.
- a*x*x + b*x + c = 0
- This function returns a list of roots. Note that the returned list
- is neither guaranteed to be sorted nor to contain unique values!
- """
- if abs(a) < epsilon:
- if abs(b) < epsilon:
- # We have a non-equation; therefore, we have no valid solution
- roots = []
- else:
- # We have a linear equation with 1 root.
- roots = [-c/b]
- else:
- # We have a true quadratic equation. Apply the quadratic formula to find two roots.
- DD = b*b - 4.0*a*c
- if DD >= 0.0:
- rDD = sqrt(DD)
- roots = [(-b+rDD)/2.0/a, (-b-rDD)/2.0/a]
- else:
- # complex roots, ignore
- roots = []
- return roots
-
-
-def solveCubic(a, b, c, d,
- abs=abs, pow=pow, sqrt=sqrt, cos=cos, acos=acos, pi=pi):
- """Solve a cubic equation where a, b, c and d are real.
- a*x*x*x + b*x*x + c*x + d = 0
- This function returns a list of roots. Note that the returned list
- is neither guaranteed to be sorted nor to contain unique values!
- """
- #
- # adapted from:
- # CUBIC.C - Solve a cubic polynomial
- # public domain by Ross Cottrell
- # found at: http://www.strangecreations.com/library/snippets/Cubic.C
- #
- if abs(a) < epsilon:
- # don't just test for zero; for very small values of 'a' solveCubic()
- # returns unreliable results, so we fall back to quad.
- return solveQuadratic(b, c, d)
- a = float(a)
- a1 = b/a
- a2 = c/a
- a3 = d/a
-
- Q = (a1*a1 - 3.0*a2)/9.0
- R = (2.0*a1*a1*a1 - 9.0*a1*a2 + 27.0*a3)/54.0
- R2_Q3 = R*R - Q*Q*Q
-
- if R2_Q3 < 0:
- theta = acos(R/sqrt(Q*Q*Q))
- rQ2 = -2.0*sqrt(Q)
- x0 = rQ2*cos(theta/3.0) - a1/3.0
- x1 = rQ2*cos((theta+2.0*pi)/3.0) - a1/3.0
- x2 = rQ2*cos((theta+4.0*pi)/3.0) - a1/3.0
- return [x0, x1, x2]
- else:
- if Q == 0 and R == 0:
- x = 0
- else:
- x = pow(sqrt(R2_Q3)+abs(R), 1/3.0)
- x = x + Q/x
- if R >= 0.0:
- x = -x
- x = x - a1/3.0
- return [x]
-
-
-#
-# Conversion routines for points to parameters and vice versa
-#
-
-def calcQuadraticParameters(pt1, pt2, pt3):
- x2, y2 = pt2
- x3, y3 = pt3
- cx, cy = pt1
- bx = (x2 - cx) * 2.0
- by = (y2 - cy) * 2.0
- ax = x3 - cx - bx
- ay = y3 - cy - by
- return (ax, ay), (bx, by), (cx, cy)
-
-
-def calcCubicParameters(pt1, pt2, pt3, pt4):
- x2, y2 = pt2
- x3, y3 = pt3
- x4, y4 = pt4
- dx, dy = pt1
- cx = (x2 -dx) * 3.0
- cy = (y2 -dy) * 3.0
- bx = (x3 - x2) * 3.0 - cx
- by = (y3 - y2) * 3.0 - cy
- ax = x4 - dx - cx - bx
- ay = y4 - dy - cy - by
- return (ax, ay), (bx, by), (cx, cy), (dx, dy)
-
-
-def calcQuadraticPoints(a, b, c):
- ax, ay = a
- bx, by = b
- cx, cy = c
- x1 = cx
- y1 = cy
- x2 = (bx * 0.5) + cx
- y2 = (by * 0.5) + cy
- x3 = ax + bx + cx
- y3 = ay + by + cy
- return (x1, y1), (x2, y2), (x3, y3)
-
-
-def calcCubicPoints(a, b, c, d):
- ax, ay = a
- bx, by = b
- cx, cy = c
- dx, dy = d
- x1 = dx
- y1 = dy
- x2 = (cx / 3.0) + dx
- y2 = (cy / 3.0) + dy
- x3 = (bx + cx) / 3.0 + x2
- y3 = (by + cy) / 3.0 + y2
- x4 = ax + dx + cx + bx
- y4 = ay + dy + cy + by
- return (x1, y1), (x2, y2), (x3, y3), (x4, y4)
-
-
-def _segmentrepr(obj):
- """
- >>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], [0.1, 2.2]]]])
- '(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))'
- """
- try:
- it = iter(obj)
- except TypeError:
- return str(obj)
- else:
- return "(%s)" % ", ".join([_segmentrepr(x) for x in it])
-
-
-def printSegments(segments):
- """Helper for the doctests, displaying each segment in a list of
- segments on a single line as a tuple.
- """
- for segment in segments:
- print _segmentrepr(segment)
-
-if __name__ == "__main__":
- import doctest
- doctest.testmod()