new module bezierTools.py
git-svn-id: svn://svn.code.sf.net/p/fonttools/code/trunk@396 4cde692c-a291-49d1-8350-778aa11640f8
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Lib/fontTools/misc/bezierTools.py
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Lib/fontTools/misc/bezierTools.py
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"""fontTools.misc.bezierTools.py -- tools for working with bezier path segments."""
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__all__ = ["calcQuadraticBounds", "calcCubicBounds", "splitLine", "splitQuadratic",
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"splitCubic", "solveQuadratic", "solveCubic"]
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from fontTools.misc.arrayTools import calcBounds
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import Numeric
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def calcQuadraticBounds(pt1, pt2, pt3):
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"""Return the bounding rectangle for a qudratic bezier segment.
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pt1 and pt3 are the "anchor" points, pt2 is the "handle"."""
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# convert points to Numeric arrays
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pt1, pt2, pt3 = Numeric.array((pt1, pt2, pt3))
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# calc quadratic parameters
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c = pt1
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b = (pt2 - c) * 2.0
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a = pt3 - c - b
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# calc first derivative
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ax, ay = a * 2
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bx, by = b
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roots = []
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if ax != 0:
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roots.append(-bx/ax)
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if ay != 0:
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roots.append(-by/ay)
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points = [a*t*t + b*t + c for t in roots if 0 <= t < 1] + [pt1, pt3]
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return calcBounds(points)
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def calcCubicBounds(pt1, pt2, pt3, pt4):
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"""Return the bounding rectangle for a cubic bezier segment.
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pt1 and pt4 are the "anchor" points, pt2 and pt3 are the "handles"."""
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# convert points to Numeric arrays
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pt1, pt2, pt3, pt4 = Numeric.array((pt1, pt2, pt3, pt4))
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# calc cubic parameters
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d = pt1
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c = (pt2 - d) * 3.0
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b = (pt3 - pt2) * 3.0 - c
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a = pt4 - d - c - b
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# calc first derivative
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ax, ay = a * 3.0
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bx, by = b * 2.0
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cx, cy = c
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xRoots = [t for t in solveQuadratic(ax, bx, cx) if 0 <= t < 1]
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yRoots = [t for t in solveQuadratic(ay, by, cy) if 0 <= t < 1]
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roots = xRoots + yRoots
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points = [(a*t*t*t + b*t*t + c * t + d) for t in roots] + [pt1, pt4]
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return calcBounds(points)
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def splitLine(pt1, pt2, where, isHorizontal):
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"""Split the line between pt1 and pt2 at position 'where', which
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is an x coordinate if isHorizontal is False, a y coordinate if
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isHorizontal is True. Return a list of two line segments if the
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line was successfully split, or a list containing the original
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line."""
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pt1, pt2 = Numeric.array((pt1, pt2))
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a = (pt2 - pt1)
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b = pt1
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ax = a[isHorizontal]
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if ax == 0:
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return [(pt1, pt2)]
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t = float(where - b[isHorizontal]) / ax
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midPt = a * t + b
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return [(pt1, midPt), (midPt, pt2)]
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def splitQuadratic(pt1, pt2, pt3, where, isHorizontal):
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"""Split the quadratic curve between pt1, pt2 and pt3 at position 'where',
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which is an x coordinate if isHorizontal is False, a y coordinate if
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isHorizontal is True. Return a list of curve segments."""
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pt1, pt2, pt3 = Numeric.array((pt1, pt2, pt3))
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c = pt1
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b = (pt2 - c) * 2.0
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a = pt3 - c - b
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solutions = solveQuadratic(a[isHorizontal], b[isHorizontal],
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c[isHorizontal] - where)
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solutions = [t for t in solutions if 0 <= t < 1]
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solutions.sort()
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if not solutions:
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return [(pt1, pt2, pt3)]
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segments = []
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solutions.insert(0, 0.0)
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solutions.append(1.0)
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for i in range(len(solutions) - 1):
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t1 = solutions[i]
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t2 = solutions[i+1]
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delta = (t2 - t1)
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# calc new a, b and c
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a1 = a * delta**2
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b1 = (2*a*t1 + b) * delta
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c1 = a*t1**2 + b*t1 + c
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# calc new points
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pt1 = c1
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pt2 = (b1 * 0.5) + c1
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pt3 = a1 + b1 + c1
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segments.append((pt1, pt2, pt3))
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return segments
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def splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal):
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"""Split the cubic curve between pt1, pt2, pt3 and pt4 at position 'where',
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which is an x coordinate if isHorizontal is False, a y coordinate if
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isHorizontal is True. Return a list of curve segments."""
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pt1, pt2, pt3, pt4 = Numeric.array((pt1, pt2, pt3, pt4))
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d = pt1
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c = (pt2 - d) * 3.0
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b = (pt3 - pt2) * 3.0 - c
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a = pt4 - d - c - b
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solutions = solveCubic(a[isHorizontal], b[isHorizontal], c[isHorizontal],
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d[isHorizontal] - where)
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solutions = [t for t in solutions if 0 <= t < 1]
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solutions.sort()
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if not solutions:
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return [(pt1, pt2, pt3, pt4)]
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segments = []
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solutions.insert(0, 0.0)
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solutions.append(1.0)
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for i in range(len(solutions) - 1):
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t1 = solutions[i]
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t2 = solutions[i+1]
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delta = (t2 - t1)
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# calc new a, b, c and d
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a1 = a * delta**3
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b1 = (3*a*t1 + b) * delta**2
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c1 = (2*b*t1 + c + 3*a*t1**2) * delta
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d1 = a*t1**3 + b*t1**2 + c*t1 + d
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# calc new points
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pt1 = d1
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pt2 = (c1 / 3.0) + d1
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pt3 = (b1 + c1) / 3.0 + pt2
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pt4 = a1 + d1 + c1 + b1
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segments.append((pt1, pt2, pt3, pt4))
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return segments
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#
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# Equation solvers.
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#
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from math import sqrt, acos, cos, pi
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def solveQuadratic(a, b, c,
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sqrt=sqrt):
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"""Solve a quadratic equation where a, b and c are real.
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a*x*x + b*x + c = 0
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This function returns a list of roots.
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"""
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if a == 0.0:
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if b == 0.0:
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# We have a non-equation; therefore, we have no valid solution
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roots = []
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else:
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# We have a linear equation with 1 root.
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roots = [-c/b]
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else:
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# We have a true quadratic equation. Apply the quadratic formula to find two roots.
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DD = b*b - 4.0*a*c
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if DD >= 0.0:
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roots = [(-b+sqrt(DD))/2.0/a, (-b-sqrt(DD))/2.0/a]
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else:
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# complex roots, ignore
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roots = []
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return roots
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def solveCubic(a, b, c, d,
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abs=abs, pow=pow, sqrt=sqrt, cos=cos, acos=acos, pi=pi):
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"""Solve a cubic equation where a, b, c and d are real.
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a*x*x*x + b*x*x + c*x + d = 0
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This function returns a list of roots.
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"""
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#
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# adapted from:
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# CUBIC.C - Solve a cubic polynomial
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# public domain by Ross Cottrell
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# found at: http://www.strangecreations.com/library/snippets/Cubic.C
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#
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if abs(a) < 1e-6:
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# don't just test for zero; for very small values of 'a' solveCubic()
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# returns unreliable results, so we fall back to quad.
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return solveQuadratic(b, c, d)
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a1 = b/a
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a2 = c/a
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a3 = d/a
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Q = (a1*a1 - 3.0*a2)/9.0
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R = (2.0*a1*a1*a1 - 9.0*a1*a2 + 27.0*a3)/54.0
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R2_Q3 = R*R - Q*Q*Q
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if R2_Q3 <= 0:
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theta = acos(R/sqrt(Q*Q*Q))
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x0 = -2.0*sqrt(Q)*cos(theta/3.0) - a1/3.0
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x1 = -2.0*sqrt(Q)*cos((theta+2.0*pi)/3.0) - a1/3.0
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x2 = -2.0*sqrt(Q)*cos((theta+4.0*pi)/3.0) - a1/3.0
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return [x0, x1, x2]
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else:
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x = pow(sqrt(R2_Q3)+abs(R), 1/3.0)
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x = x + Q/x
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if R >= 0.0:
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x = -x
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x = x - a1/3.0
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return [x]
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