| """fontTools.misc.bezierTools.py -- tools for working with bezier path segments. |
| """ |
| |
| from __future__ import print_function, division, absolute_import |
| from fontTools.misc.py23 import * |
| |
| __all__ = [ |
| "calcQuadraticBounds", |
| "calcCubicBounds", |
| "splitLine", |
| "splitQuadratic", |
| "splitCubic", |
| "splitQuadraticAtT", |
| "splitCubicAtT", |
| "solveQuadratic", |
| "solveCubic", |
| ] |
| |
| from fontTools.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) |
| >>> 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 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, 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 |
| |
| if ax == 0: |
| return [(pt1, pt2)] |
| |
| t = (where - (bx, by)[isHorizontal]) / ax |
| 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 = 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 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 = 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 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): |
| """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() |