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Diffstat (limited to 'misc/pylib/robofab/misc/bezierTools.py')
-rw-r--r-- | misc/pylib/robofab/misc/bezierTools.py | 416 |
1 files changed, 0 insertions, 416 deletions
diff --git a/misc/pylib/robofab/misc/bezierTools.py b/misc/pylib/robofab/misc/bezierTools.py deleted file mode 100644 index 9872060b1..000000000 --- a/misc/pylib/robofab/misc/bezierTools.py +++ /dev/null @@ -1,416 +0,0 @@ -"""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() |