diff options
Diffstat (limited to 'Lib/fontTools/misc/bezierTools.py')
| -rw-r--r-- | Lib/fontTools/misc/bezierTools.py | 638 |
1 files changed, 539 insertions, 99 deletions
diff --git a/Lib/fontTools/misc/bezierTools.py b/Lib/fontTools/misc/bezierTools.py index 659de34e..2cf2640c 100644 --- a/Lib/fontTools/misc/bezierTools.py +++ b/Lib/fontTools/misc/bezierTools.py @@ -2,9 +2,12 @@ """fontTools.misc.bezierTools.py -- tools for working with Bezier path segments. """ -from fontTools.misc.arrayTools import calcBounds -from fontTools.misc.py23 import * +from fontTools.misc.arrayTools import calcBounds, sectRect, rectArea +from fontTools.misc.transform import Identity import math +from collections import namedtuple + +Intersection = namedtuple("Intersection", ["pt", "t1", "t2"]) __all__ = [ @@ -25,6 +28,14 @@ __all__ = [ "splitCubicAtT", "solveQuadratic", "solveCubic", + "quadraticPointAtT", + "cubicPointAtT", + "linePointAtT", + "segmentPointAtT", + "lineLineIntersections", + "curveLineIntersections", + "curveCurveIntersections", + "segmentSegmentIntersections", ] @@ -42,23 +53,31 @@ def calcCubicArcLength(pt1, pt2, pt3, pt4, tolerance=0.005): Returns: Arc length value. """ - return calcCubicArcLengthC(complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4), tolerance) + return calcCubicArcLengthC( + complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4), tolerance + ) def _split_cubic_into_two(p0, p1, p2, p3): - mid = (p0 + 3 * (p1 + p2) + p3) * .125 - deriv3 = (p3 + p2 - p1 - p0) * .125 - return ((p0, (p0 + p1) * .5, mid - deriv3, mid), - (mid, mid + deriv3, (p2 + p3) * .5, p3)) + mid = (p0 + 3 * (p1 + p2) + p3) * 0.125 + deriv3 = (p3 + p2 - p1 - p0) * 0.125 + return ( + (p0, (p0 + p1) * 0.5, mid - deriv3, mid), + (mid, mid + deriv3, (p2 + p3) * 0.5, p3), + ) + def _calcCubicArcLengthCRecurse(mult, p0, p1, p2, p3): - arch = abs(p0-p3) - box = abs(p0-p1) + abs(p1-p2) + abs(p2-p3) - if arch * mult >= box: - return (arch + box) * .5 - else: - one,two = _split_cubic_into_two(p0,p1,p2,p3) - return _calcCubicArcLengthCRecurse(mult, *one) + _calcCubicArcLengthCRecurse(mult, *two) + arch = abs(p0 - p3) + box = abs(p0 - p1) + abs(p1 - p2) + abs(p2 - p3) + if arch * mult >= box: + return (arch + box) * 0.5 + else: + one, two = _split_cubic_into_two(p0, p1, p2, p3) + return _calcCubicArcLengthCRecurse(mult, *one) + _calcCubicArcLengthCRecurse( + mult, *two + ) + def calcCubicArcLengthC(pt1, pt2, pt3, pt4, tolerance=0.005): """Calculates the arc length for a cubic Bezier segment. @@ -70,7 +89,7 @@ def calcCubicArcLengthC(pt1, pt2, pt3, pt4, tolerance=0.005): Returns: Arc length value. """ - mult = 1. + 1.5 * tolerance # The 1.5 is a empirical hack; no math + mult = 1.0 + 1.5 * tolerance # The 1.5 is a empirical hack; no math return _calcCubicArcLengthCRecurse(mult, pt1, pt2, pt3, pt4) @@ -85,7 +104,7 @@ def _dot(v1, v2): def _intSecAtan(x): # In : sympy.integrate(sp.sec(sp.atan(x))) # Out: x*sqrt(x**2 + 1)/2 + asinh(x)/2 - return x * math.sqrt(x**2 + 1)/2 + math.asinh(x)/2 + return x * math.sqrt(x ** 2 + 1) / 2 + math.asinh(x) / 2 def calcQuadraticArcLength(pt1, pt2, pt3): @@ -141,16 +160,16 @@ def calcQuadraticArcLengthC(pt1, pt2, pt3): d = d1 - d0 n = d * 1j scale = abs(n) - if scale == 0.: - return abs(pt3-pt1) - origDist = _dot(n,d0) + if scale == 0.0: + return abs(pt3 - pt1) + origDist = _dot(n, d0) if abs(origDist) < epsilon: - if _dot(d0,d1) >= 0: - return abs(pt3-pt1) + if _dot(d0, d1) >= 0: + return abs(pt3 - pt1) a, b = abs(d0), abs(d1) - return (a*a + b*b) / (a+b) - x0 = _dot(d,d0) / origDist - x1 = _dot(d,d1) / origDist + return (a * a + b * b) / (a + b) + x0 = _dot(d, d0) / origDist + x1 = _dot(d, d1) / origDist Len = abs(2 * (_intSecAtan(x1) - _intSecAtan(x0)) * origDist / (scale * (x1 - x0))) return Len @@ -190,13 +209,17 @@ def approximateQuadraticArcLengthC(pt1, pt2, pt3): # to be integrated with the best-matching fifth-degree polynomial # approximation of it. # - #https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Legendre_quadrature + # https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Legendre_quadrature # abs(BezierCurveC[2].diff(t).subs({t:T})) for T in sorted(.5, .5±sqrt(3/5)/2), # weighted 5/18, 8/18, 5/18 respectively. - v0 = abs(-0.492943519233745*pt1 + 0.430331482911935*pt2 + 0.0626120363218102*pt3) - v1 = abs(pt3-pt1)*0.4444444444444444 - v2 = abs(-0.0626120363218102*pt1 - 0.430331482911935*pt2 + 0.492943519233745*pt3) + v0 = abs( + -0.492943519233745 * pt1 + 0.430331482911935 * pt2 + 0.0626120363218102 * pt3 + ) + v1 = abs(pt3 - pt1) * 0.4444444444444444 + v2 = abs( + -0.0626120363218102 * pt1 - 0.430331482911935 * pt2 + 0.492943519233745 * pt3 + ) return v0 + v1 + v2 @@ -220,14 +243,18 @@ def calcQuadraticBounds(pt1, pt2, pt3): (0.0, 0.0, 100, 100) """ (ax, ay), (bx, by), (cx, cy) = calcQuadraticParameters(pt1, pt2, pt3) - ax2 = ax*2.0 - ay2 = ay*2.0 + ax2 = ax * 2.0 + ay2 = ay * 2.0 roots = [] if ax2 != 0: - roots.append(-bx/ax2) + 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] + 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) @@ -256,7 +283,9 @@ def approximateCubicArcLength(pt1, pt2, pt3, pt4): >>> approximateCubicArcLength((0, 0), (50, 0), (100, -50), (-50, 0)) # cusp 154.80848416537057 """ - return approximateCubicArcLengthC(complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4)) + return approximateCubicArcLengthC( + complex(*pt1), complex(*pt2), complex(*pt3), complex(*pt4) + ) def approximateCubicArcLengthC(pt1, pt2, pt3, pt4): @@ -276,11 +305,21 @@ def approximateCubicArcLengthC(pt1, pt2, pt3, pt4): # abs(BezierCurveC[3].diff(t).subs({t:T})) for T in sorted(0, .5±(3/7)**.5/2, .5, 1), # weighted 1/20, 49/180, 32/90, 49/180, 1/20 respectively. - v0 = abs(pt2-pt1)*.15 - v1 = abs(-0.558983582205757*pt1 + 0.325650248872424*pt2 + 0.208983582205757*pt3 + 0.024349751127576*pt4) - v2 = abs(pt4-pt1+pt3-pt2)*0.26666666666666666 - v3 = abs(-0.024349751127576*pt1 - 0.208983582205757*pt2 - 0.325650248872424*pt3 + 0.558983582205757*pt4) - v4 = abs(pt4-pt3)*.15 + v0 = abs(pt2 - pt1) * 0.15 + v1 = abs( + -0.558983582205757 * pt1 + + 0.325650248872424 * pt2 + + 0.208983582205757 * pt3 + + 0.024349751127576 * pt4 + ) + v2 = abs(pt4 - pt1 + pt3 - pt2) * 0.26666666666666666 + v3 = abs( + -0.024349751127576 * pt1 + - 0.208983582205757 * pt2 + - 0.325650248872424 * pt3 + + 0.558983582205757 * pt4 + ) + v4 = abs(pt4 - pt3) * 0.15 return v0 + v1 + v2 + v3 + v4 @@ -313,7 +352,13 @@ def calcCubicBounds(pt1, pt2, pt3, pt4): 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] + 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) @@ -356,8 +401,8 @@ def splitLine(pt1, pt2, where, isHorizontal): pt1x, pt1y = pt1 pt2x, pt2y = pt2 - ax = (pt2x - pt1x) - ay = (pt2y - pt1y) + ax = pt2x - pt1x + ay = pt2y - pt1y bx = pt1x by = pt1y @@ -410,9 +455,10 @@ def splitQuadratic(pt1, pt2, pt3, where, isHorizontal): ((50, 50), (75, 50), (100, 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]) + 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) @@ -446,9 +492,10 @@ def splitCubic(pt1, pt2, pt3, pt4, where, isHorizontal): ((92.5259, 25), (95.202, 17.5085), (97.7062, 9.17517), (100, 1.77636e-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]) + 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) @@ -512,17 +559,17 @@ def _splitQuadraticAtT(a, b, c, *ts): cx, cy = c for i in range(len(ts) - 1): t1 = ts[i] - t2 = ts[i+1] - delta = (t2 - t1) + t2 = ts[i + 1] + delta = t2 - t1 # calc new a, b and c - delta_2 = delta*delta + delta_2 = delta * delta a1x = ax * delta_2 a1y = ay * delta_2 - b1x = (2*ax*t1 + bx) * delta - b1y = (2*ay*t1 + by) * delta - t1_2 = t1*t1 - c1x = ax*t1_2 + bx*t1 + cx - c1y = ay*t1_2 + by*t1 + cy + b1x = (2 * ax * t1 + bx) * delta + b1y = (2 * ay * t1 + by) * delta + t1_2 = t1 * t1 + 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)) @@ -540,24 +587,26 @@ def _splitCubicAtT(a, b, c, d, *ts): dx, dy = d for i in range(len(ts) - 1): t1 = ts[i] - t2 = ts[i+1] - delta = (t2 - t1) + t2 = ts[i + 1] + delta = t2 - t1 - delta_2 = delta*delta - delta_3 = delta*delta_2 - t1_2 = t1*t1 - t1_3 = t1*t1_2 + delta_2 = delta * delta + delta_3 = delta * delta_2 + t1_2 = t1 * t1 + t1_3 = t1 * t1_2 # 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)) + 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 @@ -569,8 +618,7 @@ def _splitCubicAtT(a, b, c, d, *ts): from math import sqrt, acos, cos, pi -def solveQuadratic(a, b, c, - sqrt=sqrt): +def solveQuadratic(a, b, c, sqrt=sqrt): """Solve a quadratic equation. Solves *a*x*x + b*x + c = 0* where a, b and c are real. @@ -590,13 +638,13 @@ def solveQuadratic(a, b, c, roots = [] else: # We have a linear equation with 1 root. - roots = [-c/b] + 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 + 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] + roots = [(-b + rDD) / 2.0 / a, (-b - rDD) / 2.0 / a] else: # complex roots, ignore roots = [] @@ -646,52 +694,52 @@ def solveCubic(a, b, c, d): # 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 + 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 + Q = (a1 * a1 - 3.0 * a2) / 9.0 + R = (2.0 * a1 * a1 * a1 - 9.0 * a1 * a2 + 27.0 * a3) / 54.0 - R2 = R*R - Q3 = Q*Q*Q + R2 = R * R + Q3 = Q * Q * Q R2 = 0 if R2 < epsilon else R2 Q3 = 0 if abs(Q3) < epsilon else Q3 R2_Q3 = R2 - Q3 - if R2 == 0. and Q3 == 0.: - x = round(-a1/3.0, epsilonDigits) + if R2 == 0.0 and Q3 == 0.0: + x = round(-a1 / 3.0, epsilonDigits) return [x, x, x] - elif R2_Q3 <= epsilon * .5: + elif R2_Q3 <= epsilon * 0.5: # The epsilon * .5 above ensures that Q3 is not zero. - theta = acos(max(min(R/sqrt(Q3), 1.0), -1.0)) - rQ2 = -2.0*sqrt(Q) - a1_3 = a1/3.0 - x0 = rQ2*cos(theta/3.0) - a1_3 - x1 = rQ2*cos((theta+2.0*pi)/3.0) - a1_3 - x2 = rQ2*cos((theta+4.0*pi)/3.0) - a1_3 + theta = acos(max(min(R / sqrt(Q3), 1.0), -1.0)) + rQ2 = -2.0 * sqrt(Q) + a1_3 = a1 / 3.0 + x0 = rQ2 * cos(theta / 3.0) - a1_3 + x1 = rQ2 * cos((theta + 2.0 * pi) / 3.0) - a1_3 + x2 = rQ2 * cos((theta + 4.0 * pi) / 3.0) - a1_3 x0, x1, x2 = sorted([x0, x1, x2]) # Merge roots that are close-enough if x1 - x0 < epsilon and x2 - x1 < epsilon: - x0 = x1 = x2 = round((x0 + x1 + x2) / 3., epsilonDigits) + x0 = x1 = x2 = round((x0 + x1 + x2) / 3.0, epsilonDigits) elif x1 - x0 < epsilon: - x0 = x1 = round((x0 + x1) / 2., epsilonDigits) + x0 = x1 = round((x0 + x1) / 2.0, epsilonDigits) x2 = round(x2, epsilonDigits) elif x2 - x1 < epsilon: x0 = round(x0, epsilonDigits) - x1 = x2 = round((x1 + x2) / 2., epsilonDigits) + x1 = x2 = round((x1 + x2) / 2.0, epsilonDigits) else: x0 = round(x0, epsilonDigits) x1 = round(x1, epsilonDigits) x2 = round(x2, epsilonDigits) return [x0, x1, x2] else: - x = pow(sqrt(R2_Q3)+abs(R), 1/3.0) - x = x + Q/x + x = pow(sqrt(R2_Q3) + abs(R), 1 / 3.0) + x = x + Q / x if R >= 0.0: x = -x - x = round(x - a1/3.0, epsilonDigits) + x = round(x - a1 / 3.0, epsilonDigits) return [x] @@ -699,6 +747,7 @@ def solveCubic(a, b, c, d): # Conversion routines for points to parameters and vice versa # + def calcQuadraticParameters(pt1, pt2, pt3): x2, y2 = pt2 x3, y3 = pt3 @@ -753,17 +802,406 @@ def calcCubicPoints(a, b, c, d): return (x1, y1), (x2, y2), (x3, y3), (x4, y4) +# +# Point at time +# + + +def linePointAtT(pt1, pt2, t): + """Finds the point at time `t` on a line. + + Args: + pt1, pt2: Coordinates of the line as 2D tuples. + t: The time along the line. + + Returns: + A 2D tuple with the coordinates of the point. + """ + return ((pt1[0] * (1 - t) + pt2[0] * t), (pt1[1] * (1 - t) + pt2[1] * t)) + + +def quadraticPointAtT(pt1, pt2, pt3, t): + """Finds the point at time `t` on a quadratic curve. + + Args: + pt1, pt2, pt3: Coordinates of the curve as 2D tuples. + t: The time along the curve. + + Returns: + A 2D tuple with the coordinates of the point. + """ + x = (1 - t) * (1 - t) * pt1[0] + 2 * (1 - t) * t * pt2[0] + t * t * pt3[0] + y = (1 - t) * (1 - t) * pt1[1] + 2 * (1 - t) * t * pt2[1] + t * t * pt3[1] + return (x, y) + + +def cubicPointAtT(pt1, pt2, pt3, pt4, t): + """Finds the point at time `t` on a cubic curve. + + Args: + pt1, pt2, pt3, pt4: Coordinates of the curve as 2D tuples. + t: The time along the curve. + + Returns: + A 2D tuple with the coordinates of the point. + """ + x = ( + (1 - t) * (1 - t) * (1 - t) * pt1[0] + + 3 * (1 - t) * (1 - t) * t * pt2[0] + + 3 * (1 - t) * t * t * pt3[0] + + t * t * t * pt4[0] + ) + y = ( + (1 - t) * (1 - t) * (1 - t) * pt1[1] + + 3 * (1 - t) * (1 - t) * t * pt2[1] + + 3 * (1 - t) * t * t * pt3[1] + + t * t * t * pt4[1] + ) + return (x, y) + + +def segmentPointAtT(seg, t): + if len(seg) == 2: + return linePointAtT(*seg, t) + elif len(seg) == 3: + return quadraticPointAtT(*seg, t) + elif len(seg) == 4: + return cubicPointAtT(*seg, t) + raise ValueError("Unknown curve degree") + + +# +# Intersection finders +# + + +def _line_t_of_pt(s, e, pt): + sx, sy = s + ex, ey = e + px, py = pt + if not math.isclose(sx, ex): + return (px - sx) / (ex - sx) + if not math.isclose(sy, ey): + return (py - sy) / (ey - sy) + # Line is a point! + return -1 + + +def _both_points_are_on_same_side_of_origin(a, b, origin): + xDiff = (a[0] - origin[0]) * (b[0] - origin[0]) + yDiff = (a[1] - origin[1]) * (b[1] - origin[1]) + return not (xDiff <= 0.0 and yDiff <= 0.0) + + +def lineLineIntersections(s1, e1, s2, e2): + """Finds intersections between two line segments. + + Args: + s1, e1: Coordinates of the first line as 2D tuples. + s2, 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[0] + >>> intersection.pt + (374.44882952482897, 313.73458370177315) + >>> (intersection.t1, intersection.t2) + (0.45069111555824454, 0.5408153767394238) + """ + s1x, s1y = s1 + e1x, e1y = e1 + s2x, s2y = s2 + e2x, e2y = e2 + if ( + math.isclose(s2x, e2x) and math.isclose(s1x, e1x) and not math.isclose(s1x, s2x) + ): # Parallel vertical + return [] + if ( + math.isclose(s2y, e2y) and math.isclose(s1y, e1y) and not math.isclose(s1y, s2y) + ): # Parallel horizontal + return [] + if math.isclose(s2x, e2x) and math.isclose(s2y, e2y): # Line segment is tiny + return [] + if math.isclose(s1x, e1x) and math.isclose(s1y, e1y): # Line segment is tiny + return [] + if math.isclose(e1x, s1x): + x = s1x + slope34 = (e2y - s2y) / (e2x - s2x) + y = slope34 * (x - s2x) + s2y + pt = (x, y) + return [ + Intersection( + pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt) + ) + ] + if math.isclose(s2x, e2x): + x = s2x + slope12 = (e1y - s1y) / (e1x - s1x) + y = slope12 * (x - s1x) + s1y + pt = (x, y) + return [ + Intersection( + pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt) + ) + ] + + slope12 = (e1y - s1y) / (e1x - s1x) + slope34 = (e2y - s2y) / (e2x - s2x) + if math.isclose(slope12, slope34): + return [] + x = (slope12 * s1x - s1y - slope34 * s2x + s2y) / (slope12 - slope34) + y = slope12 * (x - s1x) + s1y + pt = (x, y) + if _both_points_are_on_same_side_of_origin( + pt, e1, s1 + ) and _both_points_are_on_same_side_of_origin(pt, s2, e2): + return [ + Intersection( + pt=pt, t1=_line_t_of_pt(s1, e1, pt), t2=_line_t_of_pt(s2, e2, pt) + ) + ] + return [] + + +def _alignment_transformation(segment): + # Returns a transformation which aligns a segment horizontally at the + # origin. Apply this transformation to curves and root-find to find + # intersections with the segment. + start = segment[0] + end = segment[-1] + angle = math.atan2(end[1] - start[1], end[0] - start[0]) + return Identity.rotate(-angle).translate(-start[0], -start[1]) + + +def _curve_line_intersections_t(curve, line): + aligned_curve = _alignment_transformation(line).transformPoints(curve) + if len(curve) == 3: + a, b, c = calcQuadraticParameters(*aligned_curve) + intersections = solveQuadratic(a[1], b[1], c[1]) + elif len(curve) == 4: + a, b, c, d = calcCubicParameters(*aligned_curve) + intersections = solveCubic(a[1], b[1], c[1], d[1]) + else: + raise ValueError("Unknown curve degree") + return sorted(i for i in intersections if 0.0 <= i <= 1) + + +def curveLineIntersections(curve, line): + """Finds intersections between a curve and a line. + + Args: + 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[0].pt + (84.90010344084885, 189.87306176459828) + """ + if len(curve) == 3: + pointFinder = quadraticPointAtT + elif len(curve) == 4: + pointFinder = cubicPointAtT + else: + raise ValueError("Unknown curve degree") + intersections = [] + for t in _curve_line_intersections_t(curve, line): + pt = pointFinder(*curve, t) + intersections.append(Intersection(pt=pt, t1=t, t2=_line_t_of_pt(*line, pt))) + return intersections + + +def _curve_bounds(c): + if len(c) == 3: + return calcQuadraticBounds(*c) + elif len(c) == 4: + return calcCubicBounds(*c) + raise ValueError("Unknown curve degree") + + +def _split_segment_at_t(c, t): + if len(c) == 2: + s, e = c + midpoint = linePointAtT(s, e, t) + return [(s, midpoint), (midpoint, e)] + if len(c) == 3: + return splitQuadraticAtT(*c, t) + elif len(c) == 4: + return splitCubicAtT(*c, t) + raise ValueError("Unknown curve degree") + + +def _curve_curve_intersections_t( + curve1, curve2, precision=1e-3, range1=None, range2=None +): + bounds1 = _curve_bounds(curve1) + bounds2 = _curve_bounds(curve2) + + if not range1: + range1 = (0.0, 1.0) + if not range2: + range2 = (0.0, 1.0) + + # If bounds don't intersect, go home + intersects, _ = sectRect(bounds1, bounds2) + if not intersects: + return [] + + def midpoint(r): + return 0.5 * (r[0] + r[1]) + + # If they do overlap but they're tiny, approximate + if rectArea(bounds1) < precision and rectArea(bounds2) < precision: + return [(midpoint(range1), midpoint(range2))] + + c11, c12 = _split_segment_at_t(curve1, 0.5) + c11_range = (range1[0], midpoint(range1)) + c12_range = (midpoint(range1), range1[1]) + + c21, c22 = _split_segment_at_t(curve2, 0.5) + c21_range = (range2[0], midpoint(range2)) + c22_range = (midpoint(range2), range2[1]) + + found = [] + found.extend( + _curve_curve_intersections_t( + c11, c21, precision, range1=c11_range, range2=c21_range + ) + ) + found.extend( + _curve_curve_intersections_t( + c12, c21, precision, range1=c12_range, range2=c21_range + ) + ) + found.extend( + _curve_curve_intersections_t( + c11, c22, precision, range1=c11_range, range2=c22_range + ) + ) + found.extend( + _curve_curve_intersections_t( + c12, c22, precision, range1=c12_range, range2=c22_range + ) + ) + + unique_key = lambda ts: (int(ts[0] / precision), int(ts[1] / precision)) + seen = set() + unique_values = [] + + for ts in found: + key = unique_key(ts) + if key in seen: + continue + seen.add(key) + unique_values.append(ts) + + return unique_values + + +def curveCurveIntersections(curve1, curve2): + """Finds intersections between a curve and a curve. + + Args: + 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[0].pt + (81.7831487395506, 109.88904552375288) + """ + intersection_ts = _curve_curve_intersections_t(curve1, curve2) + return [ + Intersection(pt=segmentPointAtT(curve1, ts[0]), t1=ts[0], t2=ts[1]) + for ts in intersection_ts + ] + + +def segmentSegmentIntersections(seg1, seg2): + """Finds intersections between two segments. + + Args: + 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[0].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[0].pt + (84.90010344084885, 189.87306176459828) + + """ + # Arrange by degree + swapped = False + if len(seg2) > len(seg1): + seg2, seg1 = seg1, seg2 + swapped = True + if len(seg1) > 2: + if len(seg2) > 2: + intersections = curveCurveIntersections(seg1, seg2) + else: + intersections = curveLineIntersections(seg1, seg2) + elif len(seg1) == 2 and len(seg2) == 2: + intersections = lineLineIntersections(*seg1, *seg2) + else: + raise ValueError("Couldn't work out which intersection function to use") + if not swapped: + return intersections + return [Intersection(pt=i.pt, t1=i.t2, t2=i.t1) for i in intersections] + + def _segmentrepr(obj): """ - >>> _segmentrepr([1, [2, 3], [], [[2, [3, 4], [0.1, 2.2]]]]) - '(1, (2, 3), (), ((2, (3, 4), (0.1, 2.2))))' + >>> _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 "%g" % obj else: - return "(%s)" % ", ".join([_segmentrepr(x) for x in it]) + return "(%s)" % ", ".join(_segmentrepr(x) for x in it) def printSegments(segments): @@ -773,7 +1211,9 @@ def printSegments(segments): for segment in segments: print(_segmentrepr(segment)) + if __name__ == "__main__": import sys import doctest + sys.exit(doctest.testmod().failed) |