def _tValueForPointOnQuadCurve(point, pts, isHorizontal=0):
quadSegments = decomposeQuadraticSegment(pts[1:])
previousOnCurve = pts[0]
solutionsDict = dict()
for index, (pt1, pt2) in enumerate(quadSegments):
a, b, c = bezierTools.calcQuadraticParameters(previousOnCurve, pt1,
pt2)
subSolutions = bezierTools.solveQuadratic(
a[isHorizontal], b[isHorizontal],
c[isHorizontal] - point[isHorizontal])
subSolutions = [t for t in subSolutions if 0 <= t < 1]
for t in subSolutions:
solutionsDict[(t, index)] = _getQuadPoint(t, previousOnCurve, pt1,
pt2)
previousOnCurve = pt2
solutions = solutionsDict.keys()
if not solutions and not isHorizontal:
return _tValueForPointOnQuadCurve(point, pts, isHorizontal=1)
if len(solutions) > 1:
intersectionLenghts = {}
for t in solutions:
tp = solutionsDict[t]
dist = _distance(tp, point)
intersectionLenghts[dist] = t
minDist = min(intersectionLenghts.keys())
solutions = [intersectionLenghts[minDist]]
return solutions
def getExtremaForQuadratic(pt1, pt2, pt3, h=True, v=False): (ax, ay), (bx, by), c = calcQuadraticParameters(pt1, pt2, pt3) ax *= 2.0 ay *= 2.0 points = [] vectors = [] if h: roots = [t for t in solveLinear(ay, by) if 0 < t < 1] points, vectors = get_extrema_points_vectors_quad(roots, pt1, pt2, pt3) if v: roots = [t for t in solveLinear(ax, bx) if 0 < t < 1] v_points, v_vectors = get_extrema_points_vectors_quad(roots, pt1, pt2, pt3) points += v_points vectors += v_vectors return points, vectors
def getExtremaForQuadratic(pt1, pt2, pt3, h=True, v=False):
(ax, ay), (bx, by), c = calcQuadraticParameters(pt1, pt2, pt3)
ax *= 2.0
ay *= 2.0
points = []
vectors = []
if h:
roots = [t for t in solveLinear(ay, by) if 0 < t < 1]
points, vectors = get_extrema_points_vectors_quad(roots, pt1, pt2, pt3)
if v:
roots = [t for t in solveLinear(ax, bx) if 0 < t < 1]
v_points, v_vectors = get_extrema_points_vectors_quad(
roots, pt1, pt2, pt3)
points += v_points
vectors += v_vectors
return points, vectors
def qcurveIntersections(x1, y1, x2, y2, *pts):
"""
Computes intersection between a cubic spline and a line segment.
Adapted from: https://www.particleincell.com/2013/cubic-line-intersection/
Takes four defcon points describing curve and four scalars describing line
parameters.
"""
sol = []
# PACK for fontTools
points = []
for pt in pts:
points.append((pt.x, pt.y))
p1 = (pts[0].x, pts[0].y)
for p2, p3 in decomposeQuadraticSegment(points[1:]):
bx, by = (y1 - y2), (x2 - x1)
m = x1 * y2 - x2 * y1
a, b, c = bezierTools.calcQuadraticParameters(p1, p2, p3)
# prepare for next turn
p1 = p3
pc0 = bx * a[0] - by * a[1]
pc1 = (bx * b[0] + by * b[1]) / pc0
pc2 = (bx * c[0] + by * c[1] + m) / pc0
r = bezierTools.solveQuadratic(pc0, pc1, pc2)
for t in r:
if t < 0 or t > 1:
continue
s0 = a[0] * (1 - t) ** 2 + b[0] * 2 * t * (1 - t) + c[0] * t ** 2
s1 = a[1] * (1 - t) ** 2 + b[1] * 2 * t * (1 - t) + c[1] * t ** 2
if (x2 - x1) != 0:
s = (s0 - x1) / (x2 - x1)
else:
s = (s1 - y1) / (y2 - y1)
if s < 0 or s > 1:
continue
sol.append((s0, s1, t))
return sol
def _tValueForPointOnQuadCurve(point, pts, isHorizontal=0):
quadSegments = decomposeQuadraticSegment(pts[1:])
previousOnCurve = pts[0]
solutionsDict = dict()
for index, (pt1, pt2) in enumerate(quadSegments):
a, b, c = bezierTools.calcQuadraticParameters(previousOnCurve, pt1, pt2)
subSolutions = bezierTools.solveQuadratic(a[isHorizontal], b[isHorizontal], c[isHorizontal] - point[isHorizontal])
subSolutions = [t for t in subSolutions if 0 <= t < 1]
for t in subSolutions:
solutionsDict[(t, index)] = _getQuadPoint(t, previousOnCurve, pt1, pt2)
previousOnCurve = pt2
solutions = list(solutionsDict.keys())
if not solutions and not isHorizontal:
return _tValueForPointOnQuadCurve(point, pts, isHorizontal=1)
if len(solutions) > 1:
intersectionLenghts = {}
for t in solutions:
tp = solutionsDict[t]
dist = _distance(tp, point)
intersectionLenghts[dist] = t
minDist = min(intersectionLenghts.keys())
solutions = [intersectionLenghts[minDist]]
return solutions
def getExtremaForQuadratic(
pt0: Point,
pt1: Point,
pt2: Point,
h: bool = True,
v: bool = False,
include_start_end: bool = False,
) -> List[float]:
"""
Return a list of t values at which the quadratic curve defined by pt0, pt1,
pt2 has extrema.
:param h: Calculate extrema for horizontal derivative == 0 (= what type
designers call vertical extrema!).
:type h: bool
:param v: Calculate extrema for vertical derivative == 0 (= what type
designers call horizontal extrema!).
:type v: bool
:param include_start_end: Also calculate extrema that lie at the start or
end point of the curve.
:type include_start_end: bool
"""
(ax, ay), (bx, by), _c = calcQuadraticParameters(pt0, pt1, pt2)
ax *= 2.0
ay *= 2.0
roots = []
if include_start_end:
if h:
roots = [t for t in solveLinear(ay, by) if 0 <= t <= 1]
if v:
roots += [t for t in solveLinear(ax, bx) if 0 <= t <= 1]
else:
if h:
roots = [t for t in solveLinear(ay, by) if 0 < t < 1]
if v:
roots += [t for t in solveLinear(ax, bx) if 0 < t < 1]
return roots
pt1 = (100, 100)
pt2 = (200, 20)
pt3 = (30, 580)
pt4 = (153, 654)
rect = [20, 20, 100, 100]
## loops
c = 10000
print "(loop %s)" % c
print "with numpy:"
print "calcQuadraticParameters\t\t",
n = time.time()
for i in range(c):
bezierTools.calcQuadraticParameters(pt1, pt2, pt3)
print time.time() - n
print "calcBounds\t\t\t",
n = time.time()
for i in range(c):
arrayTools.calcBounds(
[pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3])
print time.time() - n
print "pointsInRect\t\t\t",
n = time.time()
for i in range(c):
arrayTools.pointsInRect(
[pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt4],
rect)
pt2 = (200, 20)
pt3 = (30, 580)
pt4 = (153, 654)
rect = [20, 20, 100, 100]
## loops
c = 10000
print "(loop %s)"%c
print "with numpy:"
print "calcQuadraticParameters\t\t",
n = time.time()
for i in range(c):
bezierTools.calcQuadraticParameters(pt1, pt2, pt3)
print time.time() - n
print "calcBounds\t\t\t",
n = time.time()
for i in range(c):
arrayTools.calcBounds([pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3])
print time.time() - n
print "pointsInRect\t\t\t",
n = time.time()
for i in range(c):
arrayTools.pointsInRect([pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt1, pt2, pt3, pt4], rect)
print time.time() - n
print "calcQuadraticBounds\t\t",
