266 lines
8.0 KiB
Python
266 lines
8.0 KiB
Python
# Copyright 2015 Google Inc. All Rights Reserved.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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from __future__ import print_function, division, absolute_import
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__all__ = ['curve_to_quadratic', 'curves_to_quadratic']
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MAX_N = 100
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class Cu2QuError(Exception):
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pass
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class ApproxNotFoundError(Cu2QuError):
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def __init__(self, curve):
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message = "no approximation found: %s" % curve
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super(Cu2QuError, self).__init__(message)
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self.curve = curve
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def dot(v1, v2):
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"""Return the dot product of two vectors."""
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return (v1 * v2.conjugate()).real
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def calc_cubic_points(a, b, c, d):
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_1 = d
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_2 = (c / 3.0) + d
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_3 = (b + c) / 3.0 + _2
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_4 = a + d + c + b
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return _1, _2, _3, _4
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def calc_cubic_parameters(p0, p1, p2, p3):
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c = (p1 - p0) * 3.0
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b = (p2 - p1) * 3.0 - c
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d = p0
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a = p3 - d - c - b
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return a, b, c, d
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def split_cubic_into_n_iter(p0, p1, p2, p3, n):
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# Hand-coded special-cases
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if n == 2:
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return iter(split_cubic_into_two(p0, p1, p2, p3))
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if n == 3:
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return iter(split_cubic_into_three(p0, p1, p2, p3))
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if n == 4:
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a, b = split_cubic_into_two(p0, p1, p2, p3)
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return iter(split_cubic_into_two(*a) + split_cubic_into_two(*b))
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if n == 6:
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a, b = split_cubic_into_two(p0, p1, p2, p3)
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return iter(split_cubic_into_three(*a) + split_cubic_into_three(*b))
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return _split_cubic_into_n_gen(p0,p1,p2,p3,n)
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def _split_cubic_into_n_gen(p0, p1, p2, p3, n):
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a, b, c, d = calc_cubic_parameters(p0, p1, p2, p3)
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dt = 1 / n
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delta_2 = dt * dt
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delta_3 = dt * delta_2
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for i in range(n):
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t1 = i * dt
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t1_2 = t1 * 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) * dt
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d1 = a*t1*t1_2 + b*t1_2 + c*t1 + d
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yield calc_cubic_points(a1, b1, c1, d1)
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def split_cubic_into_two(p0, p1, p2, p3):
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mid = (p0 + 3 * (p1 + p2) + p3) * .125
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deriv3 = (p3 + p2 - p1 - p0) * .125
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return ((p0, (p0 + p1) * .5, mid - deriv3, mid),
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(mid, mid + deriv3, (p2 + p3) * .5, p3))
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def split_cubic_into_three(p0, p1, p2, p3, _27=1/27):
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# we define 1/27 as a keyword argument so that it will be evaluated only
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# once but still in the scope of this function
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mid1 = (8*p0 + 12*p1 + 6*p2 + p3) * _27
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deriv1 = (p3 + 3*p2 - 4*p0) * _27
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mid2 = (p0 + 6*p1 + 12*p2 + 8*p3) * _27
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deriv2 = (4*p3 - 3*p1 - p0) * _27
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return ((p0, (2*p0 + p1) / 3, mid1 - deriv1, mid1),
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(mid1, mid1 + deriv1, mid2 - deriv2, mid2),
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(mid2, mid2 + deriv2, (p2 + 2*p3) / 3, p3))
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def cubic_approx_control(p, t):
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"""Approximate a cubic bezier curve with a quadratic one.
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Returns the candidate control point."""
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p1 = p[0] + (p[1] - p[0]) * 1.5
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p2 = p[3] + (p[2] - p[3]) * 1.5
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return p1 + (p2 - p1) * t
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def calc_intersect(a, b, c, d):
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"""Calculate the intersection of ab and cd, given a, b, c, d."""
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ab = b - a
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cd = d - c
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p = ab * 1j
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try:
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h = dot(p, a - c) / dot(p, cd)
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except ZeroDivisionError:
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return None
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return c + cd * h
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def cubic_farthest_fit_inside(p0, p1, p2, p3, tolerance):
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"""Returns True if the cubic Bezier p entirely lies within a distance
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tolerance of origin, False otherwise. Assumes that p0 and p3 do fit
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within tolerance of origin, and just checks the inside of the curve."""
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# First check p2 then p1, as p2 has higher error early on.
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if abs(p2) <= tolerance and abs(p1) <= tolerance:
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return True
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# Split.
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mid = (p0 + 3 * (p1 + p2) + p3) * .125
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if abs(mid) > tolerance:
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return False
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deriv3 = (p3 + p2 - p1 - p0) * .125
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return (cubic_farthest_fit_inside(p0, (p0+p1)*.5, mid-deriv3, mid, tolerance) and
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cubic_farthest_fit_inside(mid, mid+deriv3, (p2+p3)*.5, p3, tolerance))
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def cubic_approx_quadratic(cubic, tolerance, _2_3=2/3):
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"""Return the uniq quadratic approximating cubic that maintains
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endpoint tangents if that is within tolerance, None otherwise."""
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# we define 2/3 as a keyword argument so that it will be evaluated only
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# once but still in the scope of this function
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q1 = calc_intersect(*cubic)
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if q1 is None:
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return None
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c0 = cubic[0]
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c3 = cubic[3]
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c1 = c0 + (q1 - c0) * _2_3
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c2 = c3 + (q1 - c3) * _2_3
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if not cubic_farthest_fit_inside(0,
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c1 - cubic[1],
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c2 - cubic[2],
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0, tolerance):
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return None
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return c0, q1, c3
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def cubic_approx_spline(cubic, n, tolerance, _2_3=2/3):
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"""Approximate a cubic bezier curve with a spline of n quadratics.
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Returns None if no quadratic approximation is found which lies entirely
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within a distance `tolerance` from the original curve.
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"""
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# we define 2/3 as a keyword argument so that it will be evaluated only
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# once but still in the scope of this function
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if n == 1:
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return cubic_approx_quadratic(cubic, tolerance)
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cubics = split_cubic_into_n_iter(cubic[0], cubic[1], cubic[2], cubic[3], n)
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# calculate the spline of quadratics and check errors at the same time.
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next_cubic = next(cubics)
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next_q1 = cubic_approx_control(next_cubic, 0)
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q2 = cubic[0]
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d1 = 0j
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spline = [cubic[0], next_q1]
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for i in range(1, n+1):
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# Current cubic to convert
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c0, c1, c2, c3 = next_cubic
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# Current quadratic approximation of current cubic
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q0 = q2
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q1 = next_q1
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if i < n:
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next_cubic = next(cubics)
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next_q1 = cubic_approx_control(next_cubic, i / (n-1))
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spline.append(next_q1)
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q2 = (q1 + next_q1) * .5
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else:
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q2 = c3
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# End-point deltas
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d0 = d1
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d1 = q2 - c3
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if (abs(d1) > tolerance or
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not cubic_farthest_fit_inside(d0,
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q0 + (q1 - q0) * _2_3 - c1,
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q2 + (q1 - q2) * _2_3 - c2,
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d1,
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tolerance)):
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return None
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spline.append(cubic[3])
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return spline
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def curve_to_quadratic(curve, max_err):
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"""Return a quadratic spline approximating this cubic bezier.
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Raise 'ApproxNotFoundError' if no suitable approximation can be found
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with the given parameters.
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"""
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curve = [complex(*p) for p in curve]
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for n in range(1, MAX_N + 1):
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spline = cubic_approx_spline(curve, n, max_err)
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if spline is not None:
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# done. go home
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return [(s.real, s.imag) for s in spline]
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raise ApproxNotFoundError(curve)
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def curves_to_quadratic(curves, max_errors):
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"""Return quadratic splines approximating these cubic beziers.
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Raise 'ApproxNotFoundError' if no suitable approximation can be found
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for all curves with the given parameters.
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"""
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curves = [[complex(*p) for p in curve] for curve in curves]
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assert len(max_errors) == len(curves)
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l = len(curves)
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splines = [None] * l
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last_i = i = 0
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n = 1
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while True:
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spline = cubic_approx_spline(curves[i], n, max_errors[i])
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if spline is None:
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if n == MAX_N:
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break
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n += 1
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last_i = i
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continue
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splines[i] = spline
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i = (i + 1) % l
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if i == last_i:
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# done. go home
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return [[(s.real, s.imag) for s in spline] for spline in splines]
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raise ApproxNotFoundError(curves)
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