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406 lines
12 KiB
Python
406 lines
12 KiB
Python
# cython: language_level=3
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# distutils: define_macros=CYTHON_TRACE_NOGIL=1
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# Copyright 2023 Google Inc. All Rights Reserved.
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# Copyright 2023 Behdad Esfahbod. 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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try:
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import cython
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except (AttributeError, ImportError):
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# if cython not installed, use mock module with no-op decorators and types
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from fontTools.misc import cython
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COMPILED = cython.compiled
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from fontTools.misc.bezierTools import splitCubicAtTC
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from collections import namedtuple
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import math
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from typing import (
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List,
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Tuple,
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Union,
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)
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__all__ = ["quadratic_to_curves"]
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# Copied from cu2qu
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@cython.cfunc
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@cython.returns(cython.int)
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@cython.locals(
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tolerance=cython.double,
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p0=cython.complex,
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p1=cython.complex,
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p2=cython.complex,
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p3=cython.complex,
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)
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@cython.locals(mid=cython.complex, deriv3=cython.complex)
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def cubic_farthest_fit_inside(p0, p1, p2, p3, tolerance):
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"""Check if a cubic Bezier lies within a given distance of the origin.
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"Origin" means *the* origin (0,0), not the start of the curve. Note that no
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checks are made on the start and end positions of the curve; this function
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only checks the inside of the curve.
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Args:
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p0 (complex): Start point of curve.
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p1 (complex): First handle of curve.
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p2 (complex): Second handle of curve.
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p3 (complex): End point of curve.
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tolerance (double): Distance from origin.
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Returns:
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bool: True if the cubic Bezier ``p`` entirely lies within a distance
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``tolerance`` of the origin, False otherwise.
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"""
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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) * 0.125
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if abs(mid) > tolerance:
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return False
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deriv3 = (p3 + p2 - p1 - p0) * 0.125
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return cubic_farthest_fit_inside(
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p0, (p0 + p1) * 0.5, mid - deriv3, mid, tolerance
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) and cubic_farthest_fit_inside(mid, mid + deriv3, (p2 + p3) * 0.5, p3, tolerance)
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@cython.locals(
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p0=cython.complex,
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p1=cython.complex,
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p2=cython.complex,
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p1_2_3=cython.complex,
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)
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def elevate_quadratic(p0, p1, p2):
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"""Given a quadratic bezier curve, return its degree-elevated cubic."""
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# https://pomax.github.io/bezierinfo/#reordering
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p1_2_3 = p1 * (2 / 3)
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return (
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p0,
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(p0 * (1 / 3) + p1_2_3),
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(p2 * (1 / 3) + p1_2_3),
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p2,
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)
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@cython.cfunc
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@cython.locals(
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start=cython.int,
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n=cython.int,
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k=cython.int,
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prod_ratio=cython.double,
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sum_ratio=cython.double,
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ratio=cython.double,
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t=cython.double,
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p0=cython.complex,
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p1=cython.complex,
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p2=cython.complex,
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p3=cython.complex,
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)
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def merge_curves(curves, start, n):
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"""Give a cubic-Bezier spline, reconstruct one cubic-Bezier
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that has the same endpoints and tangents and approxmates
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the spline."""
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# Reconstruct the t values of the cut segments
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prod_ratio = 1.0
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sum_ratio = 1.0
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ts = [1]
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for k in range(1, n):
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ck = curves[start + k]
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c_before = curves[start + k - 1]
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# |t_(k+1) - t_k| / |t_k - t_(k - 1)| = ratio
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assert ck[0] == c_before[3]
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ratio = abs(ck[1] - ck[0]) / abs(c_before[3] - c_before[2])
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prod_ratio *= ratio
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sum_ratio += prod_ratio
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ts.append(sum_ratio)
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# (t(n) - t(n - 1)) / (t_(1) - t(0)) = prod_ratio
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ts = [t / sum_ratio for t in ts[:-1]]
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p0 = curves[start][0]
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p1 = curves[start][1]
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p2 = curves[start + n - 1][2]
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p3 = curves[start + n - 1][3]
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# Build the curve by scaling the control-points.
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p1 = p0 + (p1 - p0) / (ts[0] if ts else 1)
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p2 = p3 + (p2 - p3) / ((1 - ts[-1]) if ts else 1)
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curve = (p0, p1, p2, p3)
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return curve, ts
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@cython.locals(
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count=cython.int,
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num_offcurves=cython.int,
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i=cython.int,
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off1=cython.complex,
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off2=cython.complex,
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on=cython.complex,
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)
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def add_implicit_on_curves(p):
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q = list(p)
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count = 0
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num_offcurves = len(p) - 2
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for i in range(1, num_offcurves):
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off1 = p[i]
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off2 = p[i + 1]
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on = off1 + (off2 - off1) * 0.5
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q.insert(i + 1 + count, on)
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count += 1
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return q
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Point = Union[Tuple[float, float], complex]
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@cython.locals(
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cost=cython.int,
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is_complex=cython.int,
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)
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def quadratic_to_curves(
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quads: List[List[Point]],
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max_err: float = 0.5,
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all_cubic: bool = False,
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) -> List[Tuple[Point, ...]]:
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"""Converts a connecting list of quadratic splines to a list of quadratic
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and cubic curves.
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A quadratic spline is specified as a list of points. Either each point is
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a 2-tuple of X,Y coordinates, or each point is a complex number with
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real/imaginary components representing X,Y coordinates.
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The first and last points are on-curve points and the rest are off-curve
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points, with an implied on-curve point in the middle between every two
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consequtive off-curve points.
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Returns:
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The output is a list of tuples of points. Points are represented
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in the same format as the input, either as 2-tuples or complex numbers.
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Each tuple is either of length three, for a quadratic curve, or four,
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for a cubic curve. Each curve's last point is the same as the next
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curve's first point.
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Args:
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quads: quadratic splines
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max_err: absolute error tolerance; defaults to 0.5
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all_cubic: if True, only cubic curves are generated; defaults to False
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"""
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is_complex = type(quads[0][0]) is complex
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if not is_complex:
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quads = [[complex(x, y) for (x, y) in p] for p in quads]
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q = [quads[0][0]]
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costs = [1]
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cost = 1
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for p in quads:
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assert q[-1] == p[0]
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for i in range(len(p) - 2):
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cost += 1
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costs.append(cost)
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costs.append(cost)
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qq = add_implicit_on_curves(p)[1:]
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costs.pop()
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q.extend(qq)
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cost += 1
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costs.append(cost)
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curves = spline_to_curves(q, costs, max_err, all_cubic)
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if not is_complex:
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curves = [tuple((c.real, c.imag) for c in curve) for curve in curves]
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return curves
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Solution = namedtuple("Solution", ["num_points", "error", "start_index", "is_cubic"])
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@cython.locals(
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i=cython.int,
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j=cython.int,
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k=cython.int,
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start=cython.int,
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i_sol_count=cython.int,
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j_sol_count=cython.int,
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this_sol_count=cython.int,
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tolerance=cython.double,
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err=cython.double,
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error=cython.double,
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i_sol_error=cython.double,
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j_sol_error=cython.double,
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all_cubic=cython.int,
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is_cubic=cython.int,
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count=cython.int,
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p0=cython.complex,
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p1=cython.complex,
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p2=cython.complex,
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p3=cython.complex,
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v=cython.complex,
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u=cython.complex,
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)
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def spline_to_curves(q, costs, tolerance=0.5, all_cubic=False):
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"""
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q: quadratic spline with alternating on-curve / off-curve points.
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costs: cumulative list of encoding cost of q in terms of number of
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points that need to be encoded. Implied on-curve points do not
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contribute to the cost. If all points need to be encoded, then
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costs will be range(1, len(q)+1).
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"""
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assert len(q) >= 3, "quadratic spline requires at least 3 points"
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# Elevate quadratic segments to cubic
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elevated_quadratics = [
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elevate_quadratic(*q[i : i + 3]) for i in range(0, len(q) - 2, 2)
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]
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# Find sharp corners; they have to be oncurves for sure.
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forced = set()
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for i in range(1, len(elevated_quadratics)):
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p0 = elevated_quadratics[i - 1][2]
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p1 = elevated_quadratics[i][0]
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p2 = elevated_quadratics[i][1]
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if abs(p1 - p0) + abs(p2 - p1) > tolerance + abs(p2 - p0):
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forced.add(i)
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# Dynamic-Programming to find the solution with fewest number of
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# cubic curves, and within those the one with smallest error.
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sols = [Solution(0, 0, 0, False)]
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impossible = Solution(len(elevated_quadratics) * 3 + 1, 0, 1, False)
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start = 0
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for i in range(1, len(elevated_quadratics) + 1):
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best_sol = impossible
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for j in range(start, i):
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j_sol_count, j_sol_error = sols[j].num_points, sols[j].error
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if not all_cubic:
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# Solution with quadratics between j:i
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this_count = costs[2 * i - 1] - costs[2 * j] + 1
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i_sol_count = j_sol_count + this_count
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i_sol_error = j_sol_error
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i_sol = Solution(i_sol_count, i_sol_error, i - j, False)
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if i_sol < best_sol:
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best_sol = i_sol
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if this_count <= 3:
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# Can't get any better than this in the path below
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continue
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# Fit elevated_quadratics[j:i] into one cubic
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try:
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curve, ts = merge_curves(elevated_quadratics, j, i - j)
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except ZeroDivisionError:
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continue
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# Now reconstruct the segments from the fitted curve
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reconstructed_iter = splitCubicAtTC(*curve, *ts)
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reconstructed = []
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# Knot errors
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error = 0
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for k, reconst in enumerate(reconstructed_iter):
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orig = elevated_quadratics[j + k]
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err = abs(reconst[3] - orig[3])
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error = max(error, err)
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if error > tolerance:
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break
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reconstructed.append(reconst)
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if error > tolerance:
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# Not feasible
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continue
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# Interior errors
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for k, reconst in enumerate(reconstructed):
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orig = elevated_quadratics[j + k]
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p0, p1, p2, p3 = tuple(v - u for v, u in zip(reconst, orig))
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if not cubic_farthest_fit_inside(p0, p1, p2, p3, tolerance):
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error = tolerance + 1
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break
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if error > tolerance:
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# Not feasible
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continue
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# Save best solution
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i_sol_count = j_sol_count + 3
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i_sol_error = max(j_sol_error, error)
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i_sol = Solution(i_sol_count, i_sol_error, i - j, True)
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if i_sol < best_sol:
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best_sol = i_sol
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if i_sol_count == 3:
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# Can't get any better than this
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break
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sols.append(best_sol)
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if i in forced:
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start = i
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# Reconstruct solution
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splits = []
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cubic = []
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i = len(sols) - 1
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while i:
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count, is_cubic = sols[i].start_index, sols[i].is_cubic
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splits.append(i)
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cubic.append(is_cubic)
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i -= count
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curves = []
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j = 0
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for i, is_cubic in reversed(list(zip(splits, cubic))):
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if is_cubic:
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curves.append(merge_curves(elevated_quadratics, j, i - j)[0])
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else:
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for k in range(j, i):
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curves.append(q[k * 2 : k * 2 + 3])
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j = i
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return curves
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def main():
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from fontTools.cu2qu.benchmark import generate_curve
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from fontTools.cu2qu import curve_to_quadratic
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tolerance = 0.05
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reconstruct_tolerance = tolerance * 1
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curve = generate_curve()
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quadratics = curve_to_quadratic(curve, tolerance)
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print(
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"cu2qu tolerance %g. qu2cu tolerance %g." % (tolerance, reconstruct_tolerance)
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)
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print("One random cubic turned into %d quadratics." % len(quadratics))
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curves = quadratic_to_curves([quadratics], reconstruct_tolerance)
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print("Those quadratics turned back into %d cubics. " % len(curves))
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print("Original curve:", curve)
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print("Reconstructed curve(s):", curves)
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if __name__ == "__main__":
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main()
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