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497 lines
16 KiB
497 lines
16 KiB
#cython: language_level=3
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#distutils: define_macros=CYTHON_TRACE_NOGIL=1
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# 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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try:
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import cython
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except 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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import math
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from .errors import Error as Cu2QuError, ApproxNotFoundError
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__all__ = ['curve_to_quadratic', 'curves_to_quadratic']
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MAX_N = 100
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NAN = float("NaN")
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if cython.compiled:
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# Yep, I'm compiled.
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COMPILED = True
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else:
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# Just a lowly interpreted script.
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COMPILED = False
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@cython.cfunc
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@cython.inline
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@cython.returns(cython.double)
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@cython.locals(v1=cython.complex, v2=cython.complex)
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def dot(v1, v2):
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"""Return the dot product of two vectors.
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Args:
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v1 (complex): First vector.
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v2 (complex): Second vector.
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Returns:
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double: Dot product.
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"""
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return (v1 * v2.conjugate()).real
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@cython.cfunc
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@cython.inline
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@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
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@cython.locals(_1=cython.complex, _2=cython.complex, _3=cython.complex, _4=cython.complex)
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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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@cython.cfunc
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@cython.inline
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@cython.locals(p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex)
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@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
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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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@cython.cfunc
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@cython.locals(p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex)
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def split_cubic_into_n_iter(p0, p1, p2, p3, n):
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"""Split a cubic Bezier into n equal parts.
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Splits the curve into `n` equal parts by curve time.
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(t=0..1/n, t=1/n..2/n, ...)
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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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Returns:
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An iterator yielding the control points (four complex values) of the
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subcurves.
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"""
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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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@cython.locals(p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex, n=cython.int)
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@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
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@cython.locals(dt=cython.double, delta_2=cython.double, delta_3=cython.double, i=cython.int)
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@cython.locals(a1=cython.complex, b1=cython.complex, c1=cython.complex, d1=cython.complex)
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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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@cython.locals(p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex)
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@cython.locals(mid=cython.complex, deriv3=cython.complex)
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def split_cubic_into_two(p0, p1, p2, p3):
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"""Split a cubic Bezier into two equal parts.
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Splits the curve into two equal parts at t = 0.5
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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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Returns:
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tuple: Two cubic Beziers (each expressed as a tuple of four complex
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values).
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"""
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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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@cython.locals(p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex, _27=cython.double)
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@cython.locals(mid1=cython.complex, deriv1=cython.complex, mid2=cython.complex, deriv2=cython.complex)
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def split_cubic_into_three(p0, p1, p2, p3, _27=1/27):
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"""Split a cubic Bezier into three equal parts.
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Splits the curve into three equal parts at t = 1/3 and t = 2/3
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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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Returns:
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tuple: Three cubic Beziers (each expressed as a tuple of four complex
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values).
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"""
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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) / 3.0, mid1 - deriv1, mid1),
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(mid1, mid1 + deriv1, mid2 - deriv2, mid2),
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(mid2, mid2 + deriv2, (p2 + 2*p3) / 3.0, p3))
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@cython.returns(cython.complex)
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@cython.locals(t=cython.double, p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex)
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@cython.locals(_p1=cython.complex, _p2=cython.complex)
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def cubic_approx_control(t, p0, p1, p2, p3):
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"""Approximate a cubic Bezier using a quadratic one.
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Args:
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t (double): Position of control point.
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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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Returns:
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complex: Location of candidate control point on quadratic curve.
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"""
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_p1 = p0 + (p1 - p0) * 1.5
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_p2 = p3 + (p2 - p3) * 1.5
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return _p1 + (_p2 - _p1) * t
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@cython.returns(cython.complex)
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@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
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@cython.locals(ab=cython.complex, cd=cython.complex, p=cython.complex, h=cython.double)
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def calc_intersect(a, b, c, d):
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"""Calculate the intersection of two lines.
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Args:
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a (complex): Start point of first line.
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b (complex): End point of first line.
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c (complex): Start point of second line.
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d (complex): End point of second line.
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Returns:
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complex: Location of intersection if one present, ``complex(NaN,NaN)``
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if no intersection was found.
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"""
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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 complex(NAN, NAN)
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return c + cd * h
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@cython.cfunc
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@cython.returns(cython.int)
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@cython.locals(tolerance=cython.double, p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex)
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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) * .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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@cython.cfunc
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@cython.locals(tolerance=cython.double, _2_3=cython.double)
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@cython.locals(q1=cython.complex, c0=cython.complex, c1=cython.complex, c2=cython.complex, c3=cython.complex)
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def cubic_approx_quadratic(cubic, tolerance, _2_3=2/3):
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"""Approximate a cubic Bezier with a single quadratic within a given tolerance.
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Args:
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cubic (sequence): Four complex numbers representing control points of
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the cubic Bezier curve.
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tolerance (double): Permitted deviation from the original curve.
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Returns:
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Three complex numbers representing control points of the quadratic
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curve if it fits within the given tolerance, or ``None`` if no suitable
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curve could be calculated.
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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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q1 = calc_intersect(*cubic)
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if math.isnan(q1.imag):
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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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@cython.cfunc
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@cython.locals(n=cython.int, tolerance=cython.double, _2_3=cython.double)
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@cython.locals(i=cython.int)
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@cython.locals(c0=cython.complex, c1=cython.complex, c2=cython.complex, c3=cython.complex)
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@cython.locals(q0=cython.complex, q1=cython.complex, next_q1=cython.complex, q2=cython.complex, d1=cython.complex)
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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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Args:
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cubic (sequence): Four complex numbers representing control points of
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the cubic Bezier curve.
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n (int): Number of quadratic Bezier curves in the spline.
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tolerance (double): Permitted deviation from the original curve.
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Returns:
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A list of ``n+2`` complex numbers, representing control points of the
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quadratic spline if it fits within the given tolerance, or ``None`` if
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no suitable spline could be calculated.
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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(0, *next_cubic)
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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(i / (n-1), *next_cubic)
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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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@cython.locals(max_err=cython.double)
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@cython.locals(n=cython.int)
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def curve_to_quadratic(curve, max_err):
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"""Approximate a cubic Bezier curve with a spline of n quadratics.
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Args:
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cubic (sequence): Four 2D tuples representing control points of
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the cubic Bezier curve.
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max_err (double): Permitted deviation from the original curve.
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Returns:
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A list of 2D tuples, representing control points of the quadratic
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spline if it fits within the given tolerance, or ``None`` if no
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suitable spline could be calculated.
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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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@cython.locals(l=cython.int, last_i=cython.int, i=cython.int)
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def curves_to_quadratic(curves, max_errors):
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"""Return quadratic Bezier splines approximating the input cubic Beziers.
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Args:
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curves: A sequence of *n* curves, each curve being a sequence of four
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2D tuples.
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max_errors: A sequence of *n* floats representing the maximum permissible
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deviation from each of the cubic Bezier curves.
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Example::
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>>> curves_to_quadratic( [
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... [ (50,50), (100,100), (150,100), (200,50) ],
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... [ (75,50), (120,100), (150,75), (200,60) ]
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... ], [1,1] )
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[[(50.0, 50.0), (75.0, 75.0), (125.0, 91.66666666666666), (175.0, 75.0), (200.0, 50.0)], [(75.0, 50.0), (97.5, 75.0), (135.41666666666666, 82.08333333333333), (175.0, 67.5), (200.0, 60.0)]]
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The returned splines have "implied oncurve points" suitable for use in
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TrueType ``glif`` outlines - i.e. in the first spline returned above,
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the first quadratic segment runs from (50,50) to
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( (75 + 125)/2 , (120 + 91.666..)/2 ) = (100, 83.333...).
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Returns:
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A list of splines, each spline being a list of 2D tuples.
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Raises:
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fontTools.cu2qu.Errors.ApproxNotFoundError: if no suitable approximation
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can be found 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
|
|
return [[(s.real, s.imag) for s in spline] for spline in splines]
|
|
|
|
raise ApproxNotFoundError(curves)
|
|
|
|
|
|
if __name__ == '__main__':
|
|
import random
|
|
import timeit
|
|
|
|
MAX_ERR = 5
|
|
|
|
def generate_curve():
|
|
return [
|
|
tuple(float(random.randint(0, 2048)) for coord in range(2))
|
|
for point in range(4)]
|
|
|
|
def setup_curve_to_quadratic():
|
|
return generate_curve(), MAX_ERR
|
|
|
|
def setup_curves_to_quadratic():
|
|
num_curves = 3
|
|
return (
|
|
[generate_curve() for curve in range(num_curves)],
|
|
[MAX_ERR] * num_curves)
|
|
|
|
def run_benchmark(
|
|
benchmark_module, module, function, setup_suffix='', repeat=5, number=1000):
|
|
setup_func = 'setup_' + function
|
|
if setup_suffix:
|
|
print('%s with %s:' % (function, setup_suffix), end='')
|
|
setup_func += '_' + setup_suffix
|
|
else:
|
|
print('%s:' % function, end='')
|
|
|
|
def wrapper(function, setup_func):
|
|
function = globals()[function]
|
|
setup_func = globals()[setup_func]
|
|
def wrapped():
|
|
return function(*setup_func())
|
|
return wrapped
|
|
results = timeit.repeat(wrapper(function, setup_func), repeat=repeat, number=number)
|
|
print('\t%5.1fus' % (min(results) * 1000000. / number))
|
|
|
|
def main():
|
|
run_benchmark('cu2qu.benchmark', 'cu2qu', 'curve_to_quadratic')
|
|
run_benchmark('cu2qu.benchmark', 'cu2qu', 'curves_to_quadratic')
|
|
|
|
random.seed(1)
|
|
main()
|