# cython: language_level=3
# distutils: define_macros=CYTHON_TRACE_NOGIL=1
# Copyright 2015 Google Inc. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
try:
import cython
COMPILED = cython.compiled
except (AttributeError, ImportError):
# if cython not installed, use mock module with no-op decorators and types
from fontTools.misc import cython
COMPILED = False
import math
from .errors import Error as Cu2QuError, ApproxNotFoundError
__all__ = ["curve_to_quadratic", "curves_to_quadratic"]
MAX_N = 100
NAN = float("NaN")
@cython.cfunc
@cython.inline
@cython.returns(cython.double)
@cython.locals(v1=cython.complex, v2=cython.complex)
def dot(v1, v2):
"""Return the dot product of two vectors.
Args:
v1 (complex): First vector.
v2 (complex): Second vector.
Returns:
double: Dot product.
"""
return (v1 * v2.conjugate()).real
@cython.cfunc
@cython.inline
@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
@cython.locals(
_1=cython.complex, _2=cython.complex, _3=cython.complex, _4=cython.complex
)
def calc_cubic_points(a, b, c, d):
_1 = d
_2 = (c / 3.0) + d
_3 = (b + c) / 3.0 + _2
_4 = a + d + c + b
return _1, _2, _3, _4
@cython.cfunc
@cython.inline
@cython.locals(
p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex
)
@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
def calc_cubic_parameters(p0, p1, p2, p3):
c = (p1 - p0) * 3.0
b = (p2 - p1) * 3.0 - c
d = p0
a = p3 - d - c - b
return a, b, c, d
@cython.cfunc
@cython.inline
@cython.locals(
p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex
)
def split_cubic_into_n_iter(p0, p1, p2, p3, n):
"""Split a cubic Bezier into n equal parts.
Splits the curve into `n` equal parts by curve time.
(t=0..1/n, t=1/n..2/n, ...)
Args:
p0 (complex): Start point of curve.
p1 (complex): First handle of curve.
p2 (complex): Second handle of curve.
p3 (complex): End point of curve.
Returns:
An iterator yielding the control points (four complex values) of the
subcurves.
"""
# Hand-coded special-cases
if n == 2:
return iter(split_cubic_into_two(p0, p1, p2, p3))
if n == 3:
return iter(split_cubic_into_three(p0, p1, p2, p3))
if n == 4:
a, b = split_cubic_into_two(p0, p1, p2, p3)
return iter(
split_cubic_into_two(a[0], a[1], a[2], a[3])
+ split_cubic_into_two(b[0], b[1], b[2], b[3])
)
if n == 6:
a, b = split_cubic_into_two(p0, p1, p2, p3)
return iter(
split_cubic_into_three(a[0], a[1], a[2], a[3])
+ split_cubic_into_three(b[0], b[1], b[2], b[3])
)
return _split_cubic_into_n_gen(p0, p1, p2, p3, n)
@cython.locals(
p0=cython.complex,
p1=cython.complex,
p2=cython.complex,
p3=cython.complex,
n=cython.int,
)
@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
@cython.locals(
dt=cython.double, delta_2=cython.double, delta_3=cython.double, i=cython.int
)
@cython.locals(
a1=cython.complex, b1=cython.complex, c1=cython.complex, d1=cython.complex
)
def _split_cubic_into_n_gen(p0, p1, p2, p3, n):
a, b, c, d = calc_cubic_parameters(p0, p1, p2, p3)
dt = 1 / n
delta_2 = dt * dt
delta_3 = dt * delta_2
for i in range(n):
t1 = i * dt
t1_2 = t1 * t1
# calc new a, b, c and d
a1 = a * delta_3
b1 = (3 * a * t1 + b) * delta_2
c1 = (2 * b * t1 + c + 3 * a * t1_2) * dt
d1 = a * t1 * t1_2 + b * t1_2 + c * t1 + d
yield calc_cubic_points(a1, b1, c1, d1)
@cython.cfunc
@cython.inline
@cython.locals(
p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex
)
@cython.locals(mid=cython.complex, deriv3=cython.complex)
def split_cubic_into_two(p0, p1, p2, p3):
"""Split a cubic Bezier into two equal parts.
Splits the curve into two equal parts at t = 0.5
Args:
p0 (complex): Start point of curve.
p1 (complex): First handle of curve.
p2 (complex): Second handle of curve.
p3 (complex): End point of curve.
Returns:
tuple: Two cubic Beziers (each expressed as a tuple of four complex
values).
"""
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),
)
@cython.cfunc
@cython.inline
@cython.locals(
p0=cython.complex,
p1=cython.complex,
p2=cython.complex,
p3=cython.complex,
)
@cython.locals(
mid1=cython.complex,
deriv1=cython.complex,
mid2=cython.complex,
deriv2=cython.complex,
)
def split_cubic_into_three(p0, p1, p2, p3):
"""Split a cubic Bezier into three equal parts.
Splits the curve into three equal parts at t = 1/3 and t = 2/3
Args:
p0 (complex): Start point of curve.
p1 (complex): First handle of curve.
p2 (complex): Second handle of curve.
p3 (complex): End point of curve.
Returns:
tuple: Three cubic Beziers (each expressed as a tuple of four complex
values).
"""
mid1 = (8 * p0 + 12 * p1 + 6 * p2 + p3) * (1 / 27)
deriv1 = (p3 + 3 * p2 - 4 * p0) * (1 / 27)
mid2 = (p0 + 6 * p1 + 12 * p2 + 8 * p3) * (1 / 27)
deriv2 = (4 * p3 - 3 * p1 - p0) * (1 / 27)
return (
(p0, (2 * p0 + p1) / 3.0, mid1 - deriv1, mid1),
(mid1, mid1 + deriv1, mid2 - deriv2, mid2),
(mid2, mid2 + deriv2, (p2 + 2 * p3) / 3.0, p3),
)
@cython.cfunc
@cython.inline
@cython.returns(cython.complex)
@cython.locals(
t=cython.double,
p0=cython.complex,
p1=cython.complex,
p2=cython.complex,
p3=cython.complex,
)
@cython.locals(_p1=cython.complex, _p2=cython.complex)
def cubic_approx_control(t, p0, p1, p2, p3):
"""Approximate a cubic Bezier using a quadratic one.
Args:
t (double): Position of control point.
p0 (complex): Start point of curve.
p1 (complex): First handle of curve.
p2 (complex): Second handle of curve.
p3 (complex): End point of curve.
Returns:
complex: Location of candidate control point on quadratic curve.
"""
_p1 = p0 + (p1 - p0) * 1.5
_p2 = p3 + (p2 - p3) * 1.5
return _p1 + (_p2 - _p1) * t
@cython.cfunc
@cython.inline
@cython.returns(cython.complex)
@cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex)
@cython.locals(ab=cython.complex, cd=cython.complex, p=cython.complex, h=cython.double)
def calc_intersect(a, b, c, d):
"""Calculate the intersection of two lines.
Args:
a (complex): Start point of first line.
b (complex): End point of first line.
c (complex): Start point of second line.
d (complex): End point of second line.
Returns:
complex: Location of intersection if one present, ``complex(NaN,NaN)``
if no intersection was found.
"""
ab = b - a
cd = d - c
p = ab * 1j
try:
h = dot(p, a - c) / dot(p, cd)
except ZeroDivisionError:
return complex(NAN, NAN)
return c + cd * h
@cython.cfunc
@cython.returns(cython.int)
@cython.locals(
tolerance=cython.double,
p0=cython.complex,
p1=cython.complex,
p2=cython.complex,
p3=cython.complex,
)
@cython.locals(mid=cython.complex, deriv3=cython.complex)
def cubic_farthest_fit_inside(p0, p1, p2, p3, tolerance):
"""Check if a cubic Bezier lies within a given distance of the origin.
"Origin" means *the* origin (0,0), not the start of the curve. Note that no
checks are made on the start and end positions of the curve; this function
only checks the inside of the curve.
Args:
p0 (complex): Start point of curve.
p1 (complex): First handle of curve.
p2 (complex): Second handle of curve.
p3 (complex): End point of curve.
tolerance (double): Distance from origin.
Returns:
bool: True if the cubic Bezier ``p`` entirely lies within a distance
``tolerance`` of the origin, False otherwise.
"""
# First check p2 then p1, as p2 has higher error early on.
if abs(p2) <= tolerance and abs(p1) <= tolerance:
return True
# Split.
mid = (p0 + 3 * (p1 + p2) + p3) * 0.125
if abs(mid) > tolerance:
return False
deriv3 = (p3 + p2 - p1 - p0) * 0.125
return cubic_farthest_fit_inside(
p0, (p0 + p1) * 0.5, mid - deriv3, mid, tolerance
) and cubic_farthest_fit_inside(mid, mid + deriv3, (p2 + p3) * 0.5, p3, tolerance)
@cython.cfunc
@cython.inline
@cython.locals(tolerance=cython.double)
@cython.locals(
q1=cython.complex,
c0=cython.complex,
c1=cython.complex,
c2=cython.complex,
c3=cython.complex,
)
def cubic_approx_quadratic(cubic, tolerance):
"""Approximate a cubic Bezier with a single quadratic within a given tolerance.
Args:
cubic (sequence): Four complex numbers representing control points of
the cubic Bezier curve.
tolerance (double): Permitted deviation from the original curve.
Returns:
Three complex numbers representing control points of the quadratic
curve if it fits within the given tolerance, or ``None`` if no suitable
curve could be calculated.
"""
q1 = calc_intersect(cubic[0], cubic[1], cubic[2], cubic[3])
if math.isnan(q1.imag):
return None
c0 = cubic[0]
c3 = cubic[3]
c1 = c0 + (q1 - c0) * (2 / 3)
c2 = c3 + (q1 - c3) * (2 / 3)
if not cubic_farthest_fit_inside(0, c1 - cubic[1], c2 - cubic[2], 0, tolerance):
return None
return c0, q1, c3
@cython.cfunc
@cython.locals(n=cython.int, tolerance=cython.double)
@cython.locals(i=cython.int)
@cython.locals(all_quadratic=cython.int)
@cython.locals(
c0=cython.complex, c1=cython.complex, c2=cython.complex, c3=cython.complex
)
@cython.locals(
q0=cython.complex,
q1=cython.complex,
next_q1=cython.complex,
q2=cython.complex,
d1=cython.complex,
)
def cubic_approx_spline(cubic, n, tolerance, all_quadratic):
"""Approximate a cubic Bezier curve with a spline of n quadratics.
Args:
cubic (sequence): Four complex numbers representing control points of
the cubic Bezier curve.
n (int): Number of quadratic Bezier curves in the spline.
tolerance (double): Permitted deviation from the original curve.
Returns:
A list of ``n+2`` complex numbers, representing control points of the
quadratic spline if it fits within the given tolerance, or ``None`` if
no suitable spline could be calculated.
"""
if n == 1:
return cubic_approx_quadratic(cubic, tolerance)
if n == 2 and all_quadratic == False:
return cubic
cubics = split_cubic_into_n_iter(cubic[0], cubic[1], cubic[2], cubic[3], n)
# calculate the spline of quadratics and check errors at the same time.
next_cubic = next(cubics)
next_q1 = cubic_approx_control(
0, next_cubic[0], next_cubic[1], next_cubic[2], next_cubic[3]
)
q2 = cubic[0]
d1 = 0j
spline = [cubic[0], next_q1]
for i in range(1, n + 1):
# Current cubic to convert
c0, c1, c2, c3 = next_cubic
# Current quadratic approximation of current cubic
q0 = q2
q1 = next_q1
if i < n:
next_cubic = next(cubics)
next_q1 = cubic_approx_control(
i / (n - 1), next_cubic[0], next_cubic[1], next_cubic[2], next_cubic[3]
)
spline.append(next_q1)
q2 = (q1 + next_q1) * 0.5
else:
q2 = c3
# End-point deltas
d0 = d1
d1 = q2 - c3
if abs(d1) > tolerance or not cubic_farthest_fit_inside(
d0,
q0 + (q1 - q0) * (2 / 3) - c1,
q2 + (q1 - q2) * (2 / 3) - c2,
d1,
tolerance,
):
return None
spline.append(cubic[3])
return spline
[docs]
@cython.locals(max_err=cython.double)
@cython.locals(n=cython.int)
@cython.locals(all_quadratic=cython.int)
def curve_to_quadratic(curve, max_err, all_quadratic=True):
"""Approximate a cubic Bezier curve with a spline of n quadratics.
Args:
cubic (sequence): Four 2D tuples representing control points of
the cubic Bezier curve.
max_err (double): Permitted deviation from the original curve.
all_quadratic (bool): If True (default) returned value is a
quadratic spline. If False, it's either a single quadratic
curve or a single cubic curve.
Returns:
If all_quadratic is True: A list of 2D tuples, representing
control points of the quadratic spline if it fits within the
given tolerance, or ``None`` if no suitable spline could be
calculated.
If all_quadratic is False: Either a quadratic curve (if length
of output is 3), or a cubic curve (if length of output is 4).
"""
curve = [complex(*p) for p in curve]
for n in range(1, MAX_N + 1):
spline = cubic_approx_spline(curve, n, max_err, all_quadratic)
if spline is not None:
# done. go home
return [(s.real, s.imag) for s in spline]
raise ApproxNotFoundError(curve)
[docs]
@cython.locals(l=cython.int, last_i=cython.int, i=cython.int)
@cython.locals(all_quadratic=cython.int)
def curves_to_quadratic(curves, max_errors, all_quadratic=True):
"""Return quadratic Bezier splines approximating the input cubic Beziers.
Args:
curves: A sequence of *n* curves, each curve being a sequence of four
2D tuples.
max_errors: A sequence of *n* floats representing the maximum permissible
deviation from each of the cubic Bezier curves.
all_quadratic (bool): If True (default) returned values are a
quadratic spline. If False, they are either a single quadratic
curve or a single cubic curve.
Example::
>>> curves_to_quadratic( [
... [ (50,50), (100,100), (150,100), (200,50) ],
... [ (75,50), (120,100), (150,75), (200,60) ]
... ], [1,1] )
[[(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)]]
The returned splines have "implied oncurve points" suitable for use in
TrueType ``glif`` outlines - i.e. in the first spline returned above,
the first quadratic segment runs from (50,50) to
( (75 + 125)/2 , (120 + 91.666..)/2 ) = (100, 83.333...).
Returns:
If all_quadratic is True, a list of splines, each spline being a list
of 2D tuples.
If all_quadratic is False, a list of curves, each curve being a quadratic
(length 3), or cubic (length 4).
Raises:
fontTools.cu2qu.Errors.ApproxNotFoundError: if no suitable approximation
can be found for all curves with the given parameters.
"""
curves = [[complex(*p) for p in curve] for curve in curves]
assert len(max_errors) == len(curves)
l = len(curves)
splines = [None] * l
last_i = i = 0
n = 1
while True:
spline = cubic_approx_spline(curves[i], n, max_errors[i], all_quadratic)
if spline is None:
if n == MAX_N:
break
n += 1
last_i = i
continue
splines[i] = spline
i = (i + 1) % l
if i == last_i:
# done. go home
return [[(s.real, s.imag) for s in spline] for spline in splines]
raise ApproxNotFoundError(curves)