209 lines
6.1 KiB
Python
209 lines
6.1 KiB
Python
# 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.
|
|
|
|
|
|
from __future__ import print_function, division, absolute_import
|
|
|
|
from math import hypot
|
|
from fontTools.misc import bezierTools
|
|
|
|
__all__ = ['curve_to_quadratic', 'curves_to_quadratic']
|
|
|
|
MAX_N = 100
|
|
|
|
|
|
class Cu2QuError(Exception):
|
|
pass
|
|
|
|
|
|
class ApproxNotFoundError(Cu2QuError):
|
|
def __init__(self, curve, error=None):
|
|
if error is None:
|
|
message = "no approximation found: %s" % curve
|
|
else:
|
|
message = ("approximation error exceeds max tolerance: %s, "
|
|
"error=%g" % (curve, error))
|
|
super(Cu2QuError, self).__init__(message)
|
|
self.curve = curve
|
|
self.error = error
|
|
|
|
|
|
def vector(p1, p2):
|
|
"""Return the vector from p1 to p2."""
|
|
return p2[0] - p1[0], p2[1] - p1[1]
|
|
|
|
|
|
def translate(p, v):
|
|
"""Translate a point by a vector."""
|
|
return p[0] + v[0], p[1] + v[1]
|
|
|
|
|
|
def scale(v, n):
|
|
"""Scale a vector."""
|
|
return v[0] * n, v[1] * n
|
|
|
|
|
|
def dist(p1, p2):
|
|
"""Calculate the distance between two points."""
|
|
return hypot(p1[0] - p2[0], p1[1] - p2[1])
|
|
|
|
|
|
def dot(v1, v2):
|
|
"""Return the dot product of two vectors."""
|
|
return v1[0] * v2[0] + v1[1] * v2[1]
|
|
|
|
|
|
def lerp(a, b, t):
|
|
"""Linearly interpolate between scalars a and b at time t."""
|
|
return a * (1 - t) + b * t
|
|
|
|
|
|
def lerp_pt(p1, p2, t):
|
|
"""Linearly interpolate between points p1 and p2 at time t."""
|
|
(x1, y1), (x2, y2) = p1, p2
|
|
return lerp(x1, x2, t), lerp(y1, y2, t)
|
|
|
|
|
|
def quadratic_bezier_at(p, t):
|
|
"""Return the point on a quadratic bezier curve at time t."""
|
|
|
|
(x1, y1), (x2, y2), (x3, y3) = p
|
|
return (
|
|
lerp(lerp(x1, x2, t), lerp(x2, x3, t), t),
|
|
lerp(lerp(y1, y2, t), lerp(y2, y3, t), t))
|
|
|
|
|
|
def cubic_bezier_at(p, t):
|
|
"""Return the point on a cubic bezier curve at time t."""
|
|
|
|
(x1, y1), (x2, y2), (x3, y3), (x4, y4) = p
|
|
return (
|
|
lerp(lerp(lerp(x1, x2, t), lerp(x2, x3, t), t),
|
|
lerp(lerp(x2, x3, t), lerp(x3, x4, t), t), t),
|
|
lerp(lerp(lerp(y1, y2, t), lerp(y2, y3, t), t),
|
|
lerp(lerp(y2, y3, t), lerp(y3, y4, t), t), t))
|
|
|
|
|
|
def cubic_approx(p, t):
|
|
"""Approximate a cubic bezier curve with a quadratic one."""
|
|
|
|
p1 = lerp_pt(p[0], p[1], 1.5)
|
|
p2 = lerp_pt(p[3], p[2], 1.5)
|
|
return p[0], lerp_pt(p1, p2, t), p[3]
|
|
|
|
|
|
def calc_intersect(p):
|
|
"""Calculate the intersection of ab and cd, given [a, b, c, d]."""
|
|
|
|
a, b, c, d = p
|
|
ab = vector(a, b)
|
|
cd = vector(c, d)
|
|
p = -ab[1], ab[0]
|
|
try:
|
|
h = dot(p, vector(c, a)) / dot(p, cd)
|
|
except ZeroDivisionError:
|
|
raise ValueError('Parallel vectors given to calc_intersect.')
|
|
return translate(c, scale(cd, h))
|
|
|
|
|
|
def cubic_approx_spline(p, n):
|
|
"""Approximate a cubic bezier curve with a spline of n quadratics.
|
|
|
|
Returns None if n is 1 and the cubic's control vectors are parallel, since
|
|
no quadratic exists with this cubic's tangents.
|
|
"""
|
|
|
|
if n == 1:
|
|
try:
|
|
p1 = calc_intersect(p)
|
|
except ValueError:
|
|
return None
|
|
return [p[0], p1, p[3]]
|
|
|
|
spline = [p[0]]
|
|
ts = [i / n for i in range(1, n)]
|
|
segments = bezierTools.splitCubicAtT(p[0], p[1], p[2], p[3], *ts)
|
|
for i in range(len(segments)):
|
|
segment = cubic_approx(segments[i], i / (n - 1))
|
|
spline.append(segment[1])
|
|
spline.append(p[3])
|
|
return spline
|
|
|
|
|
|
def curve_spline_dist(bezier, spline):
|
|
"""Max distance between a bezier and quadratic spline at sampled ts."""
|
|
|
|
TOTAL_STEPS = 20
|
|
error = 0
|
|
n = len(spline) - 2
|
|
steps = TOTAL_STEPS // n
|
|
for i in range(1, n + 1):
|
|
segment = [
|
|
spline[0] if i == 1 else segment[2],
|
|
spline[i],
|
|
spline[i + 1] if i == n else lerp_pt(spline[i], spline[i + 1], 0.5)]
|
|
for j in range(steps):
|
|
p1 = cubic_bezier_at(bezier, (j / steps + i - 1) / n)
|
|
p2 = quadratic_bezier_at(segment, j / steps)
|
|
error = max(error, dist(p1, p2))
|
|
return error
|
|
|
|
|
|
def curve_to_quadratic(p, max_err):
|
|
"""Return a quadratic spline approximating this cubic bezier, and
|
|
the error of approximation.
|
|
Raise 'ApproxNotFoundError' if no suitable approximation can be found
|
|
with the given parameters.
|
|
"""
|
|
|
|
spline, error = None, None
|
|
for n in range(1, MAX_N + 1):
|
|
spline = cubic_approx_spline(p, n)
|
|
if spline is None:
|
|
continue
|
|
error = curve_spline_dist(p, spline)
|
|
if error <= max_err:
|
|
break
|
|
else:
|
|
# no break: approximation not found or error exceeds tolerance
|
|
raise ApproxNotFoundError(p, error)
|
|
return spline, error
|
|
|
|
|
|
def curves_to_quadratic(curves, max_errors):
|
|
"""Return quadratic splines approximating these cubic beziers, and
|
|
the respective errors of approximation.
|
|
Raise 'ApproxNotFoundError' if no suitable approximation can be found
|
|
for all curves with the given parameters.
|
|
"""
|
|
|
|
num_curves = len(curves)
|
|
assert len(max_errors) == num_curves
|
|
|
|
splines = [None] * num_curves
|
|
errors = [None] * num_curves
|
|
for n in range(1, MAX_N + 1):
|
|
splines = [cubic_approx_spline(c, n) for c in curves]
|
|
if not all(splines):
|
|
continue
|
|
errors = [curve_spline_dist(c, s) for c, s in zip(curves, splines)]
|
|
if all(err <= max_err for err, max_err in zip(errors, max_errors)):
|
|
break
|
|
else:
|
|
# no break: raise if any spline is None or error exceeds tolerance
|
|
for c, s, error, max_err in zip(curves, splines, errors, max_errors):
|
|
if s is None or error > max_err:
|
|
raise ApproxNotFoundError(c, error)
|
|
return splines, errors
|