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[perimeterPen] Add Gauss-Lobatto implementation for quadratic Bezier as well

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Is twice faster than the exact algorithm.
This commit is contained in:
Behdad Esfahbod 2017-02-20 10:16:17 -06:00
parent 9f1067a991
commit 3d4bd76a76
1 changed files with 35 additions and 19 deletions
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54
Lib/fontTools/pens/perimeterPen.py
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@ -30,16 +30,11 @@ class PerimeterPen(BasePen):
self.value = 0
self._mult = 1.+1.5*tolerance # The 1.5 is a empirical hack; no math
# Choose which algorithm to use for cubic. Recursive algorithm is
# accurate to arbitrary tolerances whereas the Lobatto algorithm has
# fixed error characteristics, but is faster.
#
# The 0.0015 cutoff has been empirically determined by measuring error
# of the Lobatto approach on a realworld font.
if tolerance < 0.0015:
self._addCubic = self._addCubicRecursive
else:
self._addCubic = self._addCubicLobatto
# Choose which algorithm to use for quadratic and for cubic.
# Lobatto is faster but has fixed error characteristic with no strong
# error bound. The cutoff points are derived empirically.
self._addCubic = self._addCubicLobatto if tolerance >= 0.0015 else self._addCubicRecursive
self._addQuadratic = self._addQuadraticLobatto if tolerance >= 0.00075 else self._addQuadraticExact
def _moveTo(self, p0):
self.__startPoint = p0
@ -48,23 +43,21 @@ class PerimeterPen(BasePen):
p0 = self._getCurrentPoint()
self.value += _distance(p0, p1)
def _qCurveToOne(self, p1, p2):
def _addQuadraticExact(self, c0, c1, c2):
# Analytical solution to the length of a quadratic bezier.
# I'll explain how I arrived at this later.
p0 = self._getCurrentPoint()
_p1 = complex(*p1)
d0 = _p1 - complex(*p0)
d1 = complex(*p2) - _p1
d0 = c1 - c0
d1 = c2 - c1
d = d1 - d0
n = d * 1j
scale = abs(n)
if scale == 0.:
self._lineTo(p2)
self.value += abs(c2-c0)
return
origDist = _dot(n,d0)
if origDist == 0.:
if _dot(d0,d1) >= 0:
self._lineTo(p2)
self.value += abs(c2-c0)
return
assert 0 # TODO handle cusps
x0 = _dot(d,d0) / origDist
@ -72,6 +65,29 @@ class PerimeterPen(BasePen):
Len = abs(2 * (_intSecAtan(x1) - _intSecAtan(x0)) * origDist / (scale * (x1 - x0)))
self.value += Len
def _addQuadraticLobatto(self, c0, c1, c2):
# Approximate length of quadratic Bezier curve using Lobatto quadrature
# with n=4 points: endpoints and at t=.5±sqrt(1/5)/2
#
# This, essentially, approximates the length-of-derivative function
# to be integrated with the best-matching fifth-degree polynomial
# approximation of it.
#
# https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules
# abs(BezierCurveC[3].diff(t).subs({t:T})) for T in (0, .5-(1/5)**.5/2, .5, .5+(1/5)**.5/2, 1),
# weighted 1/20, 49/180, 32/90, 49/180, 1/20 respectively.
v0 = abs(c1-c0)*0.166666666666667
v1 = abs(-0.603005664791649*c0 + 0.372677996249965*c1 + 0.230327668541684*c2)
v2 = abs(-0.230327668541684*c0 - 0.372677996249965*c1 + 0.603005664791649*c2)
v3 = abs(c2-c1)*0.166666666666667
self.value += v0 + v1 + v2 + v3
def _qCurveToOne(self, p1, p2):
p0 = self._getCurrentPoint()
self._addQuadratic(complex(*p0), complex(*p1), complex(*p2))
def _addCubicRecursive(self, p0, p1, p2, p3):
arch = abs(p0-p3)
box = abs(p0-p1) + abs(p1-p2) + abs(p2-p3)
@ -84,7 +100,7 @@ class PerimeterPen(BasePen):
def _addCubicLobatto(self, c0, c1, c2, c3):
# Approximate length of cubic Bezier curve using Lobatto quadrature
# with n=5 points: endpoints, midpoint, and at t=.5±sqrt(21)/14
# with n=5 points: endpoints, midpoint, and at t=.5±sqrt(3/7)/2
#
# This, essentially, approximates the length-of-derivative function
# to be integrated with the best-matching seventh-degree polynomial
@ -92,7 +108,7 @@ class PerimeterPen(BasePen):
#
# https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules
# abs(beziercurvec[3].diff(t).subs({t:T})) for T in (0, .5-(3/28)**.5, .5, .5+(3/28)**.5, 1),
# abs(BezierCurveC[3].diff(t).subs({t:T})) for T in (0, .5-(3/7)**.5/2, .5, .5+(3/7)**.5/2, 1),
# weighted 1/20, 49/180, 32/90, 49/180, 1/20 respectively.
v0 = abs(c1-c0)*.15
v1 = abs(-0.558983582205757*c0 + 0.325650248872424*c1 + 0.208983582205757*c2 + 0.024349751127576*c3)