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"""Convert SVG Path's elliptical arcs to Bezier curves.
The code is mostly adapted from Blink's SVGPathNormalizer::DecomposeArcToCubic
https://github.com/chromium/chromium/blob/93831f2/third_party/
blink/renderer/core/svg/svg_path_parser.cc#L169-L278
"""
from fontTools.misc.transform import Identity, Scale
from math import atan2, ceil, cos, fabs, isfinite, pi, radians, sin, sqrt, tan
TWO_PI = 2 * pi
PI_OVER_TWO = 0.5 * pi
def _map_point(matrix, pt):
# apply Transform matrix to a point represented as a complex number
r = matrix.transformPoint((pt.real, pt.imag))
return r[0] + r[1] * 1j
class EllipticalArc(object):
def __init__(self, current_point, rx, ry, rotation, large, sweep, target_point):
self.current_point = current_point
self.rx = rx
self.ry = ry
self.rotation = rotation
self.large = large
self.sweep = sweep
self.target_point = target_point
# SVG arc's rotation angle is expressed in degrees, whereas Transform.rotate
# uses radians
self.angle = radians(rotation)
# these derived attributes are computed by the _parametrize method
self.center_point = self.theta1 = self.theta2 = self.theta_arc = None
def _parametrize(self):
# convert from endopoint to center parametrization:
# https://www.w3.org/TR/SVG/implnote.html#ArcConversionEndpointToCenter
# If rx = 0 or ry = 0 then this arc is treated as a straight line segment (a
# "lineto") joining the endpoints.
# http://www.w3.org/TR/SVG/implnote.html#ArcOutOfRangeParameters
rx = fabs(self.rx)
ry = fabs(self.ry)
if not (rx and ry):
return False
# If the current point and target point for the arc are identical, it should
# be treated as a zero length path. This ensures continuity in animations.
if self.target_point == self.current_point:
return False
mid_point_distance = (self.current_point - self.target_point) * 0.5
point_transform = Identity.rotate(-self.angle)
transformed_mid_point = _map_point(point_transform, mid_point_distance)
square_rx = rx * rx
square_ry = ry * ry
square_x = transformed_mid_point.real * transformed_mid_point.real
square_y = transformed_mid_point.imag * transformed_mid_point.imag
# Check if the radii are big enough to draw the arc, scale radii if not.
# http://www.w3.org/TR/SVG/implnote.html#ArcCorrectionOutOfRangeRadii
radii_scale = square_x / square_rx + square_y / square_ry
if radii_scale > 1:
rx *= sqrt(radii_scale)
ry *= sqrt(radii_scale)
self.rx, self.ry = rx, ry
point_transform = Scale(1 / rx, 1 / ry).rotate(-self.angle)
point1 = _map_point(point_transform, self.current_point)
point2 = _map_point(point_transform, self.target_point)
delta = point2 - point1
d = delta.real * delta.real + delta.imag * delta.imag
scale_factor_squared = max(1 / d - 0.25, 0.0)
scale_factor = sqrt(scale_factor_squared)
if self.sweep == self.large:
scale_factor = -scale_factor
delta *= scale_factor
center_point = (point1 + point2) * 0.5
center_point += complex(-delta.imag, delta.real)
point1 -= center_point
point2 -= center_point
theta1 = atan2(point1.imag, point1.real)
theta2 = atan2(point2.imag, point2.real)
theta_arc = theta2 - theta1
if theta_arc < 0 and self.sweep:
theta_arc += TWO_PI
elif theta_arc > 0 and not self.sweep:
theta_arc -= TWO_PI
self.theta1 = theta1
self.theta2 = theta1 + theta_arc
self.theta_arc = theta_arc
self.center_point = center_point
return True
def _decompose_to_cubic_curves(self):
if self.center_point is None and not self._parametrize():
return
point_transform = Identity.rotate(self.angle).scale(self.rx, self.ry)
# Some results of atan2 on some platform implementations are not exact
# enough. So that we get more cubic curves than expected here. Adding 0.001f
# reduces the count of sgements to the correct count.
num_segments = int(ceil(fabs(self.theta_arc / (PI_OVER_TWO + 0.001))))
for i in range(num_segments):
start_theta = self.theta1 + i * self.theta_arc / num_segments
end_theta = self.theta1 + (i + 1) * self.theta_arc / num_segments
t = (4 / 3) * tan(0.25 * (end_theta - start_theta))
if not isfinite(t):
return
sin_start_theta = sin(start_theta)
cos_start_theta = cos(start_theta)
sin_end_theta = sin(end_theta)
cos_end_theta = cos(end_theta)
point1 = complex(
cos_start_theta - t * sin_start_theta,
sin_start_theta + t * cos_start_theta,
)
point1 += self.center_point
target_point = complex(cos_end_theta, sin_end_theta)
target_point += self.center_point
point2 = target_point
point2 += complex(t * sin_end_theta, -t * cos_end_theta)
point1 = _map_point(point_transform, point1)
point2 = _map_point(point_transform, point2)
target_point = _map_point(point_transform, target_point)
yield point1, point2, target_point
def draw(self, pen):
for point1, point2, target_point in self._decompose_to_cubic_curves():
pen.curveTo(
(point1.real, point1.imag),
(point2.real, point2.imag),
(target_point.real, target_point.imag),
)
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