svgLib: add support for converting elliptical arcs to cubic bezier curves

Fixes https://github.com/fonttools/fonttools/issues/1141

Uses code from Chromium Blink's SVG path parser, in SVGPathNormalizer::DecomposeArcToCubic
https://github.com/chromium/chromium/blob/master/third_party/blink/renderer/core/svg/svg_path_parser.cc#L169-L278

Each elliptical arc is approximated by series of cubic bezier curves, one cubic every 90-degree portion of an arc.

Lib/fontTools/svgLib/path/arc.py

@@ -0,0 +1,157 @@
+"""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 __future__ import print_function, division, absolute_import, unicode_literals
+from fontTools.misc.py23 import *
+from fontTools.misc.py23 import isfinite
+from fontTools.misc.transform import Identity, Scale
+from math import atan2, ceil, cos, fabs, 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),
+            )

Lib/fontTools/svgLib/path/parser.py

@@ -10,8 +10,10 @@
 from __future__ import (
     print_function, division, absolute_import, unicode_literals)
 from fontTools.misc.py23 import *
+from .arc import EllipticalArc
 import re
 
+
 COMMANDS = set('MmZzLlHhVvCcSsQqTtAa')
 UPPERCASE = set('MZLHVCSQTA')
 
@@ -27,7 +29,7 @@ def _tokenize_path(pathdef):
             yield token
 
 
-def parse_path(pathdef, pen, current_pos=(0, 0)):
+def parse_path(pathdef, pen, current_pos=(0, 0), arc_class=EllipticalArc):
     """ Parse SVG path definition (i.e. "d" attribute of <path> elements)
     and call a 'pen' object's moveTo, lineTo, curveTo, qCurveTo and closePath
     methods.
@@ -35,8 +37,13 @@ def parse_path(pathdef, pen, current_pos=(0, 0)):
     If 'current_pos' (2-float tuple) is provided, the initial moveTo will
     be relative to that instead being absolute.
 
-    Arc segments (commands "A" or "a") are not currently supported, and raise
+    If the pen has an "arcTo" method, it is called with the original values
-    NotImplementedError.
+    of the elliptical arc curve commands:
+
+        pen.arcTo(rx, ry, rotation, arc_large, arc_sweep, (x, y))
+
+    Otherwise, the arcs are approximated by series of cubic Bezier segments
+    ("curveTo"), one every 90 degrees.
     """
     # In the SVG specs, initial movetos are absolute, even if
     # specified as 'm'. This is the default behavior here as well.
@@ -52,6 +59,8 @@ def parse_path(pathdef, pen, current_pos=(0, 0)):
     command = None
     last_control = None
 
+    have_arcTo = hasattr(pen, "arcTo")
+
     while elements:
 
         if elements[-1] in COMMANDS:
@@ -209,7 +218,34 @@ def parse_path(pathdef, pen, current_pos=(0, 0)):
             last_control = control
 
         elif command == 'A':
-            raise NotImplementedError('arcs are not supported')
+            rx = float(elements.pop())
+            ry = float(elements.pop())
+            rotation = float(elements.pop())
+            arc_large = bool(int(elements.pop()))
+            arc_sweep = bool(int(elements.pop()))
+            end = float(elements.pop()) + float(elements.pop()) * 1j
+
+            if not absolute:
+                end += current_pos
+
+            # if the pen supports arcs, pass the values unchanged, otherwise
+            # approximate the arc with a series of cubic bezier curves
+            if have_arcTo:
+                pen.arcTo(
+                    rx,
+                    ry,
+                    rotation,
+                    arc_large,
+                    arc_sweep,
+                    (end.real, end.imag),
+                )
+            else:
+                arc = arc_class(
+                    current_pos, rx, ry, rotation, arc_large, arc_sweep, end
+                )
+                arc.draw(pen)
+
+            current_pos = end
 
     # no final Z command, it's an open path
     if start_pos is not None: