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diff --git a/Lib/fontTools/cu2qu/ufo.py b/Lib/fontTools/cu2qu/ufo.py
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+++ b/Lib/fontTools/cu2qu/ufo.py
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+# 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.
+
+
+"""Converts cubic bezier curves to quadratic splines.
+
+Conversion is performed such that the quadratic splines keep the same end-curve
+tangents as the original cubics. The approach is iterative, increasing the
+number of segments for a spline until the error gets below a bound.
+
+Respective curves from multiple fonts will be converted at once to ensure that
+the resulting splines are interpolation-compatible.
+"""
+
+import logging
+from fontTools.pens.basePen import AbstractPen
+from fontTools.pens.pointPen import PointToSegmentPen
+from fontTools.pens.reverseContourPen import ReverseContourPen
+
+from . import curves_to_quadratic
+from .errors import (
+    UnequalZipLengthsError, IncompatibleSegmentNumberError,
+    IncompatibleSegmentTypesError, IncompatibleGlyphsError,
+    IncompatibleFontsError)
+
+
+__all__ = ['fonts_to_quadratic', 'font_to_quadratic']
+
+# The default approximation error below is a relative value (1/1000 of the EM square).
+# Later on, we convert it to absolute font units by multiplying it by a font's UPEM
+# (see fonts_to_quadratic).
+DEFAULT_MAX_ERR = 0.001
+CURVE_TYPE_LIB_KEY = "com.github.googlei18n.cu2qu.curve_type"
+
+logger = logging.getLogger(__name__)
+
+
+_zip = zip
+def zip(*args):
+    """Ensure each argument to zip has the same length. Also make sure a list is
+    returned for python 2/3 compatibility.
+    """
+
+    if len(set(len(a) for a in args)) != 1:
+        raise UnequalZipLengthsError(*args)
+    return list(_zip(*args))
+
+
+class GetSegmentsPen(AbstractPen):
+    """Pen to collect segments into lists of points for conversion.
+
+    Curves always include their initial on-curve point, so some points are
+    duplicated between segments.
+    """
+
+    def __init__(self):
+        self._last_pt = None
+        self.segments = []
+
+    def _add_segment(self, tag, *args):
+        if tag in ['move', 'line', 'qcurve', 'curve']:
+            self._last_pt = args[-1]
+        self.segments.append((tag, args))
+
+    def moveTo(self, pt):
+        self._add_segment('move', pt)
+
+    def lineTo(self, pt):
+        self._add_segment('line', pt)
+
+    def qCurveTo(self, *points):
+        self._add_segment('qcurve', self._last_pt, *points)
+
+    def curveTo(self, *points):
+        self._add_segment('curve', self._last_pt, *points)
+
+    def closePath(self):
+        self._add_segment('close')
+
+    def endPath(self):
+        self._add_segment('end')
+
+    def addComponent(self, glyphName, transformation):
+        pass
+
+
+def _get_segments(glyph):
+    """Get a glyph's segments as extracted by GetSegmentsPen."""
+
+    pen = GetSegmentsPen()
+    # glyph.draw(pen)
+    # We can't simply draw the glyph with the pen, but we must initialize the
+    # PointToSegmentPen explicitly with outputImpliedClosingLine=True.
+    # By default PointToSegmentPen does not outputImpliedClosingLine -- unless
+    # last and first point on closed contour are duplicated. Because we are
+    # converting multiple glyphs at the same time, we want to make sure
+    # this function returns the same number of segments, whether or not
+    # the last and first point overlap.
+    # https://github.com/googlefonts/fontmake/issues/572
+    # https://github.com/fonttools/fonttools/pull/1720
+    pointPen = PointToSegmentPen(pen, outputImpliedClosingLine=True)
+    glyph.drawPoints(pointPen)
+    return pen.segments
+
+
+def _set_segments(glyph, segments, reverse_direction):
+    """Draw segments as extracted by GetSegmentsPen back to a glyph."""
+
+    glyph.clearContours()
+    pen = glyph.getPen()
+    if reverse_direction:
+        pen = ReverseContourPen(pen)
+    for tag, args in segments:
+        if tag == 'move':
+            pen.moveTo(*args)
+        elif tag == 'line':
+            pen.lineTo(*args)
+        elif tag == 'curve':
+            pen.curveTo(*args[1:])
+        elif tag == 'qcurve':
+            pen.qCurveTo(*args[1:])
+        elif tag == 'close':
+            pen.closePath()
+        elif tag == 'end':
+            pen.endPath()
+        else:
+            raise AssertionError('Unhandled segment type "%s"' % tag)
+
+
+def _segments_to_quadratic(segments, max_err, stats):
+    """Return quadratic approximations of cubic segments."""
+
+    assert all(s[0] == 'curve' for s in segments), 'Non-cubic given to convert'
+
+    new_points = curves_to_quadratic([s[1] for s in segments], max_err)
+    n = len(new_points[0])
+    assert all(len(s) == n for s in new_points[1:]), 'Converted incompatibly'
+
+    spline_length = str(n - 2)
+    stats[spline_length] = stats.get(spline_length, 0) + 1
+
+    return [('qcurve', p) for p in new_points]
+
+
+def _glyphs_to_quadratic(glyphs, max_err, reverse_direction, stats):
+    """Do the actual conversion of a set of compatible glyphs, after arguments
+    have been set up.
+
+    Return True if the glyphs were modified, else return False.
+    """
+
+    try:
+        segments_by_location = zip(*[_get_segments(g) for g in glyphs])
+    except UnequalZipLengthsError:
+        raise IncompatibleSegmentNumberError(glyphs)
+    if not any(segments_by_location):
+        return False
+
+    # always modify input glyphs if reverse_direction is True
+    glyphs_modified = reverse_direction
+
+    new_segments_by_location = []
+    incompatible = {}
+    for i, segments in enumerate(segments_by_location):
+        tag = segments[0][0]
+        if not all(s[0] == tag for s in segments[1:]):
+            incompatible[i] = [s[0] for s in segments]
+        elif tag == 'curve':
+            segments = _segments_to_quadratic(segments, max_err, stats)
+            glyphs_modified = True
+        new_segments_by_location.append(segments)
+
+    if glyphs_modified:
+        new_segments_by_glyph = zip(*new_segments_by_location)
+        for glyph, new_segments in zip(glyphs, new_segments_by_glyph):
+            _set_segments(glyph, new_segments, reverse_direction)
+
+    if incompatible:
+        raise IncompatibleSegmentTypesError(glyphs, segments=incompatible)
+    return glyphs_modified
+
+
+def glyphs_to_quadratic(
+        glyphs, max_err=None, reverse_direction=False, stats=None):
+    """Convert the curves of a set of compatible of glyphs to quadratic.
+
+    All curves will be converted to quadratic at once, ensuring interpolation
+    compatibility. If this is not required, calling glyphs_to_quadratic with one
+    glyph at a time may yield slightly more optimized results.
+
+    Return True if glyphs were modified, else return False.
+
+    Raises IncompatibleGlyphsError if glyphs have non-interpolatable outlines.
+    """
+    if stats is None:
+        stats = {}
+
+    if not max_err:
+        # assume 1000 is the default UPEM
+        max_err = DEFAULT_MAX_ERR * 1000
+
+    if isinstance(max_err, (list, tuple)):
+        max_errors = max_err
+    else:
+        max_errors = [max_err] * len(glyphs)
+    assert len(max_errors) == len(glyphs)
+
+    return _glyphs_to_quadratic(glyphs, max_errors, reverse_direction, stats)
+
+
+def fonts_to_quadratic(
+        fonts, max_err_em=None, max_err=None, reverse_direction=False,
+        stats=None, dump_stats=False, remember_curve_type=True):
+    """Convert the curves of a collection of fonts to quadratic.
+
+    All curves will be converted to quadratic at once, ensuring interpolation
+    compatibility. If this is not required, calling fonts_to_quadratic with one
+    font at a time may yield slightly more optimized results.
+
+    Return True if fonts were modified, else return False.
+
+    By default, cu2qu stores the curve type in the fonts' lib, under a private
+    key "com.github.googlei18n.cu2qu.curve_type", and will not try to convert
+    them again if the curve type is already set to "quadratic".
+    Setting 'remember_curve_type' to False disables this optimization.
+
+    Raises IncompatibleFontsError if same-named glyphs from different fonts
+    have non-interpolatable outlines.
+    """
+
+    if remember_curve_type:
+        curve_types = {f.lib.get(CURVE_TYPE_LIB_KEY, "cubic") for f in fonts}
+        if len(curve_types) == 1:
+            curve_type = next(iter(curve_types))
+            if curve_type == "quadratic":
+                logger.info("Curves already converted to quadratic")
+                return False
+            elif curve_type == "cubic":
+                pass  # keep converting
+            else:
+                raise NotImplementedError(curve_type)
+        elif len(curve_types) > 1:
+            # going to crash later if they do differ
+            logger.warning("fonts may contain different curve types")
+
+    if stats is None:
+        stats = {}
+
+    if max_err_em and max_err:
+        raise TypeError('Only one of max_err and max_err_em can be specified.')
+    if not (max_err_em or max_err):
+        max_err_em = DEFAULT_MAX_ERR
+
+    if isinstance(max_err, (list, tuple)):
+        assert len(max_err) == len(fonts)
+        max_errors = max_err
+    elif max_err:
+        max_errors = [max_err] * len(fonts)
+
+    if isinstance(max_err_em, (list, tuple)):
+        assert len(fonts) == len(max_err_em)
+        max_errors = [f.info.unitsPerEm * e
+                      for f, e in zip(fonts, max_err_em)]
+    elif max_err_em:
+        max_errors = [f.info.unitsPerEm * max_err_em for f in fonts]
+
+    modified = False
+    glyph_errors = {}
+    for name in set().union(*(f.keys() for f in fonts)):
+        glyphs = []
+        cur_max_errors = []
+        for font, error in zip(fonts, max_errors):
+            if name in font:
+                glyphs.append(font[name])
+                cur_max_errors.append(error)
+        try:
+            modified |= _glyphs_to_quadratic(
+                glyphs, cur_max_errors, reverse_direction, stats)
+        except IncompatibleGlyphsError as exc:
+            logger.error(exc)
+            glyph_errors[name] = exc
+
+    if glyph_errors:
+        raise IncompatibleFontsError(glyph_errors)
+
+    if modified and dump_stats:
+        spline_lengths = sorted(stats.keys())
+        logger.info('New spline lengths: %s' % (', '.join(
+                    '%s: %d' % (l, stats[l]) for l in spline_lengths)))
+
+    if remember_curve_type:
+        for font in fonts:
+            curve_type = font.lib.get(CURVE_TYPE_LIB_KEY, "cubic")
+            if curve_type != "quadratic":
+                font.lib[CURVE_TYPE_LIB_KEY] = "quadratic"
+                modified = True
+    return modified
+
+
+def glyph_to_quadratic(glyph, **kwargs):
+    """Convenience wrapper around glyphs_to_quadratic, for just one glyph.
+    Return True if the glyph was modified, else return False.
+    """
+
+    return glyphs_to_quadratic([glyph], **kwargs)
+
+
+def font_to_quadratic(font, **kwargs):
+    """Convenience wrapper around fonts_to_quadratic, for just one font.
+    Return True if the font was modified, else return False.
+    """
+
+    return fonts_to_quadratic([font], **kwargs)