[qu2cu] Add new module to convert quadratic to cubic Beziers

Lib/fontTools/qu2cu/__init__.py

@@ -0,0 +1,15 @@
+# Copyright 2016 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 .qu2cu import *

Lib/fontTools/qu2cu/qu2cu.py

@@ -0,0 +1,148 @@
+# cython: language_level=3
+# distutils: define_macros=CYTHON_TRACE_NOGIL=1
+
+# Copyright 2023 Google Inc. All Rights Reserved.
+# Copyright 2023 Behdad Esfahbod. 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.
+
+try:
+    import cython
+except ImportError:
+    # if cython not installed, use mock module with no-op decorators and types
+    from fontTools.misc import cython
+
+from fontTools.misc.bezierTools import splitCubicIntoTwoAtTC
+from fontTools.cu2qu.cu2qu import cubic_farthest_fit_inside
+
+
+__all__ = ["quadratic_to_curves"]
+
+
+NAN = float("NaN")
+
+
+if cython.compiled:
+    # Yep, I'm compiled.
+    COMPILED = True
+else:
+    # Just a lowly interpreted script.
+    COMPILED = False
+
+
+@cython.locals(_1_3=cython.double, _2_3=cython.double)
+@cython.locals(
+    p0=cython.complex,
+    p1=cython.complex,
+    p2=cython.complex,
+    p1_2_3=cython.complex,
+)
+def elevate_quadratic(p0, p1, p2, _1_3=1 / 3, _2_3=2 / 3):
+    """Given a quadratic bezier curve, return its degree-elevated cubic."""
+
+    p1_2_3 = p1 * _2_3
+    return (
+        p0,
+        (p0 * _1_3 + p1_2_3),
+        (p2 * _1_3 + p1_2_3),
+        p2,
+    )
+
+
+@cython.locals(k=cython.double, k_1=cython.double, t=cython.double)
+@cython.locals(
+    p1=cython.complex,
+    p2=cython.complex,
+    p3=cython.complex,
+    p4=cython.complex,
+    p5=cython.complex,
+    p6=cython.complex,
+    p7=cython.complex,
+    off1=cython.complex,
+    off2=cython.complex,
+)
+def merge_two_curves(p1, p2, p3, p4, p5, p6, p7):
+    """Return the initial cubic bezier curve subdivided in two segments.
+    Input must be a sequence of 7 points, i.e. two consecutive cubic curve
+    segments sharing the middle point.
+    Inspired by:
+    https://math.stackexchange.com/questions/877725/retrieve-the-initial-cubic-b%C3%A9zier-curve-subdivided-in-two-b%C3%A9zier-curves/879213#879213
+    """
+    k = abs(p5 - p4) / abs(p4 - p3)
+    k_1 = k + 1
+    off1 = k_1 * p2 - k * p1
+    off2 = (k_1 * p6 - p7) / k
+    t = 1 / k_1
+    return (p1, off1, off2, p7), t
+
+
+def quadratic_to_curves(p, tolerance=0.5):
+    assert len(p) >= 3, "quadratic spline requires at least 3 points"
+    p = [complex(x, y) for (x, y) in p]
+
+    # if spline has more than one offcurve, insert interpolated oncurves
+    q = list(p)
+    count = 0
+    num_offcurves = len(p) - 2
+    for i in range(1, num_offcurves):
+        off1 = p[i]
+        off2 = p[i + 1]
+        on = off1 + (off2 - off1) * 0.5
+        q.insert(i + 1 + count, on)
+        count += 1
+    del p
+
+    # elevate quadratic segments to cubic, and join them together
+    curves = []
+
+    curve = elevate_quadratic(*q[:3])
+    err = 0
+    for i in range(4, len(q), 2):
+        cubic_segment = elevate_quadratic(q[i - 2], q[i - 1], q[i])
+        new_curve, t = merge_two_curves(*(curve + cubic_segment[1:]))
+
+        seg1, seg2 = splitCubicIntoTwoAtTC(*new_curve, t)
+        t_point = seg2[0]
+
+        t_err = abs(t_point - cubic_segment[0])
+        if (
+            t_err > tolerance
+            or not cubic_farthest_fit_inside(
+                *(v - u for v, u in zip(seg1, curve)), tolerance - err
+            )
+            or not cubic_farthest_fit_inside(
+                *(v - u for v, u in zip(seg2, cubic_segment)), tolerance
+            )
+        ):
+            # Error too high. Start a new segment.
+            curves.append(curve)
+            new_curve = cubic_segment
+            err = 0
+            pass
+
+        curve = new_curve
+        err += t_err
+
+    curves.append(curve)
+
+    return [tuple((c.real, c.imag) for c in curve) for curve in curves]
+
+
+def main():
+    from fontTools.cu2qu.benchmark import generate_curve
+    from fontTools.cu2qu import curve_to_quadratic
+
+    curve = generate_curve()
+    quadratics = curve_to_quadratic(curve, 0.05)
+    print(len(quadratics))
+    print(len(quadratic_to_curves(quadratics, 0.05 * 2)))

setup.py

@@ -73,6 +73,9 @@ if with_cython is True or (with_cython is None and has_cython):
     ext_modules.append(
         Extension("fontTools.cu2qu.cu2qu", ["Lib/fontTools/cu2qu/cu2qu.py"]),
     )
+    ext_modules.append(
+        Extension("fontTools.qu2cu.qu2cu", ["Lib/fontTools/qu2cu/qu2cu.py"]),
+    )
     ext_modules.append(
         Extension("fontTools.misc.bezierTools", ["Lib/fontTools/misc/bezierTools.py"]),
     )