| # cython: language_level=3 |
| # distutils: define_macros=CYTHON_TRACE_NOGIL=1 |
| |
| # 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. |
| |
| try: |
| import cython |
| |
| COMPILED = cython.compiled |
| except (AttributeError, ImportError): |
| # if cython not installed, use mock module with no-op decorators and types |
| from fontTools.misc import cython |
| |
| COMPILED = False |
| |
| import math |
| |
| from .errors import Error as Cu2QuError, ApproxNotFoundError |
| |
| |
| __all__ = ["curve_to_quadratic", "curves_to_quadratic"] |
| |
| MAX_N = 100 |
| |
| NAN = float("NaN") |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.returns(cython.double) |
| @cython.locals(v1=cython.complex, v2=cython.complex) |
| def dot(v1, v2): |
| """Return the dot product of two vectors. |
| |
| Args: |
| v1 (complex): First vector. |
| v2 (complex): Second vector. |
| |
| Returns: |
| double: Dot product. |
| """ |
| return (v1 * v2.conjugate()).real |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex) |
| @cython.locals( |
| _1=cython.complex, _2=cython.complex, _3=cython.complex, _4=cython.complex |
| ) |
| def calc_cubic_points(a, b, c, d): |
| _1 = d |
| _2 = (c / 3.0) + d |
| _3 = (b + c) / 3.0 + _2 |
| _4 = a + d + c + b |
| return _1, _2, _3, _4 |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals( |
| p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex |
| ) |
| @cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex) |
| def calc_cubic_parameters(p0, p1, p2, p3): |
| c = (p1 - p0) * 3.0 |
| b = (p2 - p1) * 3.0 - c |
| d = p0 |
| a = p3 - d - c - b |
| return a, b, c, d |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals( |
| p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex |
| ) |
| def split_cubic_into_n_iter(p0, p1, p2, p3, n): |
| """Split a cubic Bezier into n equal parts. |
| |
| Splits the curve into `n` equal parts by curve time. |
| (t=0..1/n, t=1/n..2/n, ...) |
| |
| Args: |
| p0 (complex): Start point of curve. |
| p1 (complex): First handle of curve. |
| p2 (complex): Second handle of curve. |
| p3 (complex): End point of curve. |
| |
| Returns: |
| An iterator yielding the control points (four complex values) of the |
| subcurves. |
| """ |
| # Hand-coded special-cases |
| if n == 2: |
| return iter(split_cubic_into_two(p0, p1, p2, p3)) |
| if n == 3: |
| return iter(split_cubic_into_three(p0, p1, p2, p3)) |
| if n == 4: |
| a, b = split_cubic_into_two(p0, p1, p2, p3) |
| return iter( |
| split_cubic_into_two(a[0], a[1], a[2], a[3]) |
| + split_cubic_into_two(b[0], b[1], b[2], b[3]) |
| ) |
| if n == 6: |
| a, b = split_cubic_into_two(p0, p1, p2, p3) |
| return iter( |
| split_cubic_into_three(a[0], a[1], a[2], a[3]) |
| + split_cubic_into_three(b[0], b[1], b[2], b[3]) |
| ) |
| |
| return _split_cubic_into_n_gen(p0, p1, p2, p3, n) |
| |
| |
| @cython.locals( |
| p0=cython.complex, |
| p1=cython.complex, |
| p2=cython.complex, |
| p3=cython.complex, |
| n=cython.int, |
| ) |
| @cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex) |
| @cython.locals( |
| dt=cython.double, delta_2=cython.double, delta_3=cython.double, i=cython.int |
| ) |
| @cython.locals( |
| a1=cython.complex, b1=cython.complex, c1=cython.complex, d1=cython.complex |
| ) |
| def _split_cubic_into_n_gen(p0, p1, p2, p3, n): |
| a, b, c, d = calc_cubic_parameters(p0, p1, p2, p3) |
| dt = 1 / n |
| delta_2 = dt * dt |
| delta_3 = dt * delta_2 |
| for i in range(n): |
| t1 = i * dt |
| t1_2 = t1 * t1 |
| # calc new a, b, c and d |
| a1 = a * delta_3 |
| b1 = (3 * a * t1 + b) * delta_2 |
| c1 = (2 * b * t1 + c + 3 * a * t1_2) * dt |
| d1 = a * t1 * t1_2 + b * t1_2 + c * t1 + d |
| yield calc_cubic_points(a1, b1, c1, d1) |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals( |
| p0=cython.complex, p1=cython.complex, p2=cython.complex, p3=cython.complex |
| ) |
| @cython.locals(mid=cython.complex, deriv3=cython.complex) |
| def split_cubic_into_two(p0, p1, p2, p3): |
| """Split a cubic Bezier into two equal parts. |
| |
| Splits the curve into two equal parts at t = 0.5 |
| |
| Args: |
| p0 (complex): Start point of curve. |
| p1 (complex): First handle of curve. |
| p2 (complex): Second handle of curve. |
| p3 (complex): End point of curve. |
| |
| Returns: |
| tuple: Two cubic Beziers (each expressed as a tuple of four complex |
| values). |
| """ |
| mid = (p0 + 3 * (p1 + p2) + p3) * 0.125 |
| deriv3 = (p3 + p2 - p1 - p0) * 0.125 |
| return ( |
| (p0, (p0 + p1) * 0.5, mid - deriv3, mid), |
| (mid, mid + deriv3, (p2 + p3) * 0.5, p3), |
| ) |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals( |
| p0=cython.complex, |
| p1=cython.complex, |
| p2=cython.complex, |
| p3=cython.complex, |
| ) |
| @cython.locals( |
| mid1=cython.complex, |
| deriv1=cython.complex, |
| mid2=cython.complex, |
| deriv2=cython.complex, |
| ) |
| def split_cubic_into_three(p0, p1, p2, p3): |
| """Split a cubic Bezier into three equal parts. |
| |
| Splits the curve into three equal parts at t = 1/3 and t = 2/3 |
| |
| Args: |
| p0 (complex): Start point of curve. |
| p1 (complex): First handle of curve. |
| p2 (complex): Second handle of curve. |
| p3 (complex): End point of curve. |
| |
| Returns: |
| tuple: Three cubic Beziers (each expressed as a tuple of four complex |
| values). |
| """ |
| mid1 = (8 * p0 + 12 * p1 + 6 * p2 + p3) * (1 / 27) |
| deriv1 = (p3 + 3 * p2 - 4 * p0) * (1 / 27) |
| mid2 = (p0 + 6 * p1 + 12 * p2 + 8 * p3) * (1 / 27) |
| deriv2 = (4 * p3 - 3 * p1 - p0) * (1 / 27) |
| return ( |
| (p0, (2 * p0 + p1) / 3.0, mid1 - deriv1, mid1), |
| (mid1, mid1 + deriv1, mid2 - deriv2, mid2), |
| (mid2, mid2 + deriv2, (p2 + 2 * p3) / 3.0, p3), |
| ) |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.returns(cython.complex) |
| @cython.locals( |
| t=cython.double, |
| p0=cython.complex, |
| p1=cython.complex, |
| p2=cython.complex, |
| p3=cython.complex, |
| ) |
| @cython.locals(_p1=cython.complex, _p2=cython.complex) |
| def cubic_approx_control(t, p0, p1, p2, p3): |
| """Approximate a cubic Bezier using a quadratic one. |
| |
| Args: |
| t (double): Position of control point. |
| p0 (complex): Start point of curve. |
| p1 (complex): First handle of curve. |
| p2 (complex): Second handle of curve. |
| p3 (complex): End point of curve. |
| |
| Returns: |
| complex: Location of candidate control point on quadratic curve. |
| """ |
| _p1 = p0 + (p1 - p0) * 1.5 |
| _p2 = p3 + (p2 - p3) * 1.5 |
| return _p1 + (_p2 - _p1) * t |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.returns(cython.complex) |
| @cython.locals(a=cython.complex, b=cython.complex, c=cython.complex, d=cython.complex) |
| @cython.locals(ab=cython.complex, cd=cython.complex, p=cython.complex, h=cython.double) |
| def calc_intersect(a, b, c, d): |
| """Calculate the intersection of two lines. |
| |
| Args: |
| a (complex): Start point of first line. |
| b (complex): End point of first line. |
| c (complex): Start point of second line. |
| d (complex): End point of second line. |
| |
| Returns: |
| complex: Location of intersection if one present, ``complex(NaN,NaN)`` |
| if no intersection was found. |
| """ |
| ab = b - a |
| cd = d - c |
| p = ab * 1j |
| try: |
| h = dot(p, a - c) / dot(p, cd) |
| except ZeroDivisionError: |
| return complex(NAN, NAN) |
| return c + cd * h |
| |
| |
| @cython.cfunc |
| @cython.returns(cython.int) |
| @cython.locals( |
| tolerance=cython.double, |
| p0=cython.complex, |
| p1=cython.complex, |
| p2=cython.complex, |
| p3=cython.complex, |
| ) |
| @cython.locals(mid=cython.complex, deriv3=cython.complex) |
| def cubic_farthest_fit_inside(p0, p1, p2, p3, tolerance): |
| """Check if a cubic Bezier lies within a given distance of the origin. |
| |
| "Origin" means *the* origin (0,0), not the start of the curve. Note that no |
| checks are made on the start and end positions of the curve; this function |
| only checks the inside of the curve. |
| |
| Args: |
| p0 (complex): Start point of curve. |
| p1 (complex): First handle of curve. |
| p2 (complex): Second handle of curve. |
| p3 (complex): End point of curve. |
| tolerance (double): Distance from origin. |
| |
| Returns: |
| bool: True if the cubic Bezier ``p`` entirely lies within a distance |
| ``tolerance`` of the origin, False otherwise. |
| """ |
| # First check p2 then p1, as p2 has higher error early on. |
| if abs(p2) <= tolerance and abs(p1) <= tolerance: |
| return True |
| |
| # Split. |
| mid = (p0 + 3 * (p1 + p2) + p3) * 0.125 |
| if abs(mid) > tolerance: |
| return False |
| deriv3 = (p3 + p2 - p1 - p0) * 0.125 |
| return cubic_farthest_fit_inside( |
| p0, (p0 + p1) * 0.5, mid - deriv3, mid, tolerance |
| ) and cubic_farthest_fit_inside(mid, mid + deriv3, (p2 + p3) * 0.5, p3, tolerance) |
| |
| |
| @cython.cfunc |
| @cython.inline |
| @cython.locals(tolerance=cython.double) |
| @cython.locals( |
| q1=cython.complex, |
| c0=cython.complex, |
| c1=cython.complex, |
| c2=cython.complex, |
| c3=cython.complex, |
| ) |
| def cubic_approx_quadratic(cubic, tolerance): |
| """Approximate a cubic Bezier with a single quadratic within a given tolerance. |
| |
| Args: |
| cubic (sequence): Four complex numbers representing control points of |
| the cubic Bezier curve. |
| tolerance (double): Permitted deviation from the original curve. |
| |
| Returns: |
| Three complex numbers representing control points of the quadratic |
| curve if it fits within the given tolerance, or ``None`` if no suitable |
| curve could be calculated. |
| """ |
| |
| q1 = calc_intersect(cubic[0], cubic[1], cubic[2], cubic[3]) |
| if math.isnan(q1.imag): |
| return None |
| c0 = cubic[0] |
| c3 = cubic[3] |
| c1 = c0 + (q1 - c0) * (2 / 3) |
| c2 = c3 + (q1 - c3) * (2 / 3) |
| if not cubic_farthest_fit_inside(0, c1 - cubic[1], c2 - cubic[2], 0, tolerance): |
| return None |
| return c0, q1, c3 |
| |
| |
| @cython.cfunc |
| @cython.locals(n=cython.int, tolerance=cython.double) |
| @cython.locals(i=cython.int) |
| @cython.locals(all_quadratic=cython.int) |
| @cython.locals( |
| c0=cython.complex, c1=cython.complex, c2=cython.complex, c3=cython.complex |
| ) |
| @cython.locals( |
| q0=cython.complex, |
| q1=cython.complex, |
| next_q1=cython.complex, |
| q2=cython.complex, |
| d1=cython.complex, |
| ) |
| def cubic_approx_spline(cubic, n, tolerance, all_quadratic): |
| """Approximate a cubic Bezier curve with a spline of n quadratics. |
| |
| Args: |
| cubic (sequence): Four complex numbers representing control points of |
| the cubic Bezier curve. |
| n (int): Number of quadratic Bezier curves in the spline. |
| tolerance (double): Permitted deviation from the original curve. |
| |
| Returns: |
| A list of ``n+2`` complex numbers, representing control points of the |
| quadratic spline if it fits within the given tolerance, or ``None`` if |
| no suitable spline could be calculated. |
| """ |
| |
| if n == 1: |
| return cubic_approx_quadratic(cubic, tolerance) |
| if n == 2 and all_quadratic == False: |
| return cubic |
| |
| cubics = split_cubic_into_n_iter(cubic[0], cubic[1], cubic[2], cubic[3], n) |
| |
| # calculate the spline of quadratics and check errors at the same time. |
| next_cubic = next(cubics) |
| next_q1 = cubic_approx_control( |
| 0, next_cubic[0], next_cubic[1], next_cubic[2], next_cubic[3] |
| ) |
| q2 = cubic[0] |
| d1 = 0j |
| spline = [cubic[0], next_q1] |
| for i in range(1, n + 1): |
| # Current cubic to convert |
| c0, c1, c2, c3 = next_cubic |
| |
| # Current quadratic approximation of current cubic |
| q0 = q2 |
| q1 = next_q1 |
| if i < n: |
| next_cubic = next(cubics) |
| next_q1 = cubic_approx_control( |
| i / (n - 1), next_cubic[0], next_cubic[1], next_cubic[2], next_cubic[3] |
| ) |
| spline.append(next_q1) |
| q2 = (q1 + next_q1) * 0.5 |
| else: |
| q2 = c3 |
| |
| # End-point deltas |
| d0 = d1 |
| d1 = q2 - c3 |
| |
| if abs(d1) > tolerance or not cubic_farthest_fit_inside( |
| d0, |
| q0 + (q1 - q0) * (2 / 3) - c1, |
| q2 + (q1 - q2) * (2 / 3) - c2, |
| d1, |
| tolerance, |
| ): |
| return None |
| spline.append(cubic[3]) |
| |
| return spline |
| |
| |
| @cython.locals(max_err=cython.double) |
| @cython.locals(n=cython.int) |
| @cython.locals(all_quadratic=cython.int) |
| def curve_to_quadratic(curve, max_err, all_quadratic=True): |
| """Approximate a cubic Bezier curve with a spline of n quadratics. |
| |
| Args: |
| cubic (sequence): Four 2D tuples representing control points of |
| the cubic Bezier curve. |
| max_err (double): Permitted deviation from the original curve. |
| all_quadratic (bool): If True (default) returned value is a |
| quadratic spline. If False, it's either a single quadratic |
| curve or a single cubic curve. |
| |
| Returns: |
| If all_quadratic is True: A list of 2D tuples, representing |
| control points of the quadratic spline if it fits within the |
| given tolerance, or ``None`` if no suitable spline could be |
| calculated. |
| |
| If all_quadratic is False: Either a quadratic curve (if length |
| of output is 3), or a cubic curve (if length of output is 4). |
| """ |
| |
| curve = [complex(*p) for p in curve] |
| |
| for n in range(1, MAX_N + 1): |
| spline = cubic_approx_spline(curve, n, max_err, all_quadratic) |
| if spline is not None: |
| # done. go home |
| return [(s.real, s.imag) for s in spline] |
| |
| raise ApproxNotFoundError(curve) |
| |
| |
| @cython.locals(l=cython.int, last_i=cython.int, i=cython.int) |
| @cython.locals(all_quadratic=cython.int) |
| def curves_to_quadratic(curves, max_errors, all_quadratic=True): |
| """Return quadratic Bezier splines approximating the input cubic Beziers. |
| |
| Args: |
| curves: A sequence of *n* curves, each curve being a sequence of four |
| 2D tuples. |
| max_errors: A sequence of *n* floats representing the maximum permissible |
| deviation from each of the cubic Bezier curves. |
| all_quadratic (bool): If True (default) returned values are a |
| quadratic spline. If False, they are either a single quadratic |
| curve or a single cubic curve. |
| |
| Example:: |
| |
| >>> curves_to_quadratic( [ |
| ... [ (50,50), (100,100), (150,100), (200,50) ], |
| ... [ (75,50), (120,100), (150,75), (200,60) ] |
| ... ], [1,1] ) |
| [[(50.0, 50.0), (75.0, 75.0), (125.0, 91.66666666666666), (175.0, 75.0), (200.0, 50.0)], [(75.0, 50.0), (97.5, 75.0), (135.41666666666666, 82.08333333333333), (175.0, 67.5), (200.0, 60.0)]] |
| |
| The returned splines have "implied oncurve points" suitable for use in |
| TrueType ``glif`` outlines - i.e. in the first spline returned above, |
| the first quadratic segment runs from (50,50) to |
| ( (75 + 125)/2 , (120 + 91.666..)/2 ) = (100, 83.333...). |
| |
| Returns: |
| If all_quadratic is True, a list of splines, each spline being a list |
| of 2D tuples. |
| |
| If all_quadratic is False, a list of curves, each curve being a quadratic |
| (length 3), or cubic (length 4). |
| |
| Raises: |
| fontTools.cu2qu.Errors.ApproxNotFoundError: if no suitable approximation |
| can be found for all curves with the given parameters. |
| """ |
| |
| curves = [[complex(*p) for p in curve] for curve in curves] |
| assert len(max_errors) == len(curves) |
| |
| l = len(curves) |
| splines = [None] * l |
| last_i = i = 0 |
| n = 1 |
| while True: |
| spline = cubic_approx_spline(curves[i], n, max_errors[i], all_quadratic) |
| if spline is None: |
| if n == MAX_N: |
| break |
| n += 1 |
| last_i = i |
| continue |
| splines[i] = spline |
| i = (i + 1) % l |
| if i == last_i: |
| # done. go home |
| return [[(s.real, s.imag) for s in spline] for spline in splines] |
| |
| raise ApproxNotFoundError(curves) |
