code to compute bezier segment / path lengths

lib/matplotlib/bezier.py

@@ -4,6 +4,7 @@
 
 import math
 import warnings
+from collections import deque
 
 import numpy as np
 
@@ -220,6 +221,93 @@ def point_at_t(self, t):
         """
         return tuple(self(t))
 
+    def split_at_t(self, t):
+        """
+        Split into two Bezier curves using de casteljau's algorithm.
+
+        Parameters
+        ----------
+        t : float
+            Point in [0,1] at which to split into two curves
+
+        Returns
+        -------
+        B1, B2 : BezierSegment
+            The two sub-curves.
+        """
+        new_cpoints = split_de_casteljau(self._cpoints, t)
+        return BezierSegment(new_cpoints[0]), BezierSegment(new_cpoints[1])
+
+    def control_net_length(self):
+        """Sum of lengths between control points"""
+        L = 0
+        N, d = self._cpoints.shape
+        for i in range(N - 1):
+            L += np.linalg.norm(self._cpoints[i+1] - self._cpoints[i])
+        return L
+
+    def arc_length(self, rtol=1e-4, atol=1e-6):
+        """
+        Estimate the length using iterative refinement.
+
+        Our estimate is just the average between the length of the chord and
+        the length of the control net.
+
+        Since the chord length and control net give lower and upper bounds
+        (respectively) on the length, this maximum possible error is tested
+        against an absolute tolerance threshold at each subdivision.
+
+        However, sometimes this estimator converges much faster than this error
+        estimate would suggest. Therefore, the relative change in the length
+        estimate between subdivisions is compared to a relative error tolerance
+        after each set of subdivisions.
+
+        Parameters
+        ----------
+        rtol : float, default 1e-4
+            If :code:`abs(est[i+1] - est[i]) <= rtol * est[i+1]`, we return
+            :code:`est[i+1]`.
+        atol : float, default 1e-6
+            If the distance between chord length and control length at any
+            point falls below this number, iteration is terminated.
+        """
+        chord = np.linalg.norm(self._cpoints[-1] - self._cpoints[0])
+        net = self.control_net_length()
+        max_err = (net - chord)/2
+        curr_est = chord + max_err
+        # early exit so we don't try to "split" paths of zero length
+        if max_err < atol:
+            return curr_est
+
+        prev_est = np.inf
+        curves = deque([self])
+        errs = deque([max_err])
+        lengths = deque([curr_est])
+        while np.abs(curr_est - prev_est) > rtol * curr_est:
+            # subdivide the *whole* curve before checking relative convergence
+            # again
+            prev_est = curr_est
+            num_curves = len(curves)
+            for i in range(num_curves):
+                curve = curves.popleft()
+                new_curves = curve.split_at_t(0.5)
+                max_err -= errs.popleft()
+                curr_est -= lengths.popleft()
+                for ncurve in new_curves:
+                    chord = np.linalg.norm(
+                            ncurve._cpoints[-1] - ncurve._cpoints[0])
+                    net = ncurve.control_net_length()
+                    nerr = (net - chord)/2
+                    nlength = chord + nerr
+                    max_err += nerr
+                    curr_est += nlength
+                    curves.append(ncurve)
+                    errs.append(nerr)
+                    lengths.append(nlength)
+                if max_err < atol:
+                    return curr_est
+        return curr_est
+
     @property
     def arc_area(self):
         r"""

lib/matplotlib/bezier.pyi

@@ -35,6 +35,9 @@ class BezierSegment:
     def __init__(self, control_points: ArrayLike) -> None: ...
     def __call__(self, t: ArrayLike) -> np.ndarray: ...
     def point_at_t(self, t: float) -> tuple[float, ...]: ...
+    def split_at_t(self, t: float) -> tuple[BezierSegment, BezierSegment]: ...
+    def control_net_length(self) -> float: ...
+    def arc_length(self, rtol: float=1e-4, atol: float=1e-6) -> float: ...
     @property
     def arc_area(self) -> float: ...
     @classmethod

lib/matplotlib/path.py

@@ -666,6 +666,37 @@ def intersects_bbox(self, bbox, filled=True):
         return _path.path_intersects_rectangle(
             self, bbox.x0, bbox.y0, bbox.x1, bbox.y1, filled)
 
+    def length(self, rtol=1e-4, atol=1e-6, **kwargs):
+        r"""
+        Get length of Path.
+
+        Equivalent to (but not computed as)
+
+        .. math::
+
+            \sum_{j=1}^N \int_0^1 ||B'_j(t)|| dt
+
+        where the sum is over the :math:`N` Bezier curves that comprise the
+        Path. Notice that this measure of length will assign zero weight to all
+        isolated points on the Path.
+
+        Parameters
+        ----------
+        rtol : float, default 1e-4
+            If :code:`abs(est[i+1] - est[i]) <= rtol * est[i+1]`, we return
+            :code:`est[i+1]`.
+        atol : float, default 1e-6
+            If the distance between chord length and control length at any
+            point falls below this number, iteration is terminated.
+
+        Returns
+        -------
+        length : float
+            The path length.
+        """
+        return np.sum([B.arc_length(rtol=rtol, atol=atol)
+                       for B, code in self.iter_bezier(**kwargs)])
+
     def signed_area(self):
         """
         Get signed area of the filled path.

lib/matplotlib/path.pyi

@@ -90,6 +90,7 @@ class Path:
     def get_extents(self, transform: Transform | None = ..., **kwargs) -> Bbox: ...
     def intersects_path(self, other: Path, filled: bool = ...) -> bool: ...
     def intersects_bbox(self, bbox: Bbox, filled: bool = ...) -> bool: ...
+    def length(self, rtol: float=1e-4, atol: float=1e-6, **kwargs) -> float: ...
     def signed_area(self) -> float: ...
     def interpolated(self, steps: int) -> Path: ...
     def to_polygons(

lib/matplotlib/tests/test_bezier.py

@@ -26,6 +26,15 @@ def test_split_bezier_with_large_values():
 _test_curves = [list(tc.path.iter_bezier())[-1][0] for tc in _test_curves]
 
 
+def _integral_arc_length(B):
+    dB = B.differentiate(B)
+    def integrand(t):
+        return np.linalg.norm(dB(t), axis=1)
+    x = np.linspace(0, 1, 1000)
+    y = integrand(x)
+    return np.trapezoid(y, x)
+
+
 def _integral_arc_area(B):
     """(Signed) area swept out by ray from origin to curve."""
     dB = B.differentiate(B)
@@ -42,3 +51,10 @@ def integrand(t):
 @pytest.mark.parametrize("B", _test_curves)
 def test_area_formula(B):
     assert np.isclose(_integral_arc_area(B), B.arc_area)
+
+
+@pytest.mark.parametrize("B", _test_curves)
+def test_length_iteration(B):
+    assert np.isclose(_integral_arc_length(B),
+                      B.arc_length(rtol=1e-5, atol=1e-8),
+                      rtol=1e-5, atol=1e-8)

lib/matplotlib/tests/test_path.py

@@ -88,24 +88,27 @@ def test_contains_points_negative_radius():
     np.testing.assert_equal(result, [True, False, False])
 
 
-_ExampleCurve = namedtuple('_ExampleCurve', ['path', 'extents', 'area'])
+_ExampleCurve = namedtuple('_ExampleCurve',
+                           ['path', 'extents', 'area', 'length'])
 _test_curves = [
     # interior extrema determine extents and degenerate derivative
     _ExampleCurve(Path([[0, 0], [1, 0], [1, 1], [0, 1]],
                        [Path.MOVETO, Path.CURVE4, Path.CURVE4, Path.CURVE4]),
-                  extents=(0., 0., 0.75, 1.), area=0.6),
+                  extents=(0., 0., 0.75, 1.), area=0.6, length=2.0),
     # a quadratic curve, clockwise
-    _ExampleCurve(Path([[0, 0], [0, 1], [1, 0]],
-                       [Path.MOVETO, Path.CURVE3, Path.CURVE3]),
-                  extents=(0., 0., 1., 0.5), area=-1/3),
+    _ExampleCurve(Path([[0, 0], [0, 1], [1, 0]], [Path.MOVETO, Path.CURVE3,
+                  Path.CURVE3]), extents=(0., 0., 1., 0.5), area=-1/3,
+                  length=(1/25)*(10 + 15*np.sqrt(2) + np.sqrt(5)
+                  * (np.arcsinh(2) + np.arcsinh(3)))),
     # a linear curve, degenerate vertically
     _ExampleCurve(Path([[0, 1], [1, 1]], [Path.MOVETO, Path.LINETO]),
-                  extents=(0., 1., 1., 1.), area=0.),
+                  extents=(0., 1., 1., 1.), area=0., length=1.0),
     # a point
     _ExampleCurve(Path([[1, 2]], [Path.MOVETO]), extents=(1., 2., 1., 2.),
-                  area=0.),
+                  area=0., length=0),
     # non-curved triangle
-    _ExampleCurve(Path([(1, 1), (2, 1), (1.5, 2)]), extents=(1, 1, 2, 2), area=0.5),
+    _ExampleCurve(Path([(1, 1), (2, 1), (1.5, 2)]), extents=(1, 1, 2, 2),
+                  area=0.5, length=1 + np.sqrt(0.5**2 + 1)),
 ]
 
 
@@ -156,6 +159,13 @@ def test_signed_area_unit_circle():
     assert np.isclose(circ.signed_area(), np.pi)
 
 
+@pytest.mark.parametrize('precomputed_curve', _test_curves)
+def test_length_curve(precomputed_curve):
+    path, length = precomputed_curve.path, precomputed_curve.length
+    np.testing.assert_allclose(path.length(rtol=1e-5, atol=1e-8), length,
+                               rtol=1e-5, atol=1e-8)
+
+
 def test_point_in_path_nan():
     box = np.array([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]])
     p = Path(box)