def reduce_(self):
r"""Return a degree-reduced version of the current curve.
.. _pseudo-inverse:
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse
Does this by converting the current nodes :math:`v_0, \ldots, v_n`
to new nodes :math:`w_0, \ldots, w_{n - 1}` that correspond to
reversing the :meth:`elevate` process.
This uses the `pseudo-inverse`_ of the elevation matrix. For example
when elevating from degree 2 to 3, the matrix :math:`E_2` is given by
.. math::
\mathbf{v} = \left[\begin{array}{c} v_0 \\ v_1 \\ v_2
\end{array}\right] \longmapsto \left[\begin{array}{c} v_0 \\
\frac{v_0 + 2 v_1}{3} \\ \frac{2 v_1 + v_2}{3} \\ v_2
\end{array}\right] = \frac{1}{3}
\left[\begin{array}{c c} 3 & 0 \\ 2 & 1 \\ 1 & 2
\\ 0 & 3 \end{array}\right] \mathbf{v}
and the pseudo-inverse is given by
.. math::
R_2 = \left(E_2^T E_2\right)^{-1} E_2^T = \frac{1}{20}
\left[\begin{array}{c c c c} 19 & 3 & -3 & 1 \\
-5 & 15 & 15 & -5 \\ 1 & -3 & 3 & 19
\end{array}\right].
.. warning::
Though degree-elevation preserves the start and end nodes, degree
reduction has no such guarantee. Rather, the nodes produced are
"best" in the least squares sense (when solving the normal
equations).
.. image:: ../images/curve_reduce.png
:align: center
.. testsetup:: curve-reduce, curve-reduce-approx
import numpy as np
import bezier
.. doctest:: curve-reduce
:options: +NORMALIZE_WHITESPACE
>>> nodes = np.asfortranarray([
... [-3.0, 3.0],
... [ 0.0, 2.0],
... [ 1.0, 3.0],
... [ 0.0, 6.0],
... ])
>>> curve = bezier.Curve(nodes, degree=3)
>>> reduced = curve.reduce_()
>>> reduced
<Curve (degree=2, dimension=2)>
>>> reduced.nodes
array([[-3. , 3. ],
[ 1.5, 1.5],
[ 0. , 6. ]])
.. testcleanup:: curve-reduce
import make_images
make_images.curve_reduce(curve, reduced)
In the case that the current curve **is not** degree-elevated.
.. image:: ../images/curve_reduce_approx.png
:align: center
.. doctest:: curve-reduce-approx
:options: +NORMALIZE_WHITESPACE
>>> nodes = np.asfortranarray([
... [0.0 , 2.5],
... [1.25, 5.0],
... [3.75, 7.5],
... [5.0 , 2.5],
... ])
>>> curve = bezier.Curve(nodes, degree=3)
>>> reduced = curve.reduce_()
>>> reduced
<Curve (degree=2, dimension=2)>
>>> reduced.nodes
array([[-0.125, 2.125],
[ 2.5 , 8.125],
[ 5.125, 2.875]])
.. testcleanup:: curve-reduce-approx
import make_images
make_images.curve_reduce_approx(curve, reduced)
Returns:
Curve: The degree-reduced curve.
"""
new_nodes = _curve_helpers.reduce_pseudo_inverse(
self._nodes, self._degree)
root = None if self._root is self else self._root
return Curve(new_nodes,
self._degree - 1,
start=self._start,
end=self._end,
root=root,
_copy=False)
def _call_function_under_test(nodes, degree):
from bezier import _curve_helpers
return _curve_helpers.reduce_pseudo_inverse(nodes, degree)