bezier¶

Helper for Bézier Curves, Triangles, and Higher Order Objects

This library provides:

Dive in and take a look!

Why Bézier?¶

A Bézier curve (and surface, etc.) is a parametric curve that uses the Bernstein basis:

$b_{j, n} = \binom{n}{j} s^j (1 - s)^{n - j}$

to define a curve as a linear combination:

$B(s) = \sum_{j = 0}^n b_{j, n} \cdot v_j.$

This comes from the fact that the weights sum to one:

$b_{0, n} + b_{1, n} + \cdots + b_{n, n} = \left(s + (1 - s)\right)^n = 1.$

This can be generalized to higher order by considering three, four, etc. non-negative weights that sum to one (in the above we have the two non-negative weights $$s$$ and $$1 - s$$).

Due to their simple form, Bézier curves:

• can easily model geometric objects as parametric curves, surfaces, etc.
• can be computed in an efficient and numerically stable way via de Casteljau’s algorithm
• can utilize convex optimization techniques for many algorithms (such as curve-curve intersection), since curves (and surfaces, etc.) are convex combinations of the basis

Many applications – as well as the history of their development – are described in “The Bernstein polynomial basis: A centennial retrospective”, for example;

• aids physical analysis using finite element methods (FEM) on isogeometric models by using geometric shape functions called NURBS to represent data
• used in robust control of dynamic systems; utilizes convexity to create a hull of curves

Installing¶

bezier can be installed with pip:

$python -m pip install --upgrade bezier$ python2.7 -m pip install --upgrade bezier
\$ python3.6 -m pip install --upgrade bezier

bezier is open-source, so you can alternatively grab the source code from GitHub and install from source.

Getting Started¶

For example, to create a curve:

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.5, 1.0],
...     [0.0, 1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)

The intersection (points) between two curves can also be determined:

>>> nodes2 = np.asfortranarray([
...     [0.0, 0.25,  0.5, 0.75, 1.0],
...     [0.0, 2.0 , -2.0, 2.0 , 0.0],
... ])
>>> curve2 = bezier.Curve.from_nodes(nodes2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.31101776, 0.68898224, 0. , 1. ],
[0.31101776, 0.68898224, 0. , 1. ]])
>>> s_vals = np.asfortranarray(intersections[0, :])
>>> points = curve1.evaluate_multi(s_vals)
>>> points
array([[0.31101776, 0.68898224, 0. , 1. ],
[0.42857143, 0.42857143, 0. , 0. ]])

and then we can plot these curves (along with their intersections):

>>> import matplotlib.pyplot as plt
>>> import seaborn
>>> seaborn.set()
>>>
>>> ax = curve1.plot(num_pts=256)
>>> _ = curve2.plot(num_pts=256, ax=ax)
>>> lines = ax.plot(
...     points[0, :], points[1, :],
...     marker='o', linestyle='None', color='black')
>>> _ = ax.axis('scaled')
>>> _ = ax.set_xlim(-0.125, 1.125)
>>> _ = ax.set_ylim(-0.0625, 0.625)
>>> plt.show()

For API-level documentation, check out the Bézier Package documentation.

Development¶

To work on adding a feature or to run the functional tests, see the DEVELOPMENT doc for more information on how to get started.