bezier

Helper for Bézier Curves, Triangles, and Higher Order Objects

This library provides:

Support for Bézier Curves
Support for Bézier Surfaces

Dive in and take a look!

https://raw.githubusercontent.com/dhermes/bezier/0.10.0/docs/images/surfaces6Q_and_7Q.png

Why Bézier?

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

https://raw.githubusercontent.com/dhermes/bezier/0.10.0/docs/images/bernstein_basis.png

to define a curve as a linear combination:

https://raw.githubusercontent.com/dhermes/bezier/0.10.0/docs/images/bezier_defn.png

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

https://raw.githubusercontent.com/dhermes/bezier/0.10.0/docs/images/sum_to_unity.png

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

The bezier Python package can be installed with pip:

$ python    -m pip install --upgrade bezier
$ python3.7 -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()

https://raw.githubusercontent.com/dhermes/bezier/0.10.0/docs/images/curves1_and_13.png

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

License

bezier is made available under the Apache 2.0 License. For more details, see the LICENSE.

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Release history Release notifications | RSS feed

2024.6.20

Jun 21, 2024

2023.7.28

Jul 29, 2023

2021.2.12

Feb 14, 2021

2020.5.19

May 20, 2020

2020.2.3

Feb 4, 2020

2020.1.14

Jan 15, 2020

0.11.0

Jan 12, 2020

This release

0.10.0

Aug 14, 2019

0.9.0

Oct 31, 2018

0.8.0

Mar 19, 2018

0.7.0

Feb 7, 2018

0.6.4

Jan 28, 2018

0.6.3

Jan 25, 2018

0.6.2

Jan 23, 2018

0.6.1

Jan 12, 2018

0.6.0

Jan 10, 2018

0.5.0

Sep 8, 2017

0.4.0

Jun 7, 2017

0.3.0

Jan 11, 2017

0.2.1

Dec 29, 2016

0.2.0

Dec 14, 2016

0.1.1

Nov 29, 2016

0.1.0

Nov 26, 2016

0.0.1

Nov 8, 2016

bezier 0.10.0

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