Helper for Bézier Curves, Triangles, and Higher Order Objects
This library provides:
Dive in and take a look!
A Bézier curve (and surface, etc.) is a parametric curve that uses the Bernstein basis:
to define a curve as a linear combination:
This comes from the fact that the weights sum to one:
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;
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.
For example, to create a curve:
>>> nodes1 = np.asfortranarray([ ... [0.0, 0.0], ... [0.5, 1.0], ... [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 , 0.0], ... [0.25, 2.0], ... [0.5 , -2.0], ... [0.75, 2.0], ... [1.0 , 0.0], ... ]) >>> curve2 = bezier.Curve.from_nodes(nodes2) >>> intersections = curve1.intersect(curve2) >>> intersections array([[0.31101776, 0.31101776], [0.68898224, 0.68898224], [0. , 0. ], [1. , 1. ]]) >>> s_vals = intersections[:, 0] >>> points = curve1.evaluate_multi(s_vals) >>> points array([[0.31101776, 0.42857143], [0.68898224, 0.42857143], [0. , 0. ], [1. , 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.
To work on adding a feature or to run the functional tests, see the DEVELOPMENT doc for more information on how to get started.