``bezier``
==========
Helper for B |eacute| zier Curves, Triangles, and Higher Order Objects
|linux-build| |macos-build| |windows-build| |coverage|
|docs| |zenodo| |JOSS|
.. |eacute| unicode:: U+000E9 .. LATIN SMALL LETTER E WITH ACUTE
:trim:
This library provides:
* Support for B |eacute| zier `Curves`_
* Support for B |eacute| zier `Triangles`_
Dive in and take a look!
.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/triangles6Q_and_7Q.png
:align: center
Why B |eacute| zier?
--------------------
A B |eacute| zier curve (and triangle, etc.) is a parametric curve
that uses the `Bernstein basis`_:
.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/bernstein_basis.png
:align: center
to define a curve as a linear combination:
.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/bezier_defn.png
:align: center
This comes from the fact that the weights sum to one:
.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/sum_to_unity.png
:align: center
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, B |eacute| zier curves:
* can easily model geometric objects as parametric curves, triangles, 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 triangles, 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
.. _retrospective: https://dx.doi.org/10.1016/j.cagd.2012.03.001
.. _Bernstein basis: https://en.wikipedia.org/wiki/Bernstein_polynomial
.. _de Casteljau's algorithm: https://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm
.. _FEM: https://en.wikipedia.org/wiki/Finite_element_method
.. _NURBS: https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline
Installing
----------
The ``bezier`` Python package can be installed with `pip`_:
.. code-block:: console
$ python -m pip install --upgrade bezier
$ python3.12 -m pip install --upgrade bezier
$ # To install optional dependencies, e.g. SymPy
$ python -m pip install --upgrade bezier[full]
To install a pure Python version (i.e. with no binary extension):
.. code-block:: console
$ BEZIER_NO_EXTENSION=true \
> python -m pip install --upgrade bezier --no-binary=bezier
``bezier`` is open-source, so you can alternatively grab the source
code from `GitHub`_ and install from source.
.. _pip: https://pip.pypa.io
.. _GitHub: https://github.com/dhermes/bezier/
Getting Started
---------------
For example, to create a curve:
.. code-block:: python
>>> import bezier
>>> import numpy as np
>>> 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:
.. code-block:: python
>>> 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):
.. code-block:: python
>>> 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)
.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/curves1_and_13.png
:align: center
For API-level documentation, check out the B |eacute| zier 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.
Citation
--------
For publications that use ``bezier``, there is a `JOSS paper`_ that can be
cited. The following BibTeX entry can be used:
.. code-block:: rest
@article{Hermes2017,
doi = {10.21105/joss.00267},
url = {https://doi.org/10.21105%2Fjoss.00267},
year = {2017},
month = {Aug},
publisher = {The Open Journal},
volume = {2},
number = {16},
pages = {267},
author = {Danny Hermes},
title = {Helper for B{\'{e}}zier Curves, Triangles, and Higher Order Objects},
journal = {The Journal of Open Source Software}
}
A **particular** version of this library can be cited via a Zenodo DOI; see
a full `list by version`_.
.. _JOSS paper: https://joss.theoj.org/papers/10.21105/joss.00267
.. _list by version: https://zenodo.org/search?page=1&size=20&q=conceptrecid:%22838307%22&sort=-version&all_versions=True
License
-------
``bezier`` is made available under the Apache 2.0 License. For more
details, see `the LICENSE`_.
.. _Curves: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.curve.html
.. _Triangles: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.triangle.html
.. _package: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.html
.. _DEVELOPMENT doc: https://github.com/dhermes/bezier/blob/2024.6.20/DEVELOPMENT.rst
.. _the LICENSE: https://github.com/dhermes/bezier/blob/2024.6.20/LICENSE
.. |docs| image:: https://readthedocs.org/projects/bezier/badge/?version=2024.6.20
:target: https://bezier.readthedocs.io/en/2024.6.20/
:alt: Documentation Status
.. |linux-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/linux-passing.svg?sanitize=true
:target: https://github.com/dhermes/bezier/actions/runs/9607943395
:alt: Linux Build (GitHub Actions)
.. |macos-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/macos-passing.svg?sanitize=true
:target: https://github.com/dhermes/bezier/actions/runs/9607943397
:alt: macOS Build (GitHub Actions)
.. |windows-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/windows-passing.svg?sanitize=true
:target: https://github.com/dhermes/bezier/actions/runs/9607943396
:alt: Windows Build (GitHub Actions)
.. |coverage| image:: https://s3.amazonaws.com/assets.coveralls.io/badges/coveralls_100.svg
:target: https://coveralls.io/builds/68215528
:alt: Code Coverage
.. |zenodo| image:: https://zenodo.org/badge/73047402.svg
:target: https://zenodo.org/badge/latestdoi/73047402
:alt: Zenodo DOI for ``bezier``
.. |JOSS| image:: https://joss.theoj.org/papers/10.21105/joss.00267/status.svg
:target: https://dx.doi.org/10.21105/joss.00267
:alt: "Journal of Open Source Science" DOI for ``bezier``
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"description": "``bezier``\n==========\n\n Helper for B |eacute| zier Curves, Triangles, and Higher Order Objects\n\n|linux-build| |macos-build| |windows-build| |coverage|\n\n|docs| |zenodo| |JOSS|\n\n.. |eacute| unicode:: U+000E9 .. LATIN SMALL LETTER E WITH ACUTE\n :trim:\n\nThis library provides:\n\n* Support for B |eacute| zier `Curves`_\n* Support for B |eacute| zier `Triangles`_\n\nDive in and take a look!\n\n.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/triangles6Q_and_7Q.png\n :align: center\n\nWhy B |eacute| zier?\n--------------------\n\nA B |eacute| zier curve (and triangle, etc.) is a parametric curve\nthat uses the `Bernstein basis`_:\n\n.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/bernstein_basis.png\n :align: center\n\nto define a curve as a linear combination:\n\n.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/bezier_defn.png\n :align: center\n\nThis comes from the fact that the weights sum to one:\n\n.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/sum_to_unity.png\n :align: center\n\nThis can be generalized to higher order by considering three, four, etc.\nnon-negative weights that sum to one (in the above we have the two\nnon-negative weights ``s`` and ``1 - s``).\n\nDue to their simple form, B |eacute| zier curves:\n\n* can easily model geometric objects as parametric curves, triangles, etc.\n* can be computed in an efficient and numerically stable way via\n `de Casteljau's algorithm`_\n* can utilize convex optimization techniques for many algorithms (such as\n curve-curve intersection), since curves (and triangles, etc.)\n are convex combinations of the basis\n\nMany applications -- as well as the history of their development --\nare described in\n\"The Bernstein polynomial basis: A centennial `retrospective`_\",\nfor example;\n\n* aids physical analysis using finite element methods (`FEM`_) on\n isogeometric models by using geometric shape functions called\n `NURBS`_ to represent data\n* used in robust control of dynamic systems; utilizes convexity to\n create a hull of curves\n\n.. _retrospective: https://dx.doi.org/10.1016/j.cagd.2012.03.001\n.. _Bernstein basis: https://en.wikipedia.org/wiki/Bernstein_polynomial\n.. _de Casteljau's algorithm: https://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm\n.. _FEM: https://en.wikipedia.org/wiki/Finite_element_method\n.. _NURBS: https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline\n\nInstalling\n----------\n\nThe ``bezier`` Python package can be installed with `pip`_:\n\n.. code-block:: console\n\n $ python -m pip install --upgrade bezier\n $ python3.12 -m pip install --upgrade bezier\n $ # To install optional dependencies, e.g. SymPy\n $ python -m pip install --upgrade bezier[full]\n\nTo install a pure Python version (i.e. with no binary extension):\n\n.. code-block:: console\n\n $ BEZIER_NO_EXTENSION=true \\\n > python -m pip install --upgrade bezier --no-binary=bezier\n\n``bezier`` is open-source, so you can alternatively grab the source\ncode from `GitHub`_ and install from source.\n\n.. _pip: https://pip.pypa.io\n.. _GitHub: https://github.com/dhermes/bezier/\n\nGetting Started\n---------------\n\nFor example, to create a curve:\n\n.. code-block:: python\n\n >>> import bezier\n >>> import numpy as np\n >>> nodes1 = np.asfortranarray([\n ... [0.0, 0.5, 1.0],\n ... [0.0, 1.0, 0.0],\n ... ])\n >>> curve1 = bezier.Curve(nodes1, degree=2)\n\nThe intersection (points) between two curves can\nalso be determined:\n\n.. code-block:: python\n\n >>> nodes2 = np.asfortranarray([\n ... [0.0, 0.25, 0.5, 0.75, 1.0],\n ... [0.0, 2.0 , -2.0, 2.0 , 0.0],\n ... ])\n >>> curve2 = bezier.Curve.from_nodes(nodes2)\n >>> intersections = curve1.intersect(curve2)\n >>> intersections\n array([[0.31101776, 0.68898224, 0. , 1. ],\n [0.31101776, 0.68898224, 0. , 1. ]])\n >>> s_vals = np.asfortranarray(intersections[0, :])\n >>> points = curve1.evaluate_multi(s_vals)\n >>> points\n array([[0.31101776, 0.68898224, 0. , 1. ],\n [0.42857143, 0.42857143, 0. , 0. ]])\n\nand then we can plot these curves (along with their\nintersections):\n\n.. code-block:: python\n\n >>> import seaborn\n >>> seaborn.set()\n >>>\n >>> ax = curve1.plot(num_pts=256)\n >>> _ = curve2.plot(num_pts=256, ax=ax)\n >>> lines = ax.plot(\n ... points[0, :], points[1, :],\n ... marker=\"o\", linestyle=\"None\", color=\"black\")\n >>> _ = ax.axis(\"scaled\")\n >>> _ = ax.set_xlim(-0.125, 1.125)\n >>> _ = ax.set_ylim(-0.0625, 0.625)\n\n.. image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/images/curves1_and_13.png\n :align: center\n\nFor API-level documentation, check out the B |eacute| zier Python\n`package`_ documentation.\n\nDevelopment\n-----------\n\nTo work on adding a feature or to run the functional tests, see the\n`DEVELOPMENT doc`_ for more information on how to get\nstarted.\n\nCitation\n--------\n\nFor publications that use ``bezier``, there is a `JOSS paper`_ that can be\ncited. The following BibTeX entry can be used:\n\n.. code-block:: rest\n\n @article{Hermes2017,\n doi = {10.21105/joss.00267},\n url = {https://doi.org/10.21105%2Fjoss.00267},\n year = {2017},\n month = {Aug},\n publisher = {The Open Journal},\n volume = {2},\n number = {16},\n pages = {267},\n author = {Danny Hermes},\n title = {Helper for B{\\'{e}}zier Curves, Triangles, and Higher Order Objects},\n journal = {The Journal of Open Source Software}\n }\n\nA **particular** version of this library can be cited via a Zenodo DOI; see\na full `list by version`_.\n\n.. _JOSS paper: https://joss.theoj.org/papers/10.21105/joss.00267\n.. _list by version: https://zenodo.org/search?page=1&size=20&q=conceptrecid:%22838307%22&sort=-version&all_versions=True\n\nLicense\n-------\n\n``bezier`` is made available under the Apache 2.0 License. For more\ndetails, see `the LICENSE`_.\n\n.. _Curves: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.curve.html\n.. _Triangles: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.triangle.html\n.. _package: https://bezier.readthedocs.io/en/2024.6.20/python/reference/bezier.html\n.. _DEVELOPMENT doc: https://github.com/dhermes/bezier/blob/2024.6.20/DEVELOPMENT.rst\n.. _the LICENSE: https://github.com/dhermes/bezier/blob/2024.6.20/LICENSE\n\n.. |docs| image:: https://readthedocs.org/projects/bezier/badge/?version=2024.6.20\n :target: https://bezier.readthedocs.io/en/2024.6.20/\n :alt: Documentation Status\n.. |linux-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/linux-passing.svg?sanitize=true\n :target: https://github.com/dhermes/bezier/actions/runs/9607943395\n :alt: Linux Build (GitHub Actions)\n.. |macos-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/macos-passing.svg?sanitize=true\n :target: https://github.com/dhermes/bezier/actions/runs/9607943397\n :alt: macOS Build (GitHub Actions)\n.. |windows-build| image:: https://raw.githubusercontent.com/dhermes/bezier/2024.6.20/docs/windows-passing.svg?sanitize=true\n :target: https://github.com/dhermes/bezier/actions/runs/9607943396\n :alt: Windows Build (GitHub Actions)\n.. |coverage| image:: https://s3.amazonaws.com/assets.coveralls.io/badges/coveralls_100.svg\n :target: https://coveralls.io/builds/68215528\n :alt: Code Coverage\n.. |zenodo| image:: https://zenodo.org/badge/73047402.svg\n :target: https://zenodo.org/badge/latestdoi/73047402\n :alt: Zenodo DOI for ``bezier``\n.. |JOSS| image:: https://joss.theoj.org/papers/10.21105/joss.00267/status.svg\n :target: https://dx.doi.org/10.21105/joss.00267\n :alt: \"Journal of Open Source Science\" DOI for ``bezier``\n",
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