.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/curves/plot_01_helix_frenet_frame.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_geometry_curves_plot_01_helix_frenet_frame.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_geometry_curves_plot_01_helix_frenet_frame.py:


The Frenet-Serret frame of a circular helix
==================================================

A helix has constant curvature and torsion -- both confirmed here
against their closed forms -- and the moving tangent/normal/binormal
frame is visualized in 2D projection alongside an ellipse (whose
curvature varies).

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import numpy as np

    from mathematicskit.geometry import frenet_serret_frame
    from mathematicskit.geometry.utils.curves_library import ellipse, helix
    from mathematicskit.geometry.visualizers.plots import plot_curve_frame








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Helix: constant curvature and torsion
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-32

.. code-block:: Python


    a, b = 3.0, 2.0
    t = np.linspace(0.0, 4.0 * np.pi, 1000)
    result = frenet_serret_frame(helix(a, b), t)

    expected_curvature = a / (a**2 + b**2)
    expected_torsion = b / (a**2 + b**2)
    print(f"curvature: mean={np.mean(result.curvature):.6f}, closed form={expected_curvature:.6f}")
    print(f"torsion:   mean={np.mean(result.torsion):.6f}, closed form={expected_torsion:.6f}")
    print(f"total arc length: {result.arc_length[-1]:.4f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    curvature: mean=0.230419, closed form=0.230769
    torsion:   mean=0.153480, closed form=0.153846
    total arc length: 45.3079




.. GENERATED FROM PYTHON SOURCE LINES 33-35

An ellipse: curvature varies around the curve
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-40

.. code-block:: Python


    ellipse_result = frenet_serret_frame(ellipse(3.0, 1.0), np.linspace(0.0, 2.0 * np.pi, 200, endpoint=False))
    print(f"\nellipse curvature range: [{ellipse_result.curvature.min():.4f}, {ellipse_result.curvature.max():.4f}]")

    plot_curve_frame(ellipse_result)



.. image-sg:: /api/gallery/geometry/curves/images/sphx_glr_plot_01_helix_frenet_frame_001.png
   :alt: Frenet-Serret frame (red=tangent, green=normal)
   :srcset: /api/gallery/geometry/curves/images/sphx_glr_plot_01_helix_frenet_frame_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    ellipse curvature range: [0.1111, 3.0000]

    <Axes: title={'center': 'Frenet-Serret frame (red=tangent, green=normal)'}>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.023 seconds)


.. _sphx_glr_download_api_gallery_geometry_curves_plot_01_helix_frenet_frame.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_helix_frenet_frame.ipynb <plot_01_helix_frenet_frame.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_helix_frenet_frame.py <plot_01_helix_frenet_frame.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_helix_frenet_frame.zip <plot_01_helix_frenet_frame.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_