.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/rotations/combine_rotations.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

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

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

.. _sphx_glr_examples_rotations_combine_rotations.py:


===================
Combining rotations
===================

This example demonstrates how to combine two rotations :math:`R_1` and :math:`R_2`, i.e.
from right to left like so

.. math::

    R_{12} = R_2 \cdot R_1.

To convince ourselves that this order is correct, we rotate a vector
:math:`\mathbf{v} = (0, 0, z)` by two rotations: rotation :math:`R_1` by
:math:`120^{\circ}` around :math:`[1 1 1]` followed by rotation :math:`R_2` by
:math:`180^{\circ}` around :math:`[1 1 0]`.
Rotation :math:`R_1` is carried out first, followed by rotation :math:`R_2`.

Note that a negative angle :math:`\omega` when *defining* a rotation in the axis-angle
representation :math:`(\hat{\mathbf{n}}, \omega)` is necessary for consistent
transformations between rotation representations.

.. GENERATED FROM PYTHON SOURCE LINES 41-101



.. image-sg:: /examples/rotations/images/sphx_glr_combine_rotations_001.png
   :alt: combine rotations
   :srcset: /examples/rotations/images/sphx_glr_combine_rotations_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    R1, R2 and R12:
    * As quaternions:
     Rotation (3,)
    [[ 0.5    -0.5    -0.5    -0.5   ]
     [ 0.     -0.7071 -0.7071 -0.    ]
     [-0.7071 -0.     -0.7071 -0.    ]]
    * As orientation matrices:
     [[[ 0.  1. -0.]
      [-0.  0.  1.]
      [ 1. -0.  0.]]

     [[ 0.  1. -0.]
      [ 1.  0.  0.]
      [ 0. -0. -1.]]

     [[-0.  0.  1.]
      [ 0.  1. -0.]
      [-1.  0. -0.]]]
    Point rotated by R12:
     [0, 0, 1] -> [1.0, -0.0, -0.0]






|

.. code-block:: Python


    import matplotlib.pyplot as plt

    from orix.plot import register_projections
    from orix.quaternion import Rotation
    from orix.vector import Vector3d

    plt.rcParams.update({"font.size": 12, "grid.alpha": 0.5})

    register_projections()  # Register our custom Matplotlib projections

    R1 = Rotation.from_axes_angles([1, 1, 1], -120, degrees=True)
    R2 = Rotation.from_axes_angles([1, 1, 0], -180, degrees=True)
    R12 = R2 * R1

    R_all = Rotation.stack([R1, R2, R12]).squeeze()
    print("R1, R2 and R12:\n* As quaternions:\n", R_all)
    print("* As orientation matrices:\n", R_all.to_matrix().squeeze().round(10))

    v_start = Vector3d.zvector()
    v_end = R12 * v_start
    print(
        "Point rotated by R12:\n",
        v_start.data.squeeze().tolist(),
        "->",
        v_end.data.squeeze().round(10).tolist(),
    )

    # Illustrate the steps of the rotation by plotting the vector before
    # (red), during (green), and after (blue) the rotation and the rotation
    # paths (first: cyan; second: magenta)
    v_intermediate = R1 * v_start

    v_si_path = Vector3d.from_path_ends(Vector3d.stack([v_start, v_intermediate]))
    v_sie_path = Vector3d.from_path_ends(Vector3d.stack([v_intermediate, v_end]))

    fig = plt.figure(layout="tight")
    ax0 = fig.add_subplot(121, projection="stereographic")
    ax1 = fig.add_subplot(122, projection="stereographic", hemisphere="lower")
    ax0.stereographic_grid(), ax1.stereographic_grid()
    Vector3d.stack([v_start, v_intermediate, v_end]).scatter(
        figure=fig,
        s=50,
        c=["r", "g", "b"],
        axes_labels=["e1", "e2"],
    )
    ax0.plot(v_si_path, color="c"), ax1.plot(v_si_path, color="c")
    ax0.plot(v_sie_path, color="m"), ax1.plot(v_sie_path, color="m")
    R1.axis.scatter(figure=fig, c="orange")
    R2.axis.scatter(figure=fig, c="k")
    text_kw = {
        "bbox": {"alpha": 0.5, "fc": "w", "boxstyle": "round,pad=0.1"},
        "ha": "right",
        "va": "top",
    }
    ax0.text(v_start, s="Start", **text_kw)
    ax1.text(v_intermediate, s="Intermediate", **text_kw)
    ax1.text(v_end, s="End", **text_kw)
    ax1.text(R1.axis, s=r"$\hat{\mathbf{n}}_1$", **text_kw)
    ax0.text(R2.axis, s=r"$\hat{\mathbf{n}}_2$", **text_kw)


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

   **Total running time of the script:** (0 minutes 0.381 seconds)

**Estimated memory usage:**  548 MB


.. _sphx_glr_download_examples_rotations_combine_rotations.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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