-----------------------
Tetrahedron Cross Ratio
-----------------------

.. image:: render.png
    :width: 750px
    :align: center

In this example we will showcase the geometric transformation
and mathematical properties of stereographic projection
in the context of a regular tetrahedron.
You can see the full code at
:download:`tetrahedron_cross_ratio.py <../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py>`.

.. contents::
   :local:
   :backlinks: none

Setup
=====

We start off by importing the necessary libraries.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [setup-1]
   :end-before: [setup-1]

We are also going to clear the whole scene.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [setup-2]
   :end-before: [setup-2]

Constants and Functions
=======================

Now we are going to establish the essential constants and
functions necessary for the subsequent computations and visualization.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [constants]
   :end-before: [constants]

First we need to generate a regular tetrahedron inscribed
within the unit sphere based on the position of a single vertex.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [functions-1]
   :end-before: [functions-1]

Next we need to define a function which performs
a stereographic projection of the tetrahedron onto
the plane :math:`x_3 = 0`.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [functions-2]
   :end-before: [functions-2]

Finally we need to project an edge of a halfedge object
onto the unit sphere, ensuring it maintains its smoothness.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [functions-3]
   :end-before: [functions-3]

Calculations
============

Now we are going to execute the mathematical computations
required to achieve our visualization.
We first construct an inscribed tetrahedron within
the unit sphere based on a specified vertex and
project the edges of the tetrahedron onto the
unit sphere to create a list of curves on the sphere.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [calculations-1]
   :end-before: [calculations-1]

Next, we stereographically project the entire tetrahedron
onto the plane :math:`x_3 = 0`.
With that we calculate the complex cross-ratio of the
projected points and verifying whether it equals

.. math::

    e^{\pm i \frac{\pi}{3}}

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [calculations-2]
   :end-before: [calculations-2]

Visualization
=============

Finally we can bring it all together and visualize it.
We start by defining some colors and setting up the camera.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [visualization-1]
   :end-before: [visualization-1]

Next, we define materials and collections and also set the blend mode.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [visualization-2]
   :end-before: [visualization-2]

Now we can visualize the objects in Blender.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [visualization-3]
   :end-before: [visualization-3]

We are also going to create a plane in the xy-plane.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [visualization-4]
   :end-before: [visualization-4]

Finally, we configure our Blender settings for rendering.

.. literalinclude:: ../../../../../examples/blender/geometry/tetrahedron_cross_ratio.py
   :language: python
   :start-after: [visualization-5]
   :end-before: [visualization-5]