---------------------------
Lines in a Projective Plane
---------------------------

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

In this example we will generate random lines (represented as planes)
in a projective plane and visualize their intersections with a sphere
and the projective plane in a Blender 3D scene.
It's a combination of geometric calculations and Blender scripting for 
You can see the full code at
:download:`lines_in_a_projective_plane.py <../../../../examples/blender/geometry/lines_in_a_projective_plane.py>`.

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

Setup
=====

We start off by importing the necessary libraries.

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

We are also going to clear the whole scene.

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

Blender Collections
===================

We create three Blender collections: ``rendering_coll`` for lights and cameras,
``gc_coll`` for great circles, and ``plane_coll`` for plane intersections.

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

Geometric Calculations
======================

Now we are going to generate random points on a unit sphere and
calculate the corresponding planes using these points as normals.
These planes are represented in the projective space.

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

Next we create a unit sphere (:math:`S^2`) and a projective plane.
The sphere is represented as a quadric,
and the projective plane is represented as a hyperplane.

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

Geometric Intersections
=======================

For each randomly generated plane,
we calculate the intersection of the plane with the sphere (``circ_on_sphere``)
and the intersection with the projective plane (``meet_plane``).
These intersections are then visualized.

.. literalinclude:: ../../../../examples/blender/geometry/lines_in_a_projective_plane.py
   :language: python
   :start-after: [geometric-intersections-1]
   :end-before: [geometric-intersections-1]

Finally we can visualize it in Blender.

.. literalinclude:: ../../../../examples/blender/geometry/lines_in_a_projective_plane.py
   :language: python
   :start-after: [geometric-intersections-2]
   :end-before: [geometric-intersections-2]

Add Camera and Light
====================

Awesome, so if we run this now, we see something in the Editor!
However, for rendering we need a camera and a light.
Let's add those.

.. literalinclude:: ../../../../examples/blender/geometry/lines_in_a_projective_plane.py
   :language: python
   :start-after: [add-camera-and-light]
   :end-before: [add-camera-and-light]

Add Materials
=============

Finally let's assign materials to the projective plane and the sphere.
We assign the projective plane a color.

.. literalinclude:: ../../../../examples/blender/geometry/lines_in_a_projective_plane.py
   :language: python
   :start-after: [add-materials-1]
   :end-before: [add-materials-1]

The sphere will get a color with transparency.

.. literalinclude:: ../../../../examples/blender/geometry/lines_in_a_projective_plane.py
   :language: python
   :start-after: [add-materials-2]
   :end-before: [add-materials-2]

That's it! You should be able to just put that in your blender
code editor and simply execute it.