--------------------------------
Pascal Lines and Brianchon Lines
--------------------------------

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

In this example we are going to
create and display Pascal lines,
their intersections, Brianchon lines
and diagonals on a quadric surface.
You can see the full code at
:download:`pascal.py <../../../../../examples/blender/geometry/pascal.py>`.

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

Setup
=====

We start off by importing the necessary libraries.

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

We are also going to clear the whole scene.

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

Functions
=========

Now we are going to define two helper functions
which we will use later.
First, however, we are going to create a square
which we will use as a profile curve.

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

Now we will define the first function which will
draw projective lines.

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

Next we define the helper function to draw points.

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

Math
====

With the set up complete, we can now calculate the intersections.
First we have to create a quadric.

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

With this we can create the actual lines and intersections.

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

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

Finally we can visualize all of this in Blender.
We start by creating the quadric Blender object.

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

Next we dualize the quadric.

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

And finally we visualize the lines, intersections and points.

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