------------------------------
Three triangles in perspective
------------------------------

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

Triangles are said to be in perspective with respect to a point if the three lines through
corresponding points are concurrent (intersect in one point, red triangles).
The resulting point is called the *center of perspectivity*.
Two triangles are said to be *in perspective with respect to a
line* if the three intersection points of corresponding lines are collinear.
The resulting line is called the *axis of perspectivity* (three blue lines,
one for each pair of triangles). Desargues Theorem
states, that two triangles are in perspective with respect to a point if and only if
they are in perspective with respect to a line. For three triangles in perspective, it can
be shown that the three axes of perspectivity are concurrent.
The 3d configuration can be
projected to a 2d plane.
Source code:
:download:`three_triangles_in_perspective.py <../../../../examples/blender/geometry/three_triangles_in_perspective.py>`.

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

Setup
=====

We start off by importing the necessary libraries and clearing the
scene.

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

Next we initialize some data:
one point of perspectivity, three directions for the lines of perspectivity
(where vertices of the triangles lie on), and for each vertex of a
triangle a distance that defines the position of the vertex on the
line of perspectivity. Further, for producing a shadow we set initial data
that defines a center of projection (the location of the light source), and a projection plane.
The data given in this example will produce the picture above.

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

With the given initial data we can create the lines of
perspectivity as projective subspaces.

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


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

The three triangles can be constructed in different ways using the
distances given in the initial data. Here we show two possible ways where
in both cases a resulting triangle will be represented as a list of three points.

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

Now we are able to construct the edges as subpspaces, intersect corresponding
lines, construct the axes of perspectivity and intersect those.

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

In order to construct the light blue segments as in the picture
we need to initialize those subspaces in a specific way. Each light
blue segment contains four points: two corresponding
to an edge of a triangle and two lying on the axes of perspectivity.
One can determine the outer ones, for example by computing signed distances,
use them to construct a line,
and visualize this line using `domain=[[0,1]]`.

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

Finally, we add a projection plane to catch the shadow.

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

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

Again we start with a block of initial data for our 
This allows an easy change of parameters. Further we
need a utility function to visualize a triangle, here a halfedge object
is created and then visualized.

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

Now we can visualize all our objects,

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

and add a light and a camera.

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


Finally these are the rendering setting used in this example.

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


That's it! You should be able to just put that in your blender
code editor and simply execute it. Hit the render button to render
the image at the top.