-----------------
Curvature Circles
-----------------

.. image:: curvature_circles1.png
    :width: 49%

.. image:: curvature_circles2.png
    :width: 49%

This example is about curvature circles of a (discrete) plane curve.
It illustrates the relationship between the curvature circles of the plane curve and the ones of
its counterpart on the **Möbius sphere**, obtained by inverse-stereographic projection of the planar points.

The curve in this example is a discrete **cardioid**.

The example includes 7 animations:

- 1. Showing the discrete **curvature circle** at each point
- 2. Showing the discrete **curvature circle** at each point an tracking the **evolute**
- 3. Fade in of the **Möbius sphere** (only for smooth animations)
- 4. Inverse-steregraphic projection of each vertex to the **Möbius sphere**
- 5. Visualizing **spherical circles** as intersections of the sphere with osculating planes (and their **polar points**)
- 6. Stereographic projection of the **spherical circles** as well as their **polar points**
- 7. Tracking of all the (spherical and planar) objects by iterating through the curve (only for discrete animations)


Discrete animation:

.. image:: curvature_circles_discrete.gif
    :width: 80%
    :align: center



Smooth animation:

.. image:: curvature_circles_smooth.gif
    :width: 80%
    :align: center

You can download the full script
:download:`curvature_circles.py <../../../../examples/blender/docs/examples/curvature_circles.py>`,
or find it, separated in sections, below.

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

Helper Functions
================
Functions for creation of planar and spherical curvature circles
and projections of objects (per index).


.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [helper-functions]
   :end-before: [helper-functions]

Setup
=====

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [setup]
   :end-before: [setup]

Curve and Specified Helper Functions
====================================
Initialize the parameterized curve and create functions specified for this curve.

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [curve-and-functions]
   :end-before: [curve-and-functions]

Visualization Setup
===================

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [visualization]
   :end-before: [visualization]

Animation Setup and Static Objects
==================================
Don't forget to set the output directory and increase the max_samples value.

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [animation-1]
   :end-before: [animation-1]

Main Animation Function
=======================

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [animation-2]
   :end-before: [animation-2]

Animation Callback and Keyframes
================================

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [animation-3]
   :end-before: [animation-3]


Rendering Setup
===============

.. literalinclude:: ../../../../examples/blender/docs/examples/curvature_circles.py
   :language: python
   :start-after: [rendering-setup]
   :end-before: [rendering-setup]