.. _jordan_curve:

============
Jordan Curve
============


------------
Introduction
------------

Jordan Curve is a continuous closed curve which doesn't intersect itself.

.. figure:: ../img/theory/jordan_curve.svg
   :width: 70%
   :alt: Example of jordan curves 
   :align: center

For our purpose, we use oriented jordan curve by setting a **positive** direction as **counter-clockwise**.

Although a jordan curve can be very general, we restrict and use only jordan curves which are piecewise smooth curves. We call each piece as a ``segment``, a planar bezier curve.

.. figure:: ../img/jordan_curve/jordan_splited.svg
   :width: 60%
   :alt: Example of jordan curves 
   :align: center

------------------
Creating a segment
------------------

A ``segment`` is a ``PlanarBezier`` instance. You create a new instance by passing the ``ctrlpoints`` of the curve. For example, we have a linear and a quadratic segment in the figure bellow 

.. code-block:: python
   
   from shapepy import PlanarBezier
   # Creates a liner segment between (1, 2) and (4, 0)
   linear_segment = PlanarBezier([(1, 2),
                                  (4, 0)])
   # Creates a quadratic segment
   quadratic_segment = PlanarBezier([(1, 2),
                                     (4, 2),
                                     (4, 0)])

.. image:: ../img/jordan_curve/planar_segment.svg
   :width: 50 %
   :alt: Example of linear and quadratic planar segment
   :align: center



-----------------------
Creating a jordan curve
-----------------------

There are 4 ways to create a ``JordanCurve`` instance:

* From segments: ``JordanCurve`` directly
* From vertices: ``FactoryJordan.polygon``
* From spline curve: ``FactoryJordan.spline_curve``

From vertices
-----------------------

This method creates polygonal shapes only


.. code-block:: python
   
   from shapepy import JordanCurve
   
   # Create a list of vertices
   vertices = [(1, 2), (4, 0), (-1, -1), (-3, 1)]
   # Creates a quadrilateral jordan
   jordan = FactoryJordan.polygon(vertices)

.. image:: ../img/jordan_curve/from_vertices.svg
   :width: 50 %
   :alt: Example of linear and quadratic planar segment
   :align: center


From segments
-----------------------

This method can create shape of any degree


.. code-block:: python
   
   from shapepy import Segment, JordanCurve
   segment0 = Segment([(0, 0), (4, 0)])
   segment1 = Segment([(4, 0), (4, 3), (0, 3)])
   segment2 = Segment([(0, 3), (0, 0)])
   segments = [segment0, segment1, segment2]
   jordan = JordanCurve(segments)

.. image:: ../img/jordan_curve/from_segments.svg
   :width: 50 %
   :alt: Example of jordan curve created from segments
   :align: center



From spline
---------------

For this case, we will use the package ``pynurbs``


.. code-block:: python
   
   import pynurbs
   from shapepy import Point2D, JordanCurve
   knotvector = (0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1)
   ctrlpoints = [(0, 0), (2, 0), (4, 0), (4, 3),
                 (0, 3), (0, 3/2), (0, 0)]
   ctrlpoints = [cartesian(point) for point in ctrlpoints]
   curve = pynurbs.Curve(knotvector, ctrlpoints)
   jordan = FactoryJordan.spline_curve(curve)

.. image:: ../img/jordan_curve/from_segments.svg
   :width: 50 %
   :alt: Example of jordan curve created from full curve
   :align: center