Introduction
============

The open source book `Geometry with an Introduction to Cosmic Topology <https://mphitchman.com/geometry/preface.html>`_
by Michael P. Hitchman is a fascinating introduction to non-Euclidean geometries following the Erlangen program.
The approach in the book relies heavily on Möbius transformations and their effects on *clines*.

This repo aims to provide an implementation of the clines described in the book. Currently only numerical treatment is implemented but the plan is to add symbolic treatment as well.

What is a Cline?
---------------

A **cline** is a mathematical concept (see `Definition 3.2.3 <https://mphitchman.com/geometry/section3-2.html>`_) that unifies circles and lines in the complex plane under a single equation:

.. math::

   c z \bar{z} + \alpha z + \bar{\alpha} \bar{z} + d = 0

where:

- :math:`c` and :math:`d` are real numbers
- :math:`\alpha` is a complex number
- :math:`\bar{z}` and :math:`\bar{\alpha}` represent the complex conjugates of :math:`z` and :math:`\alpha`

This geometric structure represents:

- A circle if :math:`|\alpha|^2 > c \cdot d` and :math:`c \neq 0`
- A point if :math:`|\alpha|^2 = c \cdot d` and :math:`c \neq 0`
- A line if :math:`c = 0`
- No geometric object if :math:`|\alpha|^2 < c \cdot d` and :math:`c \neq 0`

Visual Representation
--------------------

Below is a visualization showing examples of clines in the complex plane:

.. plot::
    :include-source: false
    :context: close-figs

    import matplotlib.pyplot as plt
    from cline import Cline
    import numpy as np

    # Create a figure
    fig, ax = plt.subplots(figsize=(8, 6))

    # Create several clines
    circle1 = Cline.from_circle(center=0, radius=2)
    circle2 = Cline.from_circle(center=2+1j, radius=1.5)

    # For lines, we need to ensure we're using correct parameters
    line1 = Cline.from_line(z0=-3+0j, z1=3+0j)  # Horizontal line
    line2 = Cline.from_line(z0=0-3j, z1=0+3j)   # Vertical line

    # Plot all clines
    circle1.plot(ax=ax, color='royalblue', label='Circle 1')
    circle2.plot(ax=ax, color='cornflowerblue', label='Circle 2')
    line1.plot(ax=ax, color='firebrick', label='Line 1')
    line2.plot(ax=ax, color='indianred', label='Line 2')

    # Customize the plot
    ax.set_xlim(-3, 4)
    ax.set_ylim(-3, 3)
    ax.set_aspect('equal')
    ax.grid(True, linestyle='--', alpha=0.7)
    plt.title('Examples of Clines in the Complex Plane')
    plt.legend(loc='upper right')

Features
--------

The **Cline** library provides:

1. **Unified Representation**: Represent both circles and lines with a single class
2. **Multiple Construction Methods**:

   - From direct equation parameters
   - From three points
   - From circle center and radius
   - From two points (for a line)


3. **Easy Access to Properties**:

   - Circle center and radius
   - Line normal and direction vectors
   - Distance from origin

4. **Visualization**: Plot circles and lines using Matplotlib


Why Use Clines?
--------------

Clines provide several advantages:

- **Unified Treatment**: Algorithms can work with both circles and lines
- **Transformation Properties**: Clines transform elegantly under Möbius transformations
- **Algebraic Simplicity**: Representing complex geometric relationships with simple equati