Cline

A Python library for representing circles and lines in the complex plane using the general cline equation.

The inspiration for this package comes from the open source book Geometry with an Introduction to Cosmic Topology by Michael P. Hitchman, which 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.

Documentation

Full documentation is available at: https://velikod.github.io/mathcline

Overview

A cline is a circle or line that can be represented by the unified equation:

$$cz\bar{z} + \alpha z + \bar{\alpha}\bar{z} + d = 0$$

where:

• $c$ and $d$ are real numbers
• $\alpha$ is a complex number

Based on the discriminant $\Delta = |\alpha|^2 - c \cdot d$, a cline represents:

• A circle if $\Delta > 0$ and $c \neq 0$
• A point if $\Delta = 0$ and $c \neq 0$
• A line if $c = 0$
• An invalid geometric object (no solutions) if $\Delta < 0$ and $c \neq 0$

Mathematical Formulation

Circle Representation

When $\Delta > 0$ and $c \neq 0$, the cline is a circle with:

• Center: $z_0 = -\frac{\alpha}{c}$
• Radius: $r = \frac{\sqrt{|\alpha|^2 - c \cdot d}}{|c|}$

Line Representation

When $c = 0$, the cline is a line with:

• Normal vector: $\alpha = a + bi$
• Direction vector: $v = b - ai$ (perpendicular to normal)
• Cartesian form: $ax - by + \frac{d}{2} = 0$

Three-Point Construction

The library provides a method to construct a cline from three points, automatically determining whether to create a circle or a line by checking if the points are collinear.

Installation (not yet published as a package)

Features

• Unified representation of circles and lines using the cline equation
• Create clines from various inputs:
  • Circle from center and radius
  • Line from two points
  • Any cline from three points
  • Directly from equation parameters (e.g., Cline(c=1, d=1, alpha=2+1j))
• Easy access to geometric properties:
  • Circle center and radius
  • Line normal and direction vectors
  • Distance from origin
• Visualization using Matplotlib with automatic window calibration
• Comprehensive mathematical derivations and formulations
• Detailed documentation with LaTeX-rendered equations

Examples

from cline import Cline
import matplotlib.pyplot as plt

# Create a cline directly from parameters
c1 = Cline(c=1, alpha=2+1j, d=1)

# Create a circle from center and radius
c2 = Cline.from_circle(center=1+2j, radius=2)

# Create a line from two points
c3 = Cline.from_line(0, 1+1j)

# Create a cline from three points
c4 = Cline.from_three_points(0, 1, 2j)

# Plot the clines
fig, ax = plt.subplots(figsize=(10, 10))
c1.plot(ax=ax, color='red', label='Parameters c=1, α=2+j, d=1')
c2.plot(ax=ax, color='blue', label='Circle center=1+2j, radius=2')
c3.plot(ax=ax, color='green', label='Line through 0 and 1+j')
c4.plot(ax=ax, color='purple', label='Cline through 0, 1, and 2j')
plt.legend()
plt.show()

Requirements

• Python 3.6+
• NumPy
• Matplotlib

License

MIT

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.