Spirograph Generator with Turtle Graphics

A Python application that generates beautiful spirograph patterns using turtle graphics. The program creates complex mathematical curves by simulating circles rolling inside circles.

What is a Spirograph?

A spirograph is a geometric drawing device that produces mathematical roulette curves called hypotrochoids and epitrochoids. The patterns are created by tracing a point attached to a circle that rolls inside or outside another circle.

How It Works

This implementation uses a multi-circle rolling system to create complex patterns:

1. Fixed Circle: An outer stationary circle with radius R
2. First Rolling Circle: A circle with radius r = R/k that rolls inside the fixed circle
3. Second Rolling Circle: A smaller circle with radius r2 = R/k2 that rolls inside the first rolling circle
4. Pen Offset: A pen positioned at distance h from the center of the innermost circle

The Mathematics

The spirograph pattern is generated using parametric equations:

Position Equations

For a point at time t, the position is calculated as:

$$x(t) = (R + r) \cos(t) + (r + r_2) \cos\left(t + \frac{Rt}{r} - \frac{Rt}{pr}\right) + h \cos\left(t + \frac{Rt}{r} - \frac{Rt}{pr} - \frac{Rt}{pr_2}\right)$$

$$y(t) = (R + r) \sin(t) + (r + r_2) \sin\left(t + \frac{Rt}{r} - \frac{Rt}{pr}\right) + h \sin\left(t + \frac{Rt}{r} - \frac{Rt}{pr} - \frac{Rt}{pr_2}\right)$$

Where:

• $R$ = Radius of the fixed outer circle
• $r = R/k$ = Radius of the first rolling circle
• $r_2 = R/k_2$ = Radius of the second rolling circle
• $p$ = Phase parameter affecting the rolling motion
• $h$ = Distance of the pen from the center
• $t$ = Time parameter (angle in radians)

Parameters Explained

• R (Radius): Controls the overall size of the pattern
• k (First Ratio): Determines the size of the first rolling circle. Higher values create smaller circles and more intricate patterns
• k2 (Second Ratio): Determines the size of the second rolling circle. Affects the complexity of loops
• p (Phase): Modifies the rolling motion phase, creating variations in symmetry
• h (Pen Distance): Controls the amplitude of the pattern. Larger values create larger loops

Installation

1. Clone this repository:

git clone https://github.com/yourusername/Spirograph-with-turtle.git
cd Spirograph-with-turtle

2. Install the required dependencies:

pip install -r requirements.txt

Usage

Run the spirograph generator:

python spirograph.py

Controls

• Press 'q': Stop drawing and close the window

Customizing Parameters

You can modify the parameters in the main() function in spirograph.py to create different patterns:

R = 80      # Fixed circle radius
k = 1.5     # First rolling circle ratio
k2 = 3.98   # Second rolling circle ratio
p = 0.5     # Phase parameter
h = 100     # Pen distance from center

Examples

Try these parameter combinations for interesting patterns:

R	k	k2	p	h	Description
80	1.5	3.98	0.5	100	Default: Flowing loops
80	2.0	4.0	0.5	90	Symmetrical flower
80	3.0	5.0	1.0	80	Star pattern
80	1.2	6.0	0.3	120	Complex spiral

Technical Details

• Language: Python 3.x
• Graphics Library: Turtle (built-in Python module)
• Drawing Speed: Optimized with smallest step size (0.01) for smooth curves
• Auto-completion: Detects when the pattern is complete and stops automatically

Requirements

• Python 3.6 or higher
• keyboard library (for quit functionality)

License

This project is licensed under the MIT License - see the LICENSE file for details.

Contributing

Feel free to submit issues, fork the repository, and create pull requests for any improvements.

Acknowledgments

• Based on the mathematical concept of hypotrochoids and epitrochoids
• Inspired by the classic Spirograph toy invented by Denys Fisher in 1965