Linear Transformation Visualizer

Background

We use matrices throughout multiple field of mathematics, but we seldom quite "see" what changes these matrices make. A matrix cannot have 0 as its determinant. But we can't really tell the reason behind it.

This project is an effort to visualize the mathematics that we've studied in our high school years: Linear Transformations

Highly inspired by 3Blue1Brown and his efforts in visualizing mathematics, Linear Transformation Visualizer attempts for a similar goal: for you to become clearer what mathematics means

A 3D interactive visualizer that demonstrates how linear transformations affect objects and coordinate spaces. It allows users to apply, animate, and explore transformations such as rotations, scaling, reflections, and shears — all through a real-time rendered cube and grid using OpenGL, Pygame, and a Tkinter matrix input GUI.

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Some maths insights

In linear algebra, every linear transformation can be represented by a matrix. Upon matrix multiplication, the entire space changes it's dimenison: it stretches, rotates, reflects or skews. And this program allows you to see the phenomenon!

Note:

• Determinant = Volume scaling factor
• Negative determinant → Orientation reversal

And what happens when the matrix is non-invertible? Run it to find out ;)

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Screenshots

Initial Interface

Matrix Input

Terminal Output

Transformed Result

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Instructions to Run

1. Install dependencies

Make sure you have Python 3.8+ installed.

Then run:

pip install pygame PyOpenGL PyOpenGL_accelerate numpy

2. Run the program

python linear_transformation_visualizer.py

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Code Overview

Component	Description
MatrixInputGUI	Tkinter interface for entering and applying transformation matrices
LinearTransformationVisualizer	Main OpenGL + Pygame visualizer for rendering and animating transformations
apply_transformation()	Applies the given matrix to cube, grid, and basis
update_animation()	Interpolates transformation for smooth visual transitions
draw_info_panel()	Displays determinant, type, and control instructions overlay

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⚙️ Dependencies

Library	Purpose
pygame	Window creation, input handling, and OpenGL context
PyOpenGL	3D rendering and visualization
numpy	Matrix and vector operations
tkinter	Matrix input GUI

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🧮 Example Transformations

Transformation	Matrix
Rotate Z (90°)	[[0, -1, 0], [1, 0, 0], [0, 0, 1]]
Scale XY (2×)	[[2, 0, 0], [0, 2, 0], [0, 0, 1]]
Shear X	[[1, 0.5, 0], [0, 1, 0], [0, 0, 1]]
Reflect X	[[-1, 0, 0], [0, 1, 0], [0, 0, 1]]

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Troubleshooting

• If you see an error like No module named 'OpenGL' → Run pip install PyOpenGL PyOpenGL_accelerate
• If GUI doesn’t show up → Ensure Tkinter is installed (it comes by default with most Python distributions)
• Some systems may require a restart after installing pygame and PyOpenGL.

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Author

Prashansa Shrestha *Computer Engineering Student @ Pulchowk Campus