Quaternionic Storage Visualization in MATLAB

Overview

This project visualizes quaternionic storage of 3D rotation in MATLAB through an interactive plot with a manual slider.

• Left panel: Quaternion path on S³ (projected to 3D) with trace and ±q mirrored path.
• Right panel: 3D rotated vector corresponding to the quaternion.
• Demonstrates:
  • Half-angle storage (θ/2 inside the quaternion).
  • SU(2) double-cover property (±q represent the same rotation).
  • Continuous rotation without gimbal lock.

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How to Use

1. Open QuaternionVisualization.m in MATLAB.
2. Run the script.
3. Adjust the slider at the bottom to set the rotation angle θ.
4. Observe:
   • Left: quaternion moves along S³ projection.
   • Right: rotated vector (red) vs original (blue).
5. No autoplay — this version is fully manual for user control.

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Mathematical Background

A rotation quaternion is defined as:

s = cos(theta/2)
v = [ ux*sin(theta/2), uy*sin(theta/2), uz*sin(theta/2) ]
q = [ s, v[0], v[1], v[2] ]

• θ — rotation angle in 3D space
• u — unit vector rotation axis
• Half-angle — stored as θ/2 in quaternion space.

Applying the rotation to vector v:

v' = q, v, q^{-1}

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Features

Feature	Description
S³ Projection	Quaternion trajectory shown in 3D projection
±q Mirroring	Faint mirrored path showing double cover
Trace Line	Visual trace of quaternion path
3D Rotation	Red rotated vector vs blue reference
Slider	Manual control of rotation angle

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Files Included

• QuaternionVisualization.m — main MATLAB script
• diagram.png — schematic showing S³ projection and vector rotation
• README.md — documentation (this file)

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Requirements

• MATLAB R2020a or later
• No additional toolboxes required

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Credits

Developed for quaternionic visualization and SU(2) storage exploration in MATLAB.