Complete Guide to Rotation Representations

Overview

#

This guide covers rotation representations from fundamentals to optimization. We explore Euler angles, rotation matrices, quaternions, and Lie algebra - comparing their geometric meanings and practical applications.

Part 0: Prerequisite Math Review

#

Matrix Multiplication

#

A matrix transforms a vector - moving, rotating, or scaling it:

Intuition: First row dot product with vector → first element of result.

Trigonometry Review

#

Sine and cosine describe coordinates on a unit circle:

Geometric meaning:

• $\cos\theta$ = “How much remains in original direction (X)”
• $\sin\theta$ = “How much moved to new direction (Y)”

Radians

#

One full circle = $2\pi$ radians = 360°

2D Rotation Matrix

#

Example: Rotating point $(1, 0)$ by 90°:

The point moves from right to up - counterclockwise 90° rotation confirmed!

Cross Product

#

The cross product of two 3D vectors produces a vector perpendicular to both:

Example: $(1,0,0) \times (0,1,0) = (0, 0, 1)$ — X-axis × Y-axis = Z-axis (right-hand rule)

Part 1: Euler Angles & Rotation Matrix

#

3D Rotation Matrices

#

X-axis rotation (Roll): Rotation in Y-Z plane

Y-axis rotation (Pitch): Rotation in X-Z plane

Z-axis rotation (Yaw): Rotation in X-Y plane

What Are Euler Angles?

#

Euler angles say: “Do three separate rotations in sequence.”

Key relationship:

• Euler angles = Input (recipe) → 3 numbers
• Rotation matrix = Output (result) → 3×3 = 9 numbers

Same rotation, different representations.

Gimbal Lock Problem

#

When the middle rotation (Pitch/Y) reaches 90°, you lose one degree of freedom:

At $\beta = 90°$:

Problem: $\alpha$ and $\gamma$ always appear as $(\alpha - \gamma)$ — two independent variables collapsed into one: $3 \text{ DOF} \to 2 \text{ DOF}$

Part 2: Geometric Meaning of Each Representation

#

Euler Angles: “Three Separate Turns”

#

Rotate γ around Z → Rotate β around Y → Rotate α around X

Analogy: Parking
  Step 1: Turn steering wheel (Yaw)
  Step 2: Go up slope (Pitch)
  Step 3: Tilt body (Roll)

Rotation Matrix: “New Coordinate Frame”

#

Each column represents where the original axis points after rotation:

• Column 1 = Where X-axis now points
• Column 2 = Where Y-axis now points
• Column 3 = Where Z-axis now points

Quaternion: “Axis + Angle in 4 Numbers”

#

• $(u_x, u_y, u_z)$ = Rotation axis (unit vector)
• $\theta$ = Rotation angle
• $w$ = $\cos(\theta/2)$ → Angle information
• $(x, y, z)$ = $\sin(\theta/2) \cdot$ axis → Axis information

Why half angle? From quaternion rotation formula $\mathbf{p}’ = q \cdot \mathbf{p} \cdot q^{-1}$, the quaternion acts on both sides, so angle is applied twice.

Example: Z-axis 90° rotation

Lie Algebra: “Axis + Angle in 3 Numbers”

#

• Direction $\hat{\boldsymbol{\omega}}$ = Rotation axis
• Magnitude $|\boldsymbol{\omega}|$ = Rotation angle (radians)

One vector contains both axis and angle!

Example: Z-axis 90° rotation

Exp and Log: Geometric Meaning

#

Analogy:

• exp = Unfolding a path from flat map onto a globe
• log = Flattening a path on globe to a flat map

Rodrigues’ Formula

#

Where $[\boldsymbol{\omega}]_\times$ is the skew-symmetric matrix:

This matrix multiplication equals cross product: $[\boldsymbol{\omega}]_\times \cdot \mathbf{v} = \boldsymbol{\omega} \times \mathbf{v}$

Part 3: What is Optimization?

#

Intuition

#

Optimization is like finding the lowest point in a valley while blindfolded. You can only feel the slope under your feet and take steps downhill.

1. Check which direction is downhill → Gradient
2. Take one step that direction → Update
3. Repeat until convergence

Mathematical Formulation

#

What’s Special About Rotation Optimization?

#

Rotations live on a curved surface (manifold), not flat space:

• Regular numbers: $3 + 0.5 = 3.5$ → Still valid ✓
• Rotation matrix: $R + \Delta R$ → May not be valid rotation! ✗
• Quaternion: $q + \Delta q$ → May not have magnitude 1! ✗
• Lie algebra: $\boldsymbol{\omega} + \Delta\boldsymbol{\omega}$ → Always valid! ✓

Part 4: Solving the Same Problem 4 Ways

#

Problem Definition

#

Goal: Rotate point $\mathbf{p} = (1, 0, 0)$ to be close to $\mathbf{p}^* = (0, 1, 0)$

Answer: 90° rotation around Z-axis

Method 1: Euler Angle Optimization

#

Simplified: Only optimize $\gamma$ (Z-axis rotation)

Gradient descent (η = 0.5): $0° \to 28.6° \to 53.8° \to 70.7° \to 80.1° \to \cdots \to 90°$ ✓

Problem: Works here with single axis, but with all 3 axes, gimbal lock occurs at $\beta \to 90°$.

Method 2: Direct Rotation Matrix Optimization

#

Optimize all 9 elements of R:

Problem:

Solution: Must project back onto SO(3) using SVD every iteration — computationally expensive.

Method 3: Quaternion Optimization

#

Optimize 4 numbers $q = (w, x, y, z)$:

Problem:

Solution: Normalize every iteration:

Method 4: Lie Algebra Optimization ★

#

Optimize 3 numbers $\boldsymbol{\omega} = (\omega_1, \omega_2, \omega_3)$, no constraints!

Jacobian calculation using cross products:

Each column tells “if I rotate slightly around this axis, where does the point go”:

• Column 1 (X-axis): $(0, 0, 0)$ → No effect (point is on X-axis)
• Column 2 (Y-axis): $(0, 0, -1)$ → Moves to -Z
• Column 3 (Z-axis): $(0, 1, 0)$ → Moves to +Y ← This is what we need!

Update:

Done! No additional work needed!

• Normalization? ❌ Not needed
• SVD reprojection? ❌ Not needed
• Just vector addition! ✅

Verification: $\exp([(0, 0, 0.5)]_\times)$ automatically produces valid rotation matrix with $R^TR = I$ ✓

Part 5: Final Comparison

#

Side-by-Side Comparison

#

One Step Comparison

#

Rotation Matrix — After update:

$R^TR \neq I$ ❌ → SVD reprojection needed

Quaternion — After update:

$|q| = 1.118 \neq 1$ ❌ → Normalization needed

Lie Algebra — After update:

Just a vector. Always valid. No post-processing. ✅

Conclusion

#

Practical Workflow

#

• User input → Euler angles
• Optimization computation → Lie algebra
• Real-time composition/interpolation → Quaternion
• Coordinate transform application → Rotation matrix

Use the right representation for each stage and convert between them as needed.