Rotation Matrices¶
Rotation matrices (Direction Cosine Matrices) represent rotations as 3×3 orthogonal matrices.
Overview¶
A rotation matrix is a 3×3 matrix that transforms vectors from one coordinate frame to another. Also known as Direction Cosine Matrices (DCM).
Mathematical Representation¶
A rotation matrix \(R\) satisfies:
\[R^T R = I\]
\[\det(R) = 1\]
where \(I\) is the identity matrix.
Advantages¶
- Direct transformations: Multiply matrix by vector to transform
- Intuitive: Each column/row represents a coordinate axis
- Fast computation: Matrix multiplication is highly optimized
Disadvantages¶
- Redundant: 9 parameters represent only 3 degrees of freedom
- Numerical drift: Orthogonality can degrade with repeated operations
- Storage: Requires more memory than quaternions
Operations¶
Common rotation matrix operations:
- Matrix multiplication (composition)
- Matrix transpose (inverse rotation)
- Vector transformation
- Orthogonalization (Gram-Schmidt)