Point Rotation Around an Arbitrary Axis in 3D Space

Rotating a point around an arbitrary axis in 3D space is a fundamental requirement in computer graphics and geometric computations. The axis can be defined by a reference point and a direction vector, with the core implementation involving decomposition into standardized transformation steps:

Coordinate Translation: First translate the axis to pass through the coordinate system origin by subtracting the reference point coordinates from the target point coordinates. This centers the rotation axis at the origin for simpler matrix operations.

Axis Alignment: Construct an orthogonal transformation matrix to align the direction vector with a principal axis (typically Z-axis). This can be implemented using Rodrigues' rotation formula or through vector cross-product operations to compute the alignment rotation matrix.

Rotation Execution: After alignment, perform rotation around the principal axis using standard 3D rotation matrices (e.g., Z-axis rotation matrix Rz(θ) = [cosθ -sinθ 0; sinθ cosθ 0; 0 0 1]). This step simplifies to basic trigonometric operations.

Inverse Transformation: Apply reverse transformations to restore the rotated point to the original coordinate system, including inverse rotation and backward translation to the axis's original position.

In MATLAB implementation, vector operations and matrix multiplication can efficiently execute these steps. Key functions include vector normalization (norm), cross product (cross), and matrix inversion (inv). The critical implementation aspect involves correct rotation matrix construction and proper transformation sequence management. This methodology extends to higher-dimensional rotation problems, maintaining the core approach of decomposing arbitrary rotations into translation and standard rotation components.