MATLAB Implementation of 3D Rotation with Code Descriptions

Implementing 3D rotation in MATLAB typically requires utilizing rotation matrices or built-in 3D plotting functions. The core concept involves understanding coordinate system transformations and the multiplication order of rotation matrices. ### Fundamentals of Rotation Matrices 3D rotation can be decomposed into rotations around the three coordinate axes X, Y, and Z, each corresponding to a basic rotation matrix: - X-axis rotation (Roll): When rotating by angle θ, the rotation matrix Rx(θ) modifies coordinates on the Y-Z plane. - Y-axis rotation (Pitch): With rotation angle φ, matrix Ry(φ) affects the X-Z plane. - Z-axis rotation (Yaw): Using rotation angle ψ, matrix Rz(ψ) adjusts coordinates on the X-Y plane. ### MATLAB Implementation Approach 1. Define rotation matrices: Construct individual 3x3 matrices for Rx, Ry, and Rz using trigonometric functions (sine and cosine) to calculate matrix elements. For example: Rx = [1 0 0; 0 cos(θ) -sin(θ); 0 sin(θ) cos(θ)] 2. Combine rotations: For multiple axis rotations, multiply rotation matrices in specific sequences (e.g., Z-Y-X) to obtain the final transformation matrix. The multiplication order matters: final_matrix = Rz * Ry * Rx. 3. Apply transformation: Multiply original coordinate points with the transformation matrix to obtain new rotated coordinates using matrix multiplication: new_coords = coords * transformation_matrix. ### Important Considerations - Order dependency: Different multiplication sequences yield different final orientations (e.g., ZYX vs. XYZ). - Euler angle limitations: Gimbal lock may occur when intermediate rotation angles approach ±90°. - Visualization verification: Use `plot3` for point clouds or `surf` for surface plots to visualize objects before and after rotation, helping validate results through comparative plotting. By appropriately combining rotation matrices, flexible implementation of complex 3D spatial transformations can be achieved. The implementation typically involves matrix creation using trigonometric functions, proper multiplication sequencing, and coordinate transformation through matrix operations.