Rotation Matrix
A rotation matrix represents the linear transformation of rotation.
Contents
Description
A rotation matrix is defined for a specific number of dimensions. In R2, a rotation matrix is:
where θ is the angle of counter-clockwise rotation. Note that a negative θ represents clockwise rotation.
As an example, a counter-clockwise rotation of 90 degrees (or π/2 radians) is represented by:
In R3, there are separate rotation matrices for each dimension.
where α, β, and γ represent yaw, pitch, and roll in the Z, Y, and X dimensions respectively. A complete rotation matrix in three dimensions can then be calculated as R = Rz(α)Ry(β)Rx(γ).
Properties
A rotation matrix is always an orthogonal matrix. It follows that RT = R-1.
The determinant of a rotation matrix is either 1 or -1.