ThreeDimensional Rotation Matrices（三维旋转矩阵）.pdf

Physics 216 Spring 2012 Three-Dimensional Rotation Matrices 1. Rotation matrices A real orthogonal matrix R is a matrix whose elements are real numbers and satisfies R−1 = RT (or equivalently, RRT = I, where I is the n × n identity matrix). Taking the determinant of the equation RRT = I and using the fact that det(RT) = det R, it follows that (det R)2 = 1, which implies that either det R = 1 or det R = −1. A real orthogonal n × n matrix with det R = 1 is called a special orthogonal matrix and provides a matrix representation of a n-dimensional proper rotation1 (i.e. no mirrors required!). The most general three-dimensional rotation matrix represents a counterclockwise rotation by an angle θ about a fixed axis that lies along the unit vector nˆ. The rotation matrix operates on vectors to produce rotated vectors, while the coordinate axes are held fixed. This is called an active transformation. In these notes, we shall explore the general form for the matrix representation of a three-dimensional (proper) rotations, and examine some of its properties. 2. Properties of the 3 × 3 rotation matrix A rotation in the x–y plane by an angle θ measured counterclockwise from the positive x-axis is represented by the real 2 × 2 special orthogonal matrix,2 cos θ − sin θ sin θ cos θ . If we consider this rotation as occurring in three-dimensional space, then it can be described as a counterclockwise rotation by an angle θ about the z-axis. The matrix representation of this three-dimensional rotation is given by the real 3 × 3 special orthogonal matrix, 