D Rotation

Before proceeding to a 3D rotation around an arbitrary axis, we consider the rotation around each of the coordinate axes. During the rotation around the x axis z -coordinates of the coordinate vector are the same. In fact, the rotation occurs in planes perpendicular to the x axis. Similarly, rotations about the axes y and z are in planes perpendicular to the axes y and z, respectively. The transformation of the coordinate vectors in each of these planes is the matrix of 2D rotation. This matrix and the persistence of the x coordinate rotation about the x axis can be written as 4×4 homogeneous transformation of coordinates under rotation by angle θ in the next form:

(2.9)

The rotation is considered positive in the sense of right-hand rule, i.e. clockwise when viewed from the origin in the positive direction of rotation.

Similarly, the transformation matrix for rotation around the z axis by an angle φ has the form:

(2.10)

For rotation through an angle ω around the y-axis transformation has the form:

(2.11)

Since the 3D rotations are obtained by multiplying the matrices, they are not commutative, i.e. multiplication order affects the final result.

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