Mumbai university > mechanical engineering > sem 7 > cad/cam/cae

Marks: 5M,10M

Difficulty: Medium

3D rotation transformation

Rotation: It is an important geometric transformation. The final position and orientation of a geometric entity is decided by the angle of rotation (θ) and the base point about which the rotation is to be done.

Since we know that, during rotation about z-axis, the z-coordinate of object remains constant. Also during rotation about x-axis and y-axis of object, the x-coordinate and y-coordinate respectively remains constant as x,y,z are mutually perpendicular to each other.It is the property of vector.

Hence for rotation about Z-axis, we can draw the following figure.

To develop the transformation matrix, consider Point P located in XY plane, being rotated in the counter clock wise direction to the new position P’ by an angle θ as shown in above figure.

The position P’ is given by,

P' = [X', Y', Z']

From the figure,

The new position P' is specified by

and z = z remains same.

This can be written in matrix form as :

The above matrix can also be written in Homogeneous form as :

Similarly we can find rotation matrix in 3D about y- axis as well as about x-axis. The rotational matrix are as follows, 1. Rotation about x-axis

1. Rotation about y-axis