3D Transformation

Methods for geometric transformations and object modeling in 3D are extended from 2D methods by including the considerations for the z coordinate.

Basic Transformations

Translation

We translate a 3D point by adding translation distances, tx, ty, and tz, to the original coordinate position (x,y,z):

x' = x + tx, y' = y + ty, z' = z + tz

Scaling

Scaling With Respect to the Origin

We scale a 3D object with respect to the origin by setting the scaling factors sx, sy and sz, which are multiplied to the original vertex coordinate positions (x,y,z):

x' = x * sx, y' = y * sy, z' = z * sz

Coordinate-Axes Rotations

A 3D rotation can be specified around any line in space. The easiest rotation axes to handle are the coordinate axes.

3D Rotations about an Axis which is parallel to an Axis

1. Step 1. Translate the object so that the rotation axis coincides with the parallel coordinate axis.
2. Step 2. Perform the specified rotation about that axis.
3. Step 3. Translate the object so that the rotation axis is moved back to its original position.

General 3D Rotations

1. Step 1. Translate the object so that the rotation axis passes through the coordinate origin.
2. Step 2. Rotate the object so that the axis of rotation coincides with one of the coordinate axes.
3. Step 3. Perform the specified rotation about that coordinate axis.
4. Step 4. Rotate the object so that the rotation axis is brought back to its original orientation.
5. Step 5. Translate the object so that the rotation axis is brought back to its original position.