7.6 - Model Transforms¶

Before we leave the topic of cameras, let’s apply what you have learned about a camera matrix transform to model transformations.

A model can be scaled, rotated, and translated in an infinite number of ways to arrive at a final orientation, location and size for a specific rendering of a scene. When a model is rendered it uses a single transformation matrix that contains all of the model’s location, orientation and scaling information. The following analysis explains how we can extract useful information from this final model matrix transform.

Please remember that a camera transformation moves the virtual camera to the global origin and then aligns the camera’s local coordinate system axes with the global coordinate system axes. This requires a translation followed by a rotation. Consider that an identical, but reverse, transformation happens to all models. Models are typically defined in reference to the global coordinate system. When a model is put into a scene, regardless of the transformations used to create its matrix transform, it is fundamentally rotated to a desired orientation, possibly scaled to a desired size, and then moved to a desired location. In lesson 7.2 you saw that the local coordinate system axes that defined a camera (i.e, u, v, and n) became the rotation matrix we needed. We can use this knowledge to better understand a matrix transformation that places a model into a scene.

The matrix below rotates a camera’s local coordinate system so that it aligns with the global coordinate system.

To do the opposite, which is to take a model defined at the origin and rotate it to a desired orientation, as defined by a model’s local coordinate system, u, v, and n, we need the inverse of this matrix, which is

Therefore, the transformation matrix used to place a model into a scene must be equivalent to the following three transformations:

or

Performing the matrix multiplications gives this result:

The following implications of this are very useful!

• The 4th column of the transformation matrix contains the location of the model, (tx, ty, tz)
• If there is no scaling (sx === sy === sz === 1) or if the scaling is uniform (sx === sy === sz), then
  • The 1^st column of the matrix defines the model’s local coordinate system’s “to the right” axis, u.
  • The 2^nd column of the matrix defines the model’s local coordinate system’s “up” axis, v.
  • The 3^rd column of the matrix defines the model’s local coordinate system’s “back” axis, n.
  • The magnitude of the u, v, or n axis vector is equivalent to the scaling factors.