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Affine Transformations

Let
, where and are affine spaces.

Then T is said to be an affine transformation if:

By Extension:

T preserves affine combinations on the points:

where

Observations:
Examples:
rigid body motions (translations, rotations), scales, shears, reflections.

Affine vs Linear

Theorem:
Affine transformations map parallel lines to parallel lines.

Proof: Let and .

Suppose , which implies by linearity.

Then

and

and we see that the images of both lines are parallel.

Suppose we only have T defined on points.
Define
as follows:

Note that Q and R are not unique.

The definition works for :

This can now be used to show that the definition is well defined.
If Q-R=B-C then

How do we map points/vectors through an affine transformation?
Let
and be affine spaces.
  • Let be an affine transformation.
  • Let be a frame for .
  • Let be a frame for .
  • Let P be a point in whose coordinates relative are .
    ( )

Question:
What are the coordinates of T(P) relative to ?

Fact:
An affine transformation is completely characterized by the image of a frame in the domain:

If

then we can find by substitution and gathering like terms. \

Readings: White book, Appendix A


next up previous
Next: Matrix Representation of Up: Affine Geometry and Previous: Affine Combinations

CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab

cs488@cgl.uwaterloo.ca