Affine Transformations

Let

, where and are affine spaces.

Then T is said to be an affine transformation if:

• T maps vectors to vectors and points to points
• T is a linear transformation on the vectors
• 

By Extension:

T preserves affine combinations on the points:

where

Observations:

• Affine transformations map lines to lines:

  

• Affine transformations preserve ratios of distance along a line
  (converse is also true: preserves ratios of such distances affine).

Examples:

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

Affine vs Linear

• Which is a larger class of transformations: Affine or Linear?
  • T is linear if T(aA+bB)=aT(A)+bT(B) for all a,b.
  • T is affine if T(aA+bB)=aT(A)+bT(B) if a+b=1.
• If T is linear, then T(aA+bB)=aT(A)+bT(B) when a+b=1 so T is clearly affine.
• But if T is affine and then T(aA+bB) might not equal aT(A)+bT(B).

  Thus, it is possible for T to be affine but not linear.

• Therefore, affine transformations are a superclass of linear transformations (surprise!)
• An alternate way to think of this is that the behaviour of linear transformations is restricted for a wider range of values.

  Consider T(aA+bB)

  • Linear transformations have their behaviour specified for all values of a,b.
  • Affine transformations have their behaviour specified only when a+b=1. They can behave differently for all other values of a+b.
• Most of the transformations we consider will be linear.
• Translation is the only non-linear (but affine) transformation we'll see, and we'll ``bypass'' its non-linear behavior.

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:

• There exists points Q and R such that .
• Define to be T(Q)-T(R).

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