Affine Map

An affine map is a function of the form

where:

• is a matrix representing a linear transformation
• is a fixed vector representing a translation

So an affine map is basically:

a linear transformation followed by a shift

Main idea §

A linear transformation must send the origin to the origin.

That is because if is linear, then

But many useful geometric transformations do not keep the origin fixed.

For example,

just translates every point by the same amount. This is not linear, but it is affine.

So affine maps are the natural class of transformations that include:

• rotation
• scaling
• shear
• reflection
• translation
• any combination of these

General form §

In , an affine map has the form

If , then the affine map is just a linear map.

So every linear map is affine, but not every affine map is linear.

Example §

Suppose

Then

This transformation:

• scales in the -direction by 2
• scales in the -direction by 3
• shifts everything right by 1
• shifts everything down by 4

Why affine maps matter §

Affine maps describe the kinds of transformations we usually care about in geometry and graphics.

They preserve:

• straight lines
• parallelism
• ratios along the same line

But they do not necessarily preserve:

• lengths
• angles

So affine geometry is less rigid than Euclidean geometry, but still keeps a lot of the important structure.

Connection to Homogeneous Coordinates §

Affine maps are not purely linear in ordinary coordinates because of the extra term.

But in Homogeneous Coordinates, an affine map becomes a single matrix multiplication.

For a 2D point, write

Then the affine map

can be written as

In 2D this looks like

This is why homogeneous coordinates are so useful: they let affine maps be handled with the same matrix machinery as linear maps.

Intuition §

The cleanest way to think about an affine map is:

1. first reshape space linearly
2. then move the whole result

So if a linear map tells you how vectors change, an affine map tells you how points in space change.