Part 3 - Affine functions

by Corinne Lee 45672003

An affine function is a linear function with a translation.

L(x) is the linear part and b is the translation part.

Affine Transformations in 2D

There are two cases for the equation Ax+By=C, where C is some constant.
  1. one is when B equals 0.
  2. the other is when B not equal to 0.
For the first case B=0, we have the equation Ax=C where x=A/C.
If we graph values for A and C we get the picture.
The second case is when B is not equal to zero and we get the equation
y=(C-Ax)/B this gives us non-vertical lines.

However we want to find a way to represent both of these cases. For this we need a directional vector namely the normal vector.

The curve Ax+By=0 is the line through the origin perpendicular to [A,B]. Knowing this we have the properties.
  • [A,B].[x,y]=0
  • [A,B] perpendicular to [x,y]
This means that all the lines of Ax+By=C are perpendicular
to the normal vector.
The line Ax+By=0 is perpendicular to [A,B] and at signed distance C/(A2+B2)1/2
from the line Ax+By=0.
The signed distance with c>0 are the lines in the direction of [A,B].
The signed distance with c=0 is the line going throught the origin.
The signed distance with c<0 are the lines going in the opposite direction of [A,B].
If we use the line Ax+By=C then we have this picture.
The signed distance between Ax+By=C ' and Ax+By=C '' is

(C ''-C ')/(A2+B2)1/2

In 3D the equation is similiar. It is(C ''-C ')/(A2+B2+C2)1/2.

The equations (C ''-C ')/(A2+B2)1/2and (C ''-C ')/(A2+B2+C2)1/2 will be useful in discussing the concepts of waves in 2D and 3D.