Math 309 - affine functions

Part 3 - Affine functions

by Corinne Lee 45672003

An affine function is a linear function with a translation.
A(x)=L(x)+b

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

An affine function in 2D is
Ax+By+C=0
An affine function in 3D is
Ax+By+Cz+D=0

Affine Transformations in 2D

There are two cases for the equation Ax+By=C, where C is some constant.

one is when B equals 0.
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.

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.