Affine Transformation

(this page is work in progress)

Affine transformation, is a combination of linear transformations and translation. A linear transformation can be represented by a matrix. Affine transformation can also be done with a matrix, by adding a dimension to the matrix and the vector. Example:

Suppose we want to 

This is a linear transformation:
{{a,b},
 {c,d}}  . {x,y}

result:
{a x+b y,
 c x+d y}

Now, we want to add a translation by {t1,t2}. The matrix now is:
{{a,b,t1},
 {c,d,t2},
 {0,0,1}} . {x,y,1}

Result
{t1+a x+b y,
 t2+c x+d y,
 1}

Affine transformations are the functions that map straight lines to straight lines.

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. An affine combination is a linear combination in which the sum of the coefficients is 1.