Translations and Parabolic Shearing

The first and most fundamental affine transformation is the translation.  As explained before, we let a = 1 and b be real so our transformation ax + b = x + b.  
As before, we work with the / quotients of the plane by lattice subgroups which are affinely equivalent.  The fundamental domain is given by a convex parallelogram, and each parallelogram in the space is affinely equivalent to every other, which implies that points in each lattice are associated by integer relations to similar points in every other parallelogram in the space.  The transformation is given by a 3x3 matrix which maps a vector from to a vector in the projective plane.  For the case of simple translations, the matrix is as follows:

                    
                    
The multiplication of this matrix by a vector of behaves as follows:

             =     
            
which we homogonize as

                
            
by dividing through by a factor of z.  Clearly, we see that the affine transformation is:

                +  
            
for arbitrary x', y' in the affine plane.  This mapping is area preserving as it shifts our lattice in the horizontal and/or vertical direction, but does not stretch the convex hull.

A related transformation is the composition of a parabolic deformation in one variable with a standard translation and the inverse of the parabolic deformation in one variable.  Let

                      
    
be a non-affine transformation of degree 2, and

                 T:  

be a simple translation, then

                 :  

and the composition

         T :  
        
is an affine transformation

              +  

and our group is given by the group of matrices of the form:

                    

To illustrate, in the graphics below, the lattice on the left is established as scalar multiples and linear combinations of the vectors and .  The right hand window shows curves after the parabolic deformation.  The parabolas on the right are tangent at the vertex to the vector on the left from which they were transformed.  By changing the scale factor t, we change the curvature of the resulting parabolas.  If we leave t = 0, the lattice is unchanged from the left window to the right, as the transformation matrix is then the identity mapping up to the equivalence class.  The group of affine automorphisms is given by

        
        
Where = (, ) and = (, )    , t = in this example, and and are components of vectors generated by linear combinations of and .  A special case emerges when or vanishes:               

But in all other cases, both classifications of geodesics form parabolas:

Next: Scaling by Real Exponentials

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