## Projective linear maps, vs affine maps and isometries

Written on a mac. May not work on other devices.
dot to move (in case problem with mouse; listed by quadrilateral): (1) (2) (3) (0) (x) (reset)

## Isometries vs linear maps vs affine maps vs projective linear maps

This Javascript program was written to illustrate the differences between some of the maps discussed in Geometry, MA243.

### Defintions

• An isometry of the plane has the form (x,y) -> A(x,y) + b for some orthogonal matrix A and vector b in R2. Isometries preserve distance.
• For an affine map, A can be any linear map. (Given by a 2 x 2 matrix). Affine maps preserve affine relationships.
• For a linear map, b = 0. The only difference between linear and affine is that linear has to fix an origin. I will not bother picturing these. These preserve lines through the origin, and linear relationships.
• For a projective linear map, from the projective plane P2 to the projective plane, the map should be given by matrix in PGL(3). The above picture is a representation of a peice of the plane, R2. However, this is also a piece of the projective plane. I will take this to be the peice with Z=1, where (X:Y:Z) are projective coordinates. So I include R2 in P2 via (x,y)->(x:y:1).
Projective maps preserve the cross ratio.

### Image descriptions

• An isometry can be determined by the amount of translation, and the line of reflection or degree of rotation; I will just allow rotations in this example. The green/cyan outlined square can be translated by moving one vertex, or rotated by moving another. The region will always be a square. (darker vertices can't be moved; it's determined by the others).
• An affine map can be determined by the images of three points. One point corresponds to the image of the origin, the other the two standard basis elements. Knowing these determines the affine map. Thus for the orange outlined quadrilateral, moving the orange vertices determines an affine map. The region will always be a parallelogram. (darker vertex can't be moved; it's determined by the others).
• For a projective linear map, four points are required to determine the map. These points must be a "projective frame of reference". This means that no three of them should be in a straight line.

### Matrix description

In the isometry case, the transformation has the following form (assuming I'm only allowing rotation, not reflection)

For the affine case, this is generalised to:

For the projective case, we have
Here, we turn our 2 component vector into a 3 component vector by adding a 1, i.e., considering as the z=1 subplane of R3, then apply the linear map given by the matrix multiplication, then project back to the z=1 plane.
Note that each case is contained in the next, e.g., the affine case is a special case of the projective case via:

Not very good alternative xml form: $\left(x,y\right)↦\left(x:y:1\right)↦\left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {b}_{1}\\ {a}_{21}& {a}_{22}& {b}_{2}\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ 1\end{array}\right)=\left(\begin{array}{c}x\text{'}\\ y\text{'}\\ 1\end{array}\right)↦\left(x\text{'},y\text{'}\right)=\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\end{array},\right)\right)$