Projective linear maps, vs affine maps and isometries
Written on a mac. May not work on other devices.
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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 R^{2}.
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 P^{2} 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, R^{2}. 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 R^{2} in
P^{2} 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 R^{3}, 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:
$$(x,y)\mapsto (x:y:1)\mapsto \left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {b}_{1}\\ {a}_{21}& {a}_{22}& {b}_{2}\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}x\\ y\\ 1\end{array}\right)=\left(\begin{array}{c}x\text{'}\\ y\text{'}\\ 1\end{array}\right)\mapsto (x\text{'},y\text{'})=\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+(\left(\begin{array}{c}{b}_{1}\\ {b}_{2}\end{array},)\right)$$