Central and axial collineations in an affine plane

Let be an affine plane, and let be an automorphism of . The automorphism extents to an automorphism of the projective closure of . If this extended automorphism has a center and an axis, then either the axis is the line at infinity, or the axis is an affine line and the center is on the line at infinity. This simple observation leads to the following cases.

Affine name Axis and type Description
Translation Elation with axis at infinity Each line is mapped onto a parallel line. There are no fixed points and the fixed lines form a parallel class.
Contraction Homology with axis at infinity Each line is mapped onto a parallel line. There is exactly one fixed point, and the fixed lines are exactly the lines through this point.
Shear Elation with affine axis All points on a line are fixed, and all lines parallel with this line are also fixed. There are no further fixed points or fixed lines, respectively.
Strain Homology with affine axis All points on a line are fixed, and there are no further fixed points. The fixed lines distinct from the axis form a parallel class.


Contributed by Hauke Klein
Version $Id: shears.html,v 1.1 2001/01/08 14:37:36 hauke Exp $