Introduction to Collineations

Planar transformation is every bijective transformation which maps a plane onto itself.
This means that each point of the plane is mapped onto one and only one point of the same plane, and vice versa, each point of the plane is the image of only one point in the same plane.

The planar transformation which preserves collinearity of the points (the incidence structure of points and lines) is called a planar collineation.

These transformations are such that points correspond to points and lines correspond to lines.

Points (lines) that are mapped to itself under a collineation are called fixed points (lines) of that collineation. Notice, that a line can be mapped onto itself without mapping every point onto itself. Simply, the points of the line are mapped onto some other points of the same line and therefore the line is its own image, but it is not true for all of its points. There can be at most two fixed points on such a line. If there are three fixed points on a line, then each point of that line is fixed.

Up till now you have met a few planar transformation: translation, rotation, reflection, central symmetry, dilatation... These are all planar collineations. It would be useful to remind yourself of the properties of these transformations. Also, show that collinearity of points is an invariant of those transformations. Which point is fixed under a given transformation? Determine the number of fixed lines under a given reflection?

In the following two more planar collineations will be studied - perspective collineation and perspective affinity. These transformations will be of crucial importance for the planar and spatial objects constructions, as well as for the better understanding of some spatial relations.

Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra