Perspective Collineation in the Plane
A planar perspective collineation is a plane collineation in which there is exactly one fixed straight line o, whose all points are fixed, and exactly one fixed point S∉o.
The straight line o is called an axis, and the point S is called a center of the given perspective collineation.
Since perspective collineation is a transformation in which the corresponding elements lie in the same plane, to avoid confusion, we will denote one set of points (preimage elements) with subscript 1 and its corresponding images with subscript 2. For instance, the perspective collinear image of the point T_{1} will be denoted with T_{2}, and the perspective collinear image of the straight line p_{1} will be denoted with p_{2}.
We will now explore the basic properties of perspective collineation (i.e. what determines it uniquely), by performing consecutive steps in the following figure and giving the accompanying comments. Also, the construction of one corresponding pair of points obtained by perspective collineation will be given in the following dynamic figure with a stepbystep explanation.
Figure 15 illustrates point S as the center, straight line o as the axis of the given perspective collineation. Also, an arbitrary point A_{1} is given. The only information pertaining point A_{1} is that it is not a fixed point.
Figure 15


1st step: Straight line SA_{1} is mapped onto itself, because it contains two fixed points (S and X). This line is called a ray of perspective collineation. The image A_{2} of the point A_{1} has to lie on the ray SA_{1}, since the collinearity is invariant.
2nd step:
Choose a point A_{2} on the ray, which can be any point except points S, X or A_{1}.
3rd step:
B_{1} is an arbitrary point in the plane. Can we determine its perspective collinear image B_{2} from the given elements?
4th step:
The point B_{2} has to lie on the ray SB_{1}.
5th step:
Notice the straight line p_{1}=A_{1}B_{1}.
6th step:
Because of the collinearity, straight line p_{2} passes through the point A_{2} and intersects the axis at the same point as the line p_{1}.
7th step:
The point B_{2} is uniquely determined by the intersection of the ray SB_{1} and line p_{2}.
Move the point B_{1} on the figure and notice how the construction of the point B_{2} does not depend on its position, i.e. with this construction algorithm the perspective collinear image of any point in the plane can be determined.
Move the point A_{2} on the ray and notice how its position changes the position of the point B_{2}, i.e. the image of the point B_{1}. Hence, by choosing the point A_{2} the perspective collineation is uniquely determined.

Basic properties of the perspective collineation
All lines passing through the center of perspective collineation are mapped onto themselves. They are called rays of the perspective collineation.
The pairs of corresponding points obtained by perspective collineation lie on the rays of perspective collineation.
The pairs of corresponding straight lines obtained by perspective collineation intersect at the axis of the perspective collineation. Hence, straight lines parallel to the axis of the
perspective collineation are mapped onto the
parallel straight lines.
Each perspective collineation is uniquely determined with its center S, axis o and one pair of corresponding points A_{1}, A_{2} are given.
Perspective collineation defined in such a way will be denoted as (S,o,A_{1},A_{2}).
For a defining pair of corresponding points we can choose any pair of corresponding points with respect to the given perspective collineation. For instance, the identity (S,o,A_{1}, A_{2})=(S,o,B_{1},B_{2}) applies for the perspective collineation on figure 15. Check it in the following way: Leave only the check box for 3rd, 4th and 7th step. Now the same perspective collineation is determined with (S,o,B_{1},B_{2}). The construction of the point A_{2}, the perspective collinear image of the point A_{1}, is given with the 1st, 5th, 6th and 2nd step.
Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja  3DGeomTeh  Developing project of the University of Zagreb, made with GeoGebra
