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 T1 will be denoted with T2, and the perspective collinear image of the straight line p1 will be denoted with p2.
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 step-by-step explanation.
Figure 15 illustrates point S as the center, straight line o as the axis of the given perspective collineation. Also, an arbitrary point A1 is given. The only information pertaining point A1 is that it is not a fixed point.
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Figure 15
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1st step: Straight line SA1 is mapped onto itself, because it contains two fixed points (S and X). This line is called a ray of perspective collineation. The image A2 of the point A1 has to lie on the ray SA1, since the collinearity is invariant.
2nd step:
Choose a point A2 on the ray, which can be any point except points S, X or A1.
3rd step:
B1 is an arbitrary point in the plane. Can we determine its perspective collinear image B2 from the given elements?
4th step:
The point B2 has to lie on the ray SB1.
5th step:
Notice the straight line p1=A1B1.
6th step:
Because of the collinearity, straight line p2 passes through the point A2 and intersects the axis at the same point as the line p1.
7th step:
The point B2 is uniquely determined by the intersection of the ray SB1 and line p2.
Move the point B1 on the figure and notice how the construction of the point B2 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 A2 on the ray and notice how its position changes the position of the point B2, i.e. the image of the point B1. Hence, by choosing the point A2 the perspective collineation is uniquely determined.
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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 A1, A2 are given.
Perspective collineation defined in such a way will be denoted as (S,o,A1,A2).
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,A1, A2)=(S,o,B1,B2) 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,B1,B2). The construction of the point A2, the perspective collinear image of the point A1, 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
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