Perspective collinear image of a straight line

The construction for determining the perspective collinear image of a given point is given on the figure 15. Determining the image of any other plane figure is based on this basic construction.

The perspective collineation (S,o,X1,X2) and the straight line a1 are given on the figure 16. Construct the perspective collinear image of the straight line a1, i.e. the straight line a2.

Figure 16

1st step: An arbitrary point A1∈a1 is chosen and its perspective collinear image A2 is constructed (following the construction algorithm in figure 15).

2nd step: Since every straight line is uniquely determined with two of its points, the line a2 can be constructed as a straight line passing through the point A2 and the intersection point of the line a1 and the axis of perspective collineation.
- Move the point A1∈a1 and notice that the image of the straight line a1, i.e. the line a2, does not depend on the chosen position of the point A1.
- The position of the points X1 and X2 has uniquely determined the perspective collineation. Move those points in various positions (on the same side or opposite sides of the axis or center) to change the given perspective collineation. Notice that the construction algorithm remains unchanged.

3rd step: By moving the point Y bring the line a1 in position parallel to the axis. Notice that then its image line a2 is also parallel to the axis. By changing the position of the starting points X1 and X2, observe that this statement applies to any perspective collineation.

We already know that the perspective collineation maps points to the points of the same plane. Thus, each point may be treated as an element of the domain, as well as an element of the codomain. In the previous examples, domain elements were marked by index 1, and codomain by index 2. For example, we have observed the following point and lines transformations: A1→A2, X1→X2, p1→p2, a1→a2 …
On the other hand, the perspective collineation is a bijective transformation for which the inverse transformation is also a perspective collineation with the same provisional elements but with reversed indices, the field with the index 2 becomes domain filed, and the field with index 1 the codomain field of the inverse mapping. Thus, for the inverse mapping we have the following transformations: A2→A1, X2→X1, p2→p1, a2→a1 …
In all the following examples, these two mapping will be treated equally, i.e. we will map from the field 1 to two field 2 and vice versa. In doing, so, one should always pay attention to the proper denotation of the indices.

The perspective collinear images of two straight lines a2 and b2 are given in figure 17 and 18. Those lines intersect at point O2 in figure 17, and are parallel in figure 18 (intersection point is at infinity and denoted with O2).

Figure 17

Figure 18

The perspective collinear image of the pair of intersecting lines in figure 17 is constructed so that first the image of their intersection point O1 is determined. Then the property that a pair of corresponding straight lines obtained by perspective collineation intersects at the axis of the perspective collineation is used.
On the figure movable points are X1, X2 (for changing the perspective collineation) and O2 (changing the position of lines a2, b2). Notice the following:

  • In general, the perspective collineation does NOT preserve the angle measure between two lines, i.e. it is not a conformal transformation.

    In figure 18 the perspective collinear image of a pair of parallel lines is is determined by first constructing the images of any points A2∈a2 and B2∈b2. Then the property that a pair of corresponding straight lines obtained by perspective collineation intersects at the axis is used. The lines a1 and b1 intersect at the point O1 which is the image of the point at infinity O2. Notice the following property:

  • In general, theperspective collineation does NOT preserve parallelism.

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