Vanishing and neutral line of the perspective collineation
As can be conclided from the example in figure 18, the perspective collinear image of points at infinity are not points at infinity. Since all points at infinity of a plane lie on its line at infinity, it can be concluded that the perspective collineation maps the line at infinity into a finite line.
The perspective collineation (S, o, A_{1}, A_{2}) is given on the figure 19. Let the line at infinity be denoted with n_{1}^{∞}. Construct its perspective collinear image.
Figure 19


1st step:
Let us choose two arbitrary lines a_{1} and b_{1} that are parallel. Their intersection point lies at infinity and in this case we will denote it with N_{1}^{∞}.
2nd step:
The images of the lines a_{1} and b_{1} are constructed as on the figure 18 (only starting from the field with index 1). The gained lines a_{2} and b_{2} intersect at the point N_{2}, which is the image of the point at infinity N_{1}^{∞}.
3rd step: The perspective collinear image of the line at infinity n_{1}^{∞} is a line n_{2}, which passes through point N_{2} and the intersection point of line n_{1}^{∞} and the axis. Since, the line n_{1}^{∞} is at infinity then it intersects the axis at infinity. Therefore, the line n_{2} is parallel with the axis. The line n_{2} is called the vanishing line of the given perspective collineation.
4th step: The intersection point N_{1}^{∞} of the parallel lines a_{1} and b_{1} was an arbitrarily chosen point. Move the point G to change the direction of the line b_{1}, i.e. of the point N_{1}^{∞}. Notice that the images of all points at infinity lie on the vanishing line n_{2}.
Change the chosen perspective collineation by moving the provisioned points A_{1} or A_{2} and notice that the vanishing line is always parallel to the axis.

Perspective collinear image of the line at infinity is called the vanishing line of the given perspective collineation.
On the other hand, since the perspective collineation is a bijective transformation, the line at infinity will be a perspective collinear image of some line not lying at infinity. In other words, there exists a line whose all points are mapped into the line at infinity.
The perspective collineation (S,o,A_{1},A_{2}) is given on the figure 20. Let us now denote the line at infinity by i_{2}^{∞} (the image line). Construct the perspective collinear image of the line at infinity i_{2}^{∞}, i.e. its perspective collinear preimage.
Figure 20


1st step:
Let us choose two arbitrary parallel lines a_{2} and b_{2} intersecting at the point I_{2}^{∞} at infinity.
2nd step:
The images of the straight lines a_{1} and b_{1} are constructed as on figure 18 (i.e. lines mapped onto the lines a_{2} and b_{2}). Its intersection point I_{1} is mapped to the point at infinity I_{2}^{∞}.
3rd step: The line i_{1}, that corresponds to the line i_{2}^{∞} in the perspective collineation, passes through the point I_{1} and is parallel to the axis. The line i_{1} is called the neutral line of the perspective collineation.
4th step: Since the point I_{2}^{∞} was an arbitrary chosen point, move the point G to change the direction of the line b_{2} (point I_{2}^{∞}) and notice that all the points at infinity correspond to the points on the neutral line i_{1} by given perspective collineation.
Change the chosen perspective collineation by moving the points A_{1} or A_{2} and notice that the neutral line is always parallel to the axis.

The line that the perspective collineation maps to the line at infinity is called the neutral line of the perspective collineation.
Created by Sonja Gorjanc and Ema Jurkin, translated by Helena Halas and Iva Kodrnja  3DGeomTeh  Developing project of the University of Zagreb, made with GeoGebra
