### 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, A1, A2) is given on the figure 19. Let the line at infinity be denoted with n1. Construct its perspective collinear image.

 Figure 19 1st step: Let us choose two arbitrary lines a1 and b1 that are parallel. Their intersection point lies at infinity and in this case we will denote it with N1∞. 2nd step: The images of the lines a1 and b1 are constructed as on the figure 18 (only starting from the field with index 1). The gained lines a2 and b2 intersect at the point N2, which is the image of the point at infinity N1∞. 3rd step: The perspective collinear image of the line at infinity n1∞ is a line n2, which passes through point N2 and the intersection point of line n1∞ and the axis. Since, the line n1∞ is at infinity then it intersects the axis at infinity. Therefore, the line n2 is parallel with the axis. The line n2 is called the vanishing line of the given perspective collineation. 4th step: The intersection point N1∞ of the parallel lines a1 and b1 was an arbitrarily chosen point. Move the point G to change the direction of the line b1, i.e. of the point N1∞. Notice that the images of all points at infinity lie on the vanishing line n2. Change the chosen perspective collineation by moving the provisioned points A1 or A2 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,A1,A2) is given on the figure 20. Let us now denote the line at infinity by i2 (the image line). Construct the perspective collinear image of the line at infinity i2, i.e. its perspective collinear preimage.

 Figure 20 1st step: Let us choose two arbitrary parallel lines a2 and b2 intersecting at the point I2∞ at infinity. 2nd step: The images of the straight lines a1 and b1 are constructed as on figure 18 (i.e. lines mapped onto the lines a2 and b2). Its intersection point I1 is mapped to the point at infinity I2∞. 3rd step: The line i1, that corresponds to the line i2∞ in the perspective collineation, passes through the point I1 and is parallel to the axis. The line i1 is called the neutral line of the perspective collineation. 4th step: Since the point I2∞ was an arbitrary chosen point, move the point G to change the direction of the line b2 (point I2∞) and notice that all the points at infinity correspond to the points on the neutral line i1 by given perspective collineation. Change the chosen perspective collineation by moving the points A1 or A2 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