### Perspective collinear image of a line segment

Perspective collineation maps points onto points and straight lines onto straight lines, but what happens when we map a line segment? A line segment is determined with its endpoints and it is clear how they will be mapped, but what happens with a set of countinuosly coonected points between them?
In general, we distinguish four cases of perspective collinear image of the given line segment A1B1:
• If the line segment A1B1 does not intersect the neutral line, then its perspective collinear image is a line segment A2B2.
• If one of the endpoints A1 (or B1) lies on the neutral line, then the perspective collinear image of the line segment A1B1 is a ray starting from the point A2 (or B2).
• If the line segment A1B1 intersects the neutral line in a point which is not its end point, then its perspective collinear image splits into two opposite rays that lie on the same straight line starting from the points A2 and B2.
• If the line segment A1B1 lies on the neutral line, then its perspective collinear image is a line segment at infinity.

Let a perspective collineation (S,o,X1,X2) and line segment A1B1 lying on the line p1 be given. The construction of the perspective collinear image of the line segment is given on the figure 21. Move the line segment and check the above given statements.

Figure 21

Notice that the perspective collineation does NOT preserve the distance between the points (it is not an isometry) and also does not preserve the segment ratios (for instance the midpoints).

What can you conclude about the mapping of some ray? In which case is a ray perspective collinear image of some ray?

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