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:
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 21Notice 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?