Affine image of a line segment
In the previous section it is shown that the affinity maps points onto points and points at infinity onto points at infinity. Therefore, the affine image of a line segment can not "brake" as it was the case with perspective collineation. Hence, affine image of a line segment is a line segment.
An affinity (o,A_{1},A_{2}) and line segment A_{1}C_{1} are given in Figure 29.
Figure 29


The affine image C_{2} of the point C_{1} is constructed. The line segment A_{2}C_{2} is the image of the line segment A_{1}C_{1}. Notice that every point B_{1}∈A_{1}C_{1} is mapped to the point B_{2}∈A_{2}C_{2}.
 Move the point C_{1} or the pair of points A_{1}, A_{2} that determines the affinity and observe how the image A_{2}C_{2} changes. Notice the following:
In general, the affinity does NOT preserve distance between points.
Line segment parallel to the affine axis is mapped to a line segment of the same length.
For collinear points A, B and C, such that the point B lies between points A and C, the number (ABC)=AC:BC is called the segment ratio. This number represents the ratio in which the point B divides the line segment AC.
 Move the point B_{1} and observe how the segment ratios (A_{1}B_{1}C_{1}) and (A_{2}B_{2}C_{2}) change. Notice the following:
The segment ratio is invariant under the affinity of the affinity, i.e. (A_{1}B_{1}C_{1}) = (A_{2}B_{2}C_{2})
and specially it follows:
Midpoint of a line segment is invariant under the affinity of the affinity.

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