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,A1,A2) and line segment A1C1 are given in Figure 29.

Figure 29

The affine image C2 of the point C1 is constructed.
The line segment A2C2 is the image of the line segment A1C1. Notice that every point B1∈A1C1 is mapped to the point B2∈A2C2.
- Move the point C1 or the pair of points A1, A2 that determines the affinity and observe how the image A2C2 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 B1 and observe how the segment ratios (A1B1C1) and (A2B2C2) change. Notice the following:

  • The segment ratio is invariant under the affinity of the affinity, i.e.
    (A1B1C1) = (A2B2C2)

    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