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.
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Figure 29
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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.
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Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra
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