Affine image of a straight line
An affinity (o, X1, X2) and straight line a1 are given in Figure 26. Construct the straight line a2 i.e. the affine image of the straight line a1.
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Figure 26
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1st step:
The point A1∈a1 is chosen and its affine image A2 is constructed (the construction is the same as the one in Figure 25).
2nd step:
The affine image a2 of the straight line a1 passes through the point A1 and through the intersection of the straight line a1 and the affine axis o.
- Move the point A1 and notice that the affine image a2 does not depend on the choice of the point A1.
- Change the position of the points X1 and X2 with respect to the axis o and the point A1. Observe the changes of the image a2.
- By moving the point Y change the position of the straight line a1.
Notice that the construction does not depend on its position, i.e. the construction is valid for any position of the straight line a1.
- Move the point Y so that it coincides with the point Z. Now the straight line a1 is parallel to the axis o. In which position is the image a2? Why?
- By moving the point X change the direction of the affine rays (i.e. the position of the affine center). Notice that the construction is valid for any affinity.
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The inverse of any affinity is again an affinity. Therefore, the principle of construction is the same (again as for the perspective collineation one should pay attention to the proper denotation of the indexes).
In the next two examples this property will be used.
An affinity (o, X1, X2) and the pair of straight lines a2, b2 intersecting at point O2 are given in Figure 27. Construct the affine images a1 and b1.
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Figure 27
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1st step:
The affine image O1 of the intersection point O2 is constructed.
2nd step:
The lines a1 and b1 pass through the point O1. The straight lines a1 and a2 (b1 and b2) intersect at a point on the affine axis o.
- By moving the point O2 change the position of the straight lines a2 and b2. Notice that the construction is valid for any pair of intersecting straight lines.
- Notice that the angle between lines a1 and b1 is different from the angle between lines a2 and b2.
In general, the affinity does NOT preserve the angle measure between the straight lines, i.e this transformation is not conformal.
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An affinity (o, X1, X2) and the pair of parallel straight lines a2, b2 are given in Figure 28. Construct the affine images a1 and b1.
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Figure 28
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1st step:
The point A2∈a2 is chosen and then, by using the affine image A1, the affine image a1 of the straight line a2 is constructed (see the construction in Figure 26).
2nd step:
The line at infinity is a ray of every affinity in a plane because the center of every affinity is a point at infinity and all corresponding affine rays are incident with its center. Hence, in every affinity the points at infinity are mapped again onto points at infinity
Therefore, the given affinity maps the point O2∞ (the point at infinity of the straight line a2) onto the point O1∞ (the point at infinity of the straight line a1). Thus, the line b1 is parallel to the line a1.
- Check the construction with the corresponding pair of points B1∈b1, B2∈b2.
- By moving the point H change the direction of the straight lines a2 and b2.
The affinity preserves the parallelism, i.e. parallel straight lines are mapped into parallel straight lines .
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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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