Affine image of a straight line
An affinity (o, X_{1}, X_{2}) and straight line a_{1} are given in Figure 26. Construct the straight line a_{2} i.e. the affine image of the straight line a_{1}.
Figure 26


1st step:
The point A_{1}∈a_{1} is chosen and its affine image A_{2} is constructed (the construction is the same as the one in Figure 25).
2nd step:
The affine image a_{2} of the straight line a_{1} passes through the point A_{1} and through the intersection of the straight line a_{1} and the affine axis o.
 Move the point A_{1} and notice that the affine image a_{2} does not depend on the choice of the point A_{1}.
 Change the position of the points X_{1} and X_{2} with respect to the axis o and the point A_{1}. Observe the changes of the image a_{2}.
 By moving the point Y change the position of the straight line a_{1}. Notice that the construction does not depend on its position, i.e. the construction is valid for any position of the straight line a_{1}.
 Move the point Y so that it coincides with the point Z. Now the straight line a_{1} is parallel to the axis o. In which position is the image a_{2}? 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.

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, X_{1}, X_{2}) and the pair of straight lines a_{2}, b_{2} intersecting at point O_{2} are given in Figure 27. Construct the affine images a_{1} and b_{1}.
Figure 27


1st step:
The affine image O_{1} of the intersection point O_{2} is constructed.
2nd step:
The lines a_{1} and b_{1} pass through the point O_{1}. The straight lines a_{1} and a_{2} (b_{1} and b_{2}) intersect at a point on the affine axis o.
 By moving the point O_{2} change the position of the straight lines a_{2} and b_{2}. Notice that the construction is valid for any pair of intersecting straight lines.
 Notice that the angle between lines a_{1} and b_{1} is different from the angle between lines a_{2} and b_{2}.
In general, the affinity does NOT preserve the angle measure between the straight lines, i.e this transformation is not conformal.

An affinity (o, X_{1}, X_{2}) and the pair of parallel straight lines a_{2}, b_{2} are given in Figure 28. Construct the affine images a_{1} and b_{1}.
Figure 28


1st step:
The point A_{2}∈a_{2} is chosen and then, by using the affine image A_{1}, the affine image a_{1} of the straight line a_{2} 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 O_{2}^{∞} (the point at infinity of the straight line a_{2}) onto the point O_{1}^{∞} (the point at infinity of the straight line a_{1}). Thus, the line b_{1} is parallel to the line a_{1}.
 Check the construction with the corresponding pair of points B_{1}∈b_{1}, B_{2}∈b_{2}.
 By moving the point H change the direction of the straight lines a_{2} and b_{2}.
The affinity preserves the parallelism, i.e. parallel straight lines are mapped into parallel straight lines .

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