Affine image of a polygon
Affine image of a polygon is again a polygon with the same number of sides and vertices. Since the affinity does not preserve distance and angle measurement, regular polygons will not be mapped into regular polygons. However, the parallelism and segment ratios are invariant under the affinity and therefore, the properties of polygons related to those properties will be preserved.
An affinity (o,X_{1},X_{2}) and equilateral triangle Δ A_{1}B_{1}C_{1} are given in Figure 30. Construct its affine image Δ A_{2}B_{2}C_{2}.
Figure 30


 The construction of an affine image of a polygon starts with finding the affine image of one of its vertices (see the construction in Figure 25).
 Affine images of the remaining vertices are constructed based on the properties of the affinity: Lines on which lie the corresponding sides intersect at a point on the affine axis, pair of corresponding points lie on affine rays.
 Affine image of the equilateral triangle Δ A_{1}B_{1}C_{1} is a scalene triangle Δ A_{2}B_{2}C_{2}.
 Change the position and size of the triangle Δ A_{1}B_{1}C_{1} by moving the points A_{1} and C_{1}. Put the triangle in different positions with respect to the determining elements of the affinity and observe how its affine image changes.
 The centroid of a triangle is the only triangle point that is an invariance of the affinity. Why?
 Change the chosen affinity by moving the points X_{1} and X_{2} and notice that the construction is valid for any affinity.

An affinity (o,X_{1},X_{2}) and square A_{1}B_{1}C_{1}D_{1} are given in Figure 31. Construct its affine image (the parallelogram A_{2}B_{2}C_{2}D_{2}).
Figure 31


 As in the previous example, first the point A_{2} is constructed and then the remaining vertices are constructed using the properties of affinity.
 The obtained affine image A_{2}B_{2}C_{2}D_{2} is a parallelogram. Why?
The position and size of the square can be changed by moving the points A_{1} and D_{1}. Put the square in different positions with respect to the determining elements of the affinity and observe how the affine image changes.
Change the chosen affinity by moving the points X_{1} and X_{2}.
 Under which conditions is an image of a square a rectangle?
The circle passing through intersection points of the axis and lines A_{1}B_{1}, A_{1}D_{1} and with the center at the affine center is constructed. This circle also passes through point A_{1}. Why?
If you place the point A_{2} on the circle (for instance, by moving the point D_{1} or X_{2}), then the affine image of the square A_{1}B_{1}C_{1}D_{1} will be a rectangle. Why? (Thales' theorem)
Will the remaining pairs of corresponding vertices also lie on such circles?
 The intersection of the square diagonals is mapped onto the intersection of the diagonals of the corresponding parallelogram. Why?

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