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,X1,X2) and equilateral triangle Δ A1B1C1 are given in Figure 30.
Construct its affine image Δ A2B2C2.
- 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 A1B1C1 is a scalene triangle Δ A2B2C2.
- Change the position and size of the triangle Δ A1B1C1 by moving the points A1 and C1. 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 X1 and X2 and notice that the construction is valid for any affinity.
An affinity (o,X1,X2) and square A1B1C1D1 are given in Figure 31. Construct its affine image (the parallelogram A2B2C2D2).
- As in the previous example, first the point A2 is constructed and then the remaining vertices are constructed using the properties of affinity.
- The obtained affine image A2B2C2D2 is a parallelogram. Why?
The position and size of the square can be changed by moving the points A1 and D1. 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 X1 and X2.
- Under which conditions is an image of a square a rectangle?
The circle passing through intersection points of the axis and lines A1B1, A1D1 and with the center at the affine center is constructed. This circle also passes through point A1. Why?
If you place the point A2 on the circle (for instance, by moving the point D1 or X2), then the affine image of the square A1B1C1D1 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