Math 403 - Aﬃne Geometry - February 23, 2007

We have seen that there are several ”geometries”, such as Euclidean geometry, non-Euclidean (hyper- bolic) geometry, three-point geometry, each having its own set of axioms. Another point of view is to describe a geometry as the study of the properties of a space that remain invariant (i.e. unchanged) under a certain group of transformations of the space. Different groups of transformations correspond to different geometries. This idea was introduced by the mathematician Felix Klein in a talk he gave at the University of Erlanger in 1872 and has become known as Klein’s Erlanger Program. In Chapter 2, we turn to the study of transformations of the plane. Although Tondeur does not use this terminology, the transformations studied in this Chapter fall into the category of affine transformations. Defining affine geometry as those properties which are invariant under affine transformations, we will see that the definitions and theorems in Chapter 1 all belong to affine geometry. There are several ways to define affine transformations. At http://mathworld.wolfram.com/AffineTransformation.html, you can find the following definition: “An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). ” We can give a very different but equivalent definition by using coordinates and matrix algebra:

Deﬁnition An aﬃne transformation of the plane is a mapping f(~x) of the form

f(~x) = A~x +~b

where A is an invertible matrix and ~b is a vector.

Properties of Aﬃne Transformations

1. An affine transformation is one-to-one and onto from the plane to plane. Therefore it is invertible. ~ 1 1 ~ 1 1~ If f(~x) = A~x + b, then a formula for its inverse is f − (~x) = A− (~x b) = A− ~x A− b. We thus see that the inverse of an affine transformation is itself an affine transformation.− −

2. An aﬃne transformation preserves lines: if A, B and C are collinear, then f(A), f(B) and f(C) are collinear.

3. An aﬃne transformation preserves “betweenness”: if B is between A and C on a line, then f(B) is between f(A) and f(C).

4. An aﬃne transformation preserves the lengths of segments lying along a line: if if A, B and C are collinear, then the following ratios are equal:

A B f(A) f(B) − = − . B C f(B) f(C) − − 5. An aﬃne transformation preserves midpoints (homework problem).

6. An aﬃne transformation preserves the property of being parallel (homework problem).

7. The image of a parallelogram under an aﬃne transformation is another parallelogram.

8. The image of a triangle under an aﬃne transformation is another triangle. 9. Aﬃne transformations preserve centroids of triangles.

10. Aﬃne transformations preserve barycentric coordinates. That is, if f is an aﬃne transformation and if P has barycentric coordinates a, b and c with respect to noncollinear points A, B and C, then f(P ) has the same barycentric coordinates a, b and c with respect to points f(A), f(B) and f(C).

11. Aﬃne transformations applied to triangles preserve the properties of being perspective with respect to a point and being perspective with respect to a line.

What properties are not aﬃne? There are many important properties in Euclidean geometry which are not preserved by aﬃne transformations:

1. Angle measure is not preserved by aﬃne transformations.

2. Length of line segments is not preserved by aﬃne transformations.

3. A circle is not preserved by aﬃne transformations; a circle typically maps to an ellipse. However, ellipses are preserved.

4. Special types of triangles such as right, isosceles, equilateral are not preserved by aﬃne transformations.

5. Rectangles (4 right angles) are not preserved by aﬃne transformations; a rectangle typically maps to a parallelogram.