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Math 403 - Affine Geometry - February 23, 2007

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Math 403 - Affine Geometry - February 23, 2007 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 transforma- tions. 5. Rectangles (4 right angles) are not preserved by affine transformations; a rectangle typically maps to a parallelogram..
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