Abstract
Orthogonal transformations leave invariant both the shape and the dimensions of geometric figures. If the demand that such transformations preserve dimension is discarded but that shapes be preserved, the set of transformations that is found is the group of similarity transformations of the plane or of space. This chapter presents an overview of similarity transformations and their properties. Under a similarity transformation, the images of three collinear points, A, B, and C, are three collinear points A', B' and C'. If B lies between A and C, B' lies between A' and C'. It may be shown that the image of a line is a line and in a similarity transformation of space, the image of a plane is a plane. The ratio of the lengths of any two line segments is equal to the ratios of the lengths of their images under a similarity transformation. This chapter describes homothetic transformations and discusses the representation of a similarity transformation as the product of a homothetic transformation and an orthogonal transformation. It also reviews the similarity transformations of the plane in coordinates and the similarity transformations in space.
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