Chapter 3 - Euclidean Geometry

Abstract

An isometry, or rigid motion, of Euclidean space is a mapping that preserves the Euclidean distance d between points. This chapter focuses on the rigid motions (isometries) of Euclidean space and illustrates how congruency theorms of triangles can be extended to other geometric objects. It emphasizes that an arbitrary isometry of Euclidean space can be uniquely expressed as an orthogonal transformation followed by a translation. A consequence is that the tangent map of an isometry F is, at every point, essentially just the orthogonal part of F. This chapter proved an analogue for curves of the various criteria for congruence of triangles in plane geometry; more specifically, it showed that a necessary and sufficient condition for two curves in R3 to be congruent is that they have the same curvature and torsion (and speed), and the unit-speed curve for position in R3 is determined by its curvature and torsion. Furthermore, the sufficiency proof shows how to find the required isometry explicitly.