Modern Geometry: Methods and Applications, Volume 1

Author: B.A. Dubrovin et al.
Publisher:
Publish Date: 1999-11-01
Features:
Fragment: CHAPTER 1 Geometry in Regions of Space. Basic Concepts? Coordinate Systems
We begin by discussing some of the concepts fundamental to geometry. In school geometry—the so-called "elementary Euclidean" geometry of the ancient Greeks—the main objects of study are various metrical properties of the simplest geometrical figures. The basic goal of that geometry is to find relationships between lengths and angles in triangles and other polygons. Knowledge of such relationships then provides a basis for the calculation of the surface areas and volumes of certain solids. The central concepts underlying school geometry are the following: the length of a straight-line segment (or of a circular arc); and the angle between two intersecting straight lines (or circular arcs). The chief aim of analytic (or coordinate) geometry is to describe geometrical figures by means of algebraic formulas referred to a Cartesian system of coordinates of the plane or three-dimensional space. The objects studied are the same as in elementary Euclidean geometry: the sole difference lies in the methodology. Again, differential geometry is the same old subject, except that there are the subtle techniques of the differential calculus and linear algebra brought into full play. Being applicable to general "smooth" geometrical objects, these techniques provide access to a wider class of such objects.
1.1. Cartesian Coordinates in Space
Our most basic conception of geometry is set out in the following two paragraphs:
(i) We do our geometry in a certain space consisting of points P, Q, ....
(ii) As in analytic geometry, we introduce a system of coordinates for the space. This is done by simply associating with each point of the space an ordered n-tuple (x_1, ..., x_n) of real numbers—the coordinates of the point—in such a way as to satisfy the following two conditions:
(a) Distinct points are assigned distinct n-tuples. In other words, points P and Q with coordinates (x_1, ..., x_n) and (y_1, ..., y_n) are one and the same point if and only if x_i' = y_i for i = 1, ..., n.
(b) Every possible n-tuple (x_1, ..., x_n) is used, i.e., is assigned to some point of the space.
1.1.1. Definition. A space furnished with a system of Cartesian coordinates satisfying conditions (a) and (b) is called an n-dimensional Cartesian space and is denoted by R^n. The integer n is called the dimension of the space. We shall often refer somewhat loosely to the n-tuples (x_1, ..., x_n) themselves as the points of the space. The simplest example of a Cartesian space is the real number line. Here each point has just one coordinate x, so that n = 1, i.e., it is a 1-dimensional Cartesian space. Other examples, familiar from analytic geometry, are provided by Cartesian coordinatization of the plane (which is then a 2-dimensional Cartesian space) and of ordinary (i.e., 3-dimensional) space (Figure 1). These Cartesian spaces are completely adequate for solving the problems of school geometry.