Reduction of Geometry to Algebra

The reduction of geometry to algebra requires the notion of a transformation group. The transformation group supplies two essential ingredients. First it is used to define the notion of equivalence in the geometry in question. For example, in Euclidean geometry, two triangles are congruent if there is distance preserving transformation carrying one to the other and they are similar if there is a similarity transformation carrying one to the other. Secondly, in each kind of geometry there are normal form theorems which can be used to simplify coordinate proofs.

Coordinate Systems in Geometry

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).

Additional Information

Other coordinate systems are possible. The book provides the students current information on the different areas of this subject. The publication is useful not only to the students but also to the research scholars and academic professionals.