Fundamentals

Abstract

The first chapter of this book is an introduction into projective geometry. Although projective geometry is, from the abstract viewpoint, nothing but linear algebra in disguise, it is a geometric language which allows a unified approach to such different things as Euclidean geometry of points, the differential geometry of ruled surfaces, or spherical kinematics. Sec. 1.1 introduces projective space and coordinates for points and hyperplanes, studies linear subspaces and quadratic varieties, and shows the relations between some of the classical geometries. In Sec. 1.2, we consider curves and surfaces from the viewpoint of projective geometry. Afterwards we describe some basic facts about algebraic varieties in affine and projective space. Finally, Sec. 1.4 deals with polynomial and rational curves and surfaces in Computer-Aided Geometric Design.