Chapter 6 - Surfaces

Abstract

A surface can be represented mathematically in three ways: by implicit equations, by explicit equations, and by parametric equations as functions of two parameters u, v. The lines of constant-u and the lines of constant-v form a net of coordinates on the surface. A relationship between the parameters u and v defines a curve on the surface. Given a surface X the partial derivatives Xu, Xv in a point P are the tangent vectors to the constant-u and constant-v curves that pass through P. The normal to the surface is given by the cross product of the above vectors. The square of an element of length on the surface is calculated by an expression known as the first fundamental form. The coefficients of the first fundamental form can be used to calculate angles between curves on a given surface, and areas. Next we consider the curve obtained by the intersection of a surface with a plane. The equation that gives the curvature of this curve involves the first fundamental form and another expression known as the second fundamental form. On a given surface, in a non-singular point there is only one normal to the surface, but infinitely many tangents. The normal and each tangent define a plane normal to the surface that cuts the surface along a certain curve. The curvature of that curve in the given point is the normal curvature of the surface in the direction of the chosen tangent. There are two directions, perpendicular one to the other, such that the normal curvature for one directions is minimal, for the other maximal. Let us note the former κmin, the latter κmax. The Gaussian curvature of the surface is κminκmax, and the mean curvature (κmin+κmax)/2. A ruled surface is a surface such that through each point of it passes a straight line fully contained in the surface.