Curves and surfaces in the context of optometry. Part 2: Surfaces

Abstract

This paper introduces the differential geom-etry of surfaces in Euclidean 3-space. The first and second fundamental forms of a surface are defined.   The  first  fundamental  form  provides a metric for calculations of length and area on the surface. The second fundamental form deter-mines surface curvature and, hence, concepts of importance in optometry such as surface power and sagitta. The principal curvatures at a point on a surface are obtained as solutions of a qua-dratic equation. The torus is used to illustrate the methods.