Date of Award

6-24-1950

Degree Type

Master's Essay - Restricted

Degree Name

Master of Science (MS)

Department

Mathematical and Statistical Sciences

Abstract

Before we begin our sojourn into this particular phase of applied mathematics, I feel it wise to define our co-subjects. We shall speak first of a geodesic in everyday parlance as being the shortest curve segment between any two points on a particular surface. If the surface be a plane, the curve segments are straight line segments. If the surface be a sphere, the curve segments are arcs of great circles. With this thought in mind, we will investigate the definition from a pure mathematics' point of view. We read in Lane's METRIC DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES that "a curve C on a surface S is a geodesic in case at each point on C the osculating plane of C contains the normal line of s." Since the definition itself might raise a question or two, it is probably wise to explain just what we mean by osculating plane. The osculating plane at any point R is simply the plane which adheres most closely, possesses highest contact, to the curve of which R is a point. In the case of a plane curve, the osculating plane is the plane of the curve.

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