Motion of Parallel Surfaces in Euclidean 3-Space E3

Abstract

This thesis is connected mainly with the study of the evolution of parallel curves and surfaces embedded in Euclidean 3-space R. This evolution depends on the intrinsic invariants of the embedded curves / surfaces. We then focused to obtain results for some special parallel curves and surfaces which are of importance for their wide applications. In cach case we derive the evolution partial differential equations for all invariant related to the immersed parallel curve / surface in space. Some examples are given and graphed to give a visual understanding of the equations obtained. The study is local which means that the velocity components depend on the curva- tures and their derivatives. The figures in this thesis were constructed using mathematica (11), which show the dynamical theory point of view. Three chapters were introduced in this thesis and its detailed description runs as follow: Chapter 1: This chapter outlines the background and rationale for the present study. It briefly reviews the basic notions, useful concepts, definitions and theories which describe the geometry of curves, surfaces and curves lying on surfaces. The chapter subsequently was written to support the aims of the present study. Chapter 2: In this chapter we describe the motion of curves, surface and curves lying on surfaces in Euclidean 3-space, where the motion is described locally in terms of curvatures and their derivatives. We then derive the evolution equations that govem the described motion. We also study and describe the motion of special surfaces such as surface of revolution and ruled surface for their importance in chapter three, where we use the results obtained to generalize the study to the model describing the motion of parallel surfaces. Chapter 3: In this chapter we explore the geometry of parallel curves, surfaces and curves lying on parallel surfaces. We investigate the properties of the geometry of such objects in R'. We work to derive the associated moving frame to the parallel object and present the frame found in terms of the original frame associated to the original curve or surface. All lo- cal invariants of the parallel object will be derived. Some examples will be given and graphed, local quantities such as curvatures, metrics and fundamental forms will be cal- culated. For a general case we construct the general model of an evolving parallel object (curve or surface). For this model we derive the evolution equations that governs the motion of the model, such as the evolution equations of the frame associated, curvatures. metric, components of the first and second fundamental forms. As an application for the evolving general model we consider the parallel surfaces of revolution and parallel ruled surface. Evolution equations for such models are found, we describe the motion of such surfaces by PDE's of the curvatures for the base surface. We show the geometric meaning of the evolution equations by graph. Finally, some examples will be given and discussed. We depend on integrability conditions in all our calculations.