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2010/1 Module Catalogue
 Module Code: MAT3020 Module Title: CURVES AND SURFACES
Module Provider: Mathematics Short Name: MS303
Level: HE3 Module Co-ordinator: BRIDGES TJ Prof (Maths)
Number of credits: 15 Number of ECTS credits: 7.5
Module Availability
Assessment Pattern
Unit(s) of Assessment
Weighting Towards Module Mark( %)
2 hour unseen examination
Coursework: assessed problem sheets
Coursework: class test
Qualifying Condition(s) 
A weighted aggregate mark of 40% is required to pass the module.
Module Overview
The module has three parts. The first part is the study of curves in 2D and 3D and their properties. The second part develops the definition of surfaces in 3D and their properties. The third part is the study of curves within surfaces in 3D, with the principal example being geodesics.
MAT1016 (Linear Algebra), MAT1015 (Calculus), MAT2007 (Ordinary Differential Equations)
Module Aims
The main aim of this lecture course is to introduce the differential geometry of curves and surfaces in three-dimensional Euclidean space.
Learning Outcomes
At the end of the module the student should have a firm grasp of the geometry of curves and surfaces in 3D, particularly the concepts of curvature for curves, the concept of geodesic curves on surfaces, and the various notions of curvature for surfaces.
Module Content

The module introduces the study of curves and surfaces in Euclidean space. The geometry of curves involves the concept of torsion (twisting out of a plane) and curvature (twisting away from a line), and the geometry of surfaces involves the mean and gaussian curvatures (the bending away from a plane). The topics covered include arc length, Frenet frames, calculus on curves and surfaces, tangent vectors of curves and surfaces, geodesics on surfaces and their role as the shortest distance between two points, the normal vector of a surface, and integration along surfaces.  Examples of surfaces are spheres, tori, ruled surfaces, surfaces of revolution, and minimal surfaces. Examples from mechanics, computer graphics and other areas are used for illustration. The module consists of five parts




    * Planar curves: representation, arc-length, parameterisation, curvature




    * Space curves: representation, arc-length, parameterisation, curvature, torsion




    * 2D surfaces in 3D: representation, tangent space, normal space, metrics, calculus




    * Paths in surfaces: geodesics, length and speed, parallel transport




    * Curvature of surfaces: mean curvature, gaussian curvature, implications of curvature
Methods of Teaching/Learning
Teaching is by lectures and example classes. Learning takes place through lectures, exercises and background reading.
Selected Texts/Journals
Andrew Pressley (2001) Elementary Differential Geometry, Springer Verlag, ISBN 1852331526
Last Updated

18 March 2011