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| Module Availability |
| Semester 2 |
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| Assessment Pattern |
Assessment Pattern
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Class tests
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25
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2 hour unseen examination
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75
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Qualifying Condition(s)
A weighted aggregate mark of 40% is required to pass the module.
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| 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. |
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| Prerequisites/Co-requisites |
MAT1016 (Linear Algebra) and MAT1015 (Calculus) |
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| Module Aims |
The main aim of this lecture course is to introduce the differential geometry of curves and surfaces in three-dimensional Euclidean space. |
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| 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. |
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| 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: length and speed, curves with zero geodesic curvature
* Curvature of surfaces: mean curvature, gaussian curvature, implications of curvature
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| Methods of Teaching/Learning |
Teaching is by lectures and example classes. Learning takes place through lectures, exercises and background reading. |
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| Selected Texts/Journals |
Andrew Pressley (2001) Elementary Differential Geometry, Springer Verlag, ISBN 1852331526 |
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| Last Updated |
| 21 April 2011 |
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