Generalisation of Clairaut’s theorem to Minkowski spaces


Saad, A. (2013) Generalisation of Clairaut’s theorem to Minkowski spaces. Unpublished PhD Thesis. Coventry: Coventry University


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Abstract

The geometry of surfaces of rotation in three dimensional Euclidean space has been studied widely. The rotational surfaces in three dimensional Euclidean space are generated by rotating an arbitrary curve about an arbitrary axis. Moreover, the geodesics on surfaces of rotation in three dimension Euclidean space have been considered and discovered. Clairaut's [1713-1765] theorem describes the geodesics on surfaces of rotation and provides a result which is very helpful in understanding all geodesics on these surfaces. On the other hand, the Minkowski spaces have shorter history. In 1908 Minkowski [1864-1909] gave his talk on four dimensional real vector space, with asymmetric form of signature (+,+,+,-). In this space there are different types of vectors/axes (space-like- time-like and null) as well as different types of curves (space-like- time-like and null). This thesis considers the different types of axes of rotations, then creates three different types of surfaces of rotation in three dimensional Minkowski space, and generates Clairaut's theorem to each type of these surfaces, it also explains the analogy between three dimensional Euclidean and Minkowskian spaces. Moreover, this thesis produces different types of surfaces of rotation in four dimensional Minkowski spaces. It also generalises Clairaut's theorem for these surfaces of rotations in four dimensional Minkowski space. Then we see how Clairaut's theorem characterization carries over to three dimensional and four dimensional Minkowski spaces.



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Item TypeThesis (PhD)
TitleGeneralisation of Clairaut’s theorem to Minkowski spaces
AuthorsSaad, A.
Uncontrolled KeywordsClairaut's theorem, Minkowski space, surfaces of rotation, axes of rotations, geodesics, geometry
Departments Engineering and Computing/Applied Mathematics Research Centre
Engineering and Computing
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Deposited on 05-Dec-2014 in Research - Coventry.
Last modified on 05-Dec-2014

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