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Description
Computer animation and a 3-D model of a perspectivity demonstrate that Euclidean, Non-Euclidean and Protective geometries are unified and also distinguished by Klein's view of each one as a space p...lus a group of transformations. For each geometry, there's a particular set of properties of figures which are "essential" - and the geometry's transformation-group preserves just these properties. In terms of these properties, a new geometry, Affine geometry, is intermediate between Euclidean and Protective geometries: a model of an Affine transformation as a succession of parallel projections between planes, shows that ratios of lengths along a line are preserved (unlike Protective) while distances are not preserved (unlike Euclidean). This sequence of successive generalisations of a geometry is taken up in the second half of the programme in terms of group inclusions: animations and models are used to prove that the Euclidean Group is a subgroup of the Affine Group, which in turn is a subgroup of the Protective Group. The latter inclusion is demonstrated by showing that any parallel projection can be achieved by a sequence of two perspectivities.
Computer animation and a 3-D model of a perspectivity demonstrate that Euclidean, Non-Euclidean and Protective geometries are unified and also distinguished by Klein's view of each one as a space p...lus a group of transformations. For each geometry, there's a particular set of properties of figures which are "essential" - and the geometry's transformation-group preserves just these properties. In terms of these properties, a new geometry, Affine geometry, is intermediate between Euclidean and Protective geometries: a model of an Affine transformation as a succession of parallel projections between planes, shows that ratios of lengths along a line are preserved (unlike Protective) while distances are not preserved (unlike Euclidean). This sequence of successive generalisations of a geometry is taken up in the second half of the programme in terms of group inclusions: animations and models are used to prove that the Euclidean Group is a subgroup of the Affine Group, which in turn is a subgroup of the Protective Group. The latter inclusion is demonstrated by showing that any parallel projection can be achieved by a sequence of two perspectivities.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 25 |
| Recording date: | 14-02-1979 |
| First transmission date: | 1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:24:16 |
| + Show more... | |
| Producer: | Jack Koumi |
| Contributors: | David Crowe; Jeremy Gray; Allan I.,1936-2013 Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Perspectivity; Non-Euclidean geometry; Euclidean geometry; Protectivity geometry |
| Master spool number: | HOU3128 |
| Production number: | FOUM017Y |
| Videofinder number: | 847 |
| Available to public: | no |
