Available online
Description
By discussing the problems involved in creating a map of the world, the programme demonstrates that it is impossible to obtain an isometry of the sphere to the plane.
By discussing the problems involved in creating a map of the world, the programme demonstrates that it is impossible to obtain an isometry of the sphere to the plane.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 02 |
| First transmission date: | 21-02-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:25:00 |
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| Producer: | Jack Koumi |
| Contributors: | Roger Duke; Colin Rourke |
| Publisher: | BBC Open University |
| Keyword(s): | Angles; Areas; Plane Maps (of the) Earth; Shortest Paths |
| Subject terms: | Cartography; Topology |
| Footage description: | Colin Rourke introduces the programme by describing several different types of maps of the earth and discussing the problem of mapping the spherical earth onto a flat surface. Roger Duke describes the properties that he would like a map to have, ideally it should preserve distances, shapes, sizes angles and areas. Colin Rourke examines the concept of distance preservation more closely and establishes that it implies the preservation of shortest paths and that the distance along any curve is preserved. This last result is used to show that distance preservation implies preservation of all the other desired characteristics. Such a map is called an isometry. Colin Rourke argues that angle preservation, shortest path preservation and area preservation imply an isometry. Roger Duke now examines various maps which have some of these properties to check to see if they have them all. First of all he describes the way in which a central or Gnononic projection is drawn. He shows that this map preserves shortest paths, as great circles project to straight lines. Angles and areas however are not preserved. A map which preserves areas is now examined. This is Lambert's horizontal projection, Roger Duke explains how this projection is drawn and shows that although areas are preserved, short distances are not. Colin Rourke explains how areas can be preserved whilst angles are not. In order to preserve angles two conditions must be met, meridians and parallels must be at right angles and the scale along the meridians must be the same as the scale along the parallels. This idea leads directly to the next projection, Mercator's projection. Roger Duke describes this map and then shows that although angles are preserved, shortest distances and areas are not. Colin Rourke now argues that a map which preserves angles and areas would be an isometry. Roger Duke demonstrates that it is not possible to preserve both angles and shortest paths. This allows Colin Rourke to deduce that the search for an isometry is bound to fail. There is not isometry from the sphere to the plane. Roger Duke demonstrates this by trying to flatten a cap fitted around the top of the globe, without stretching or cutting. Colin Rourke argues that interrupted maps are not isometries either. He then summarises the results of the programme so far. Colin Rourke discusses the problem of mapping the whole sphere onto the plane without tearing. Roger Duke explains how this is done, by constructing a Stereographic projection from the South pole. On this projection meridians are mapped to straight lines, circles of latitude are mapped to circles. This shows that Stereographic projection preserves angles. The final map explained is the Great Circle map. Roger Duke describes how to construct such a map, explaining that its centre could be at any point. This map allows an accurate measurement of any distance from the centre. Colin Rourke sums up some of the ideas from Topology and Analysis found in the programme. |
| Master spool number: | 6HT/72969 |
| Production number: | FOU9019D |
| Videofinder number: | 834 |
| Available to public: | no |
