Available online
Description
This programme looks at the standard types of surfaces generated by general equations of the second degree.
This programme looks at the standard types of surfaces generated by general equations of the second degree.
| Module code and title: | M203, Introduction to pure mathematics |
|---|---|
| Item code: | M203; 16 |
| First transmission date: | 05-06-1980 |
| Published: | 1980 |
| Rights Statement: | |
| Restrictions on use: | |
| Duration: | 00:25:00 |
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| Producer: | Martin Wright |
| Contributors: | David Crowe; Allan I.,1936-2013 Solomon |
| Publisher: | BBC Open University |
| Keyword(s): | Computer Animation; Conics; Cylinder; Ellipsoid; Elliptic Paraboloid; Hyperbolic Paraboloid; Hyperboloid; Models; Quadratic Equations; Standard Form |
| Footage description: | Allan Solomon introduces the programme by describing how a complicated equation representing an ellipsoidal surface can be reduced to a standard form. The equation is expressed in matrix form and diagonalised by an orthogonal transformation. The standard form of the equation makes it easy to visualise the type of surface it represents. Using a model of the ellipsoidal surface he demonstrates how slicing through the surface parallel to any of the co-ordinate planes he obtains an elliptical section. David Crowe now discusses the possibilities that arise as the signs of the coefficients in the standard equation are changed. Ely making the third constant negative he obtains a figure which gives an ellipse when sliced in the xy plane but a hyperbola when sliced through in the yz plane. Such a shape is called a hyperboloid. Allan Solomon now examines the surface generated by an equation with one positive co-efficient and two negative. This gives a surface which splits up into two pieces which when sliced give elliptical and hyperbolic surfaces. He demonstrates this with a model. This shape is known as a two sheet hyperboloid. David Crowe talks over a vivid computer animation describing the shapes and equations discussed in the programme so far. David Crowe discusses some of the qualities of particular central quadrics. Allan Solomon then describes non-central quadrics, represented by equations which cannot be reduced to the standard form without a linear term. He examines the various possibilities for the coefficients at b and c. When all three are positive this gives an elliptic paraboloid giving parabolas when the surface is sectioned through the x and y planes, sections parallel to the xy plane are ellipses. This surface is called an elliptic paraboloid. David Crowe next examines the surface generated by an equation with two negative coefficients. Vertice sections give parabolas, horizontal sections give hyperbolas. This surface is called a hyperbolic paraboloid. Allan Solomon reviews the findings of the programme. |
| Master spool number: | DOU3251 |
| Production number: | FOUM056T |
| Videofinder number: | 4099 |
| Available to public: | no |
