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Description
This programme shows how closed curves like circles can be represented parametrically using functions.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 01C; 1995
First transmission date: 1994
Published: 1995
Rights Statement:
Restrictions on use:
Duration: 00:24:30
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Producer: Pip Surgey
Contributors: Lynne Graham; Ian Harrison
Publisher: BBC Open University
Keyword(s): Cardioid; Circle; Ellipse; Parabola
Subject terms: Curves, Algebraic; Curves; Parameter estimation
Footage description: How can circles and ellipses be described by functions? And what about other closed curves like cardioids or astroids? Most people think of a function as given by a rule like y=x², mapping a single x value, a single y value. And although not all functions are one to one, all do have a unique y value in the codomain for each x value in the domain. But a circle is many to many. For any given x value, there are often two corresponding values. And vice versa. So a circle can't be the graph of one function. By combining both the sine and cosine functions, however, it is possible to generate a circle. This programme discusses how the two functions x=cos t, y=sin t can be used to describe circle parametrically and looks at the effect of combining both these functions. Such a representation involves using the parameter t. The standard form for a circle is deduced by eliminating the parameter t. The programme then goes on to look similarly at parametric forms for both the ellipse and parabola, again justifying these algebraically. It is shown that a parametric representation is not necessarily unique (two forms are given for the ellipse). Computer animations show how a three dimensional graph is linked to the two dimensional image set. For example, a circular helix relates to the circle. Next, the programme looks at the generation of both cardioid and trisectrix from their parametric forms. Finally, the programme considers how a function like a cycloid can be described more easily using parameters. This programme would interest anyone who is familiar with the graphs of sin x and cosine and with both the graphs and standard forms of the circle, ellipse and parabola, and who is interested in mathematics at about Advanced Level.
Master spool number: DOU8007
Production number: FOUM466R
Videofinder number: 2278
Available to public: no