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This programme looks at how all conic sections can be found as slices of some cone and are therefore perspectively equivalent. This result is used to prove a theorem in 2-dimensional geometry.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 05E
First transmission date: 1995
Published: 1995
Rights Statement:
Restrictions on use:
Duration: 00:24:02
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Producer: Jean Nunn
Contributors: Roberta Cheriyan; Jeremy Gray
Publisher: BBC Open University
Keyword(s): Asymptotes; Ceva's theorem; Eccentricity; Ellipse; Generator; Hyperbola; Parabola; Projective transformation; Projectively equivalent; Section
Subject terms: Conic sections; Conics, Spherical
Footage description: The programme focuses on the theorem that with any conic and any tangent triangle to it the construction joining each vertex of the triangle to the opposite point of tangency produces three lines that meet in a point. The proof involves two things a) the investigation of conies as slices of a cone and consequently b) the projective equivalence of all conics. Computer animations are used to explore how changing the angle of intersection of a plane with a cone produces ellipses, parabolas and hyperbolas, depending on the relation between the angle of the slice and the angle of the cone. But given the freedom to chose the angle of the cone and the position of the slice any two conics can be found as slices of one cone and therefore all conics are protectively equivalent. The therorem is then proved for a circle, using the converse of Ceva's theorem, and so holds true for all other conics.
Master spool number: DOU7934
Production number: FOUM454L
Videofinder number: 4647
Available to public: no