This programme shows how three types of curve are related to each other and why they are known as conic sections.
|Module code and title:
|M101, Mathematics: a foundation course
|M101; 18; 1985
|First transmission date:
|Restrictions on use:
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|Mike Crampin; David A. Sargent
|BBC Open University
|3-D animation; Conics; Curves; Ellipses; Hyperbolas; Image creation; Models of cones; Moving beam of light; Parabolas
|This programme shows why the series of curves called ellipses, hyperbolas, and parabolas are collectively known as conics. The programme begins by showing the image created by a moving beam of light. The light successively traces out the images of a circle, an ellipse, a parabola and a hyperbola. The presenter then reminds the viewer of the focus / directrix definitions of the ellipse parabola and hyperbola. Animations show how these curves can be drawn using the focus/directrix definition and also how the shapes of these curves vary as the value of the eccentricity changes. The presenter then explains that the beam of light shown at the start was focussed into the shape of a cone. An opaque screen placed at varying angles gave rise to the different curves. The presenter then shows some models of cones with intersecting planes. A 3-D animation shows how the shapes vary as the angle of the intersecting plane is changed. The presenter then uses a model to prove that the conic section which appears to be an ellipse actually is one. Finally, the presenter discusses other ways of defining curves as locii by properties similar to the focus directrix property and sets a problem for the student to do after the programme.
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