A nice animation of the slicing is provided as one of the 'MathCad Animations' at Old Dominion University in Virginia.
All of the conic sections can be described by quadratic equations. In the special case of a quadratic function equation, the graph is a parabola (this corresponds to the cutting plane being parallel to the side of the cone).
When the plane cuts more directly across the cone, the result is an ellipse (the circle is a special case of this), and when it is closer to being parallel to the axis of the (double) cone, the result is a hyperbola (and a special case of this is just two crossing straight lines - when the cutting plane actually includes the axis of the cone).
For more details and some exercises try the section on conics at the University of Saskatchewan's Exercises in Math Readiness site.
Various other interesting geometric properties of these curves and their
equations are illustrated by applets
and movies on the web.
If you have come across any good links to web-based material about
please do let us know and we will add them here.
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