Bill Gosper writes:
Some find it counterintuitive that slicing a cone gives an ellipse and not an oval.
What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse.
What about slicing a paraboloid of rotation? More generally how can eliminating a quadratic mumbloid vs a linear (planar) equation yield anything but a quadratic (conic)?
Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp