Here's the simplest egg-shape known to me: http://img179.imageshack.us/img179/7915/eggxw7.gif. IIRC, it is the shape used by Arnault of Zwolle (ca. 1450) in describing his lute. Its construction is so simple that I think, before giving it, I should ask if others can figure it out first. David ----- Original Message ----- From: "Torgerson, Mark D" <mdtorge@sandia.gov> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, February 15, 2007 00:36 Subject: RE: [math-fun] Re: sections of quadratic surfaces Is there a technical definition of "egg-shaped"? A way to measure the ovality? Ellipses have eccentricity, do ovals then have eggcentricity? More seriously, what constraints make sense to give a well defined "egg shape"? You have one one axis of symmetry and two curves that must fit together smoothly, very smoothly. I don't see how you can glue halves of two different ellipses together to get the smoothness constraint. (Can you?) There should be lots of other "almost ellipse" definitions that will glue together. Any takers on a simple definition or process to make an egg? Err... oval? ________________________________ From: math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com on behalf of Schroeppel, Richard Sent: Wed 2/14/2007 5:02 PM To: math-fun; math-fun@mailman.xmission.com Cc: rcs@cs.arizona.edu Subject: RE: [math-fun] Re: sections of quadratic surfaces An oval is egg-shaped. The two ends have different curvatures. There's one axis of symmetry, while an ellipse has two. When a plane cuts a cone at an angle, the cut near the vertex cuts the cone more-nearly perpendicularly than the cut far from the vertex. Naively, the near-vertex cut should produce a curve that spreads out more than the far end of the cut, which should be somewhat pointy. So you expect an oval, not an ellipse. The trick is that the far cut is "bigger", which evens out the two ends of the ellipse. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of James Propp Sent: Wed 2/14/2007 3:40 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Re: sections of quadratic surfaces 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun --------------------------------------------------------------------------------
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