Fred Lunnon wrote: [attributions have been edited for accuracy]: <<<< On 2/17/07, Emma Cohen <emma@don-eve.dyndns.org> wrote: << Jim Propp wrote: <<< 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?

...

...

Isn't this definition of a line segment just an ellipse whose minor axis is zero?

...

However, if you attempt to construct a finite line segment as the intersection of a plane with a cone --- the origin of the term "conic section" --- you will find it impossible, WFL

...

...

...

This is an interesting point. It seems that by rights, a line segment (= an ellipse with |minor axis| = 0) *should* be within the family of conic sections, as a degenerate case. So if a is fixed, the locus of (*) (x/a)^2 + (y/b)^2 = 1 seems as if it ought to approach the locus of some limiting version of the equation (*) as b decreases to 0. This seems possible, in a sense, if conic sections are thought of as occurring in projective geometry. If we replace R^3 by its "projective compactification" by adding the P^2 of the "lines at infinity", the result is P^3, projective 3-space. A cone like, say, x^2 + y^2 = z^2 is replaced by the addition of a circle at infinity to it, resulting in a cool conoid shape in P^3 whose name I forget, but which (projectively) resembles a torus of revolution with one meridional circle pinched to a point, and which gets fatter the farther away one is from that point. Then the conic sections become the intersection of one of these conoids with an "affine projective plane" in P^3. All the nondegenerate intersections are equivalent to ellipses, but the segmentlike degenerate ellipse at issue becomes a circle. (At least it's compact.) --Dan