by pat » Fri Jul 30, 2004 10:12 am
Throughout, I've tried to use italics to denote scalar quantities and bold to denote vector quantities.
An oval is all points in the plane such that the sum of the distances of the points to two given points remains constant. Formulaicly, that is, given two focal points j and k and some total distance, d, then an oval is all points x in a plane (which contains j and k), | (x - j) | + | (x - k) | = d.
Those are vector formulas. If j = k, then the oval is a circle of diameter d. One could consider these formulas to hold in any number of dimensions.
However, it may be preferable to extend the concept of oval slightly
so that in n dimensions, there are n focal points p<sub>1</sub>, p<sub>2</sub>, ..., p<sub>n</sub>.
The oval will be slightly degenerate if some set of k of those points are in the same (k-2)-dimensional space.
An alternative formulation (for ovals whose axises are aligned with the universal ones) using coordinates rather than vectors is that an oval is all points in the (x,y)-plane just that, given three non-zero numbers a,b,c, we have: (<sup>x</sup>/<sub>a</sub>)<sup>2</sup> + (<sup>y</sup>/<sub>b</sub>)<sup>2</sup> = c<sup>2</sup>. Here, it's a circle if a = b. (By suitable scaling, we can force one of the three numbers a,b,c to be one, usually we pick c = 1.)
For this formulation, the obvious extension to n dimensions is to pick n non-zero numbers c<sub>1</sub>,c<sub>2</sub>,...,c<sub>n</sub> and say that the oval is all points x such that Σ (<sup>x<sub>i</sub></sup>/<sub>c<sub>i</sub>)<sup>2</sup> = 1 (where x<sub>i</sub> is the i-th coordinate of x).
I am pretty sure that these two formulations (both natural) of hyperovals result in different shapes. The latter always has mirror symmetry when reflected across the hyperplanes perpendicular to world axises and symmetry through the origin. The former cannot have that kind of symmetry when there are an odd number of distinct points involved.