From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: INTERESTING QUESTION
Date: 26 May 1998 12:44:44 GMT

In article <1998052602142900.WAA00392@ladder01.news.aol.com>,
ENoether <enoether@aol.com> wrote:
>>3 points to determine a circle, 4 points to determine a sphere, then how
>>many points to determine a torus? What is the equation of the torus
>>through these points?
>
>This problem does not have an easy answer. There are quite a few infinite sets
>of points that do not uniquely determine a torus.

Right, but if the points are in "general position", seven should suffice.
Think about the equation you are trying to match. It should be
determined by the 3 coordinates of the center of the torus, the
direction of the central vector (across the plane of the torus), which
can be described with two spherical coordinates, and the two radii.
Conceptually, you write out that equation with seven parameters and three
coordinates (x,y,z) of the points on the torus. Each of the given
points which is supposed to be on the torus may be substituted for
(x,y,z) to give a relation among the parameters. This gives a set of
several equations which can be used to determine the values of the
parameters. 

So how many equations do you need to uniquely determine seven unknowns?
"In general" it's seven -- that is, you must specify seven points on the 
torus; then the torus is uniquely determined -- although not every
set of seven points will do, as noted by the previous poster.
(Moreover, there may be sets of seven points which lead to seven
equations without real solutions -- that is, no torus passes through
them -- and there may be sets of fewer than seven points which lead
to a unique real solution -- that is, some particular sets of six or
fewer points do lie on a torus but that torus is unique. Without
looking more carefully at the equations I cannot say whether these
possibilities actually occur.)

dave