Hello,
Consider the following ellipsoid in R^n
x_1^2/a_1^2+x_2^2/a_2^2++x_n^2/a_n^2 = 1
( can assume a_i <= a_j for i < j without loss of generality)
I am interested in finding out a regular simplex (equilateral triangle
in 3D, regular tetrahedron in 4D etc) on this surface. While I have an
intutive idea as to how the points could be placed on the surface of
the ellipsoid to form a regular simplex, I was wondering if there is a
closed form formula for these points. Any pointers on related work in n
dimensions will be appreciated. It turns out to be very trivial in 2D.
Thanks

"pd" <pannaga_ks@yahoo.comwrites in article <1158591603.311291.165840@i42g2000cwa.googlegroups. comdated 18 Sep 2006 08:00:03 -0700:
>Consider the following ellipsoid in R^n
>
>x_1^2/a_1^2+x_2^2/a_2^2++x_n^2/a_n^2 = 1
>
>( can assume a_i <= a_j for i < j without loss of generality)
>
>I am interested in finding out a regular simplex (equilateral triangle
>in 3D, regular tetrahedron in 4D etc) on this surface. While I have an
>intutive idea as to how the points could be placed on the surface of
>the ellipsoid to form a regular simplex, I was wondering if there is a
>closed form formula for these points. Any pointers on related work in n
>dimensions will be appreciated. It turns out to be very trivial in 2D.

So trivial it isn't worth mentioning, hmmmm.

The intersection of the ellipsoid with plane x_i = c_i is also trivial.
The hard part is to prove there's a c_i which "works".

Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.

In article <1158591603.311291.165840@i42g2000cwa.googlegroups. com>,
pd <pannaga_ks@yahoo.comwrote:
>Hello,
>
>Consider the following ellipsoid in R^n
>
>x_1^2/a_1^2+x_2^2/a_2^2++x_n^2/a_n^2 = 1
>
>( can assume a_i <= a_j for i < j without loss of generality)
>
>I am interested in finding out a regular simplex (equilateral triangle
>in 3D, regular tetrahedron in 4D etc) on this surface. While I have an
>intutive idea as to how the points could be placed on the surface of
>the ellipsoid to form a regular simplex, I was wondering if there is a
>closed form formula for these points. Any pointers on related
>work in n
>dimensions will be appreciated. It turns out to be very trivial in 2D.
>
>Thanks
>

Represent your ellipsoid in R^n by Q(x) = x^T A x = 1
where A is a positive definite matrix. Thus for any nonzero
vector x, x/sqrt(Q(x)) is on the ellipsoid. If U is any
n x n orthogonal matrix, the column vectors u_1, , u_n
of U form a regular n-1-simplex. Thus if
Q(u_j) = (U^T A U)_{jj} all happen to be equal, the vectors
u_j/sqrt(Q(u_j)) form a regular n-1-simplex on the ellipsoid.
The question is whether such an orthogonal matrix always
exists. Note that sum_j Q(u_j) = trace(U^T A U) = trace(A),
so if Q(u_j) are all equal they are equal to trace(A)/n.
I will prove by induction on n that such a matrix does always
exist. The case n=1 is trivial. For the induction step,
suppose it's true for n-1. Let u be any unit vector in R^n
such that Q(u) = trace(A)/n. Such a vector will exist on any
curve on the unit sphere joining an eigenvector of A for
its least eigenvalue to an eigenvector for its greatest
eigenvalue. Let V be any orthogonal matrix whose last
column is u, and consider the orthogonal matrices
U = V W where W is a block matrix of the form

[ S 0 ]
[ 0 1 ]

where S is an (n-1) x (n-1) orthogonal matrix.
Let the (n-1) x (n-1) upper left submatrix of V^T A V be
B. This of course is a positive definite matrix.
Then (U^T A U)_{jj} = (S^T B S)_{jj} for 1 <= j <= n-1
and Q(u) for j=n. By the induction hypothesis we can
choose S so that (S^T B S)_{jj} for 1 <= j <= n-1 are
all equal to trace(B)/(n-1). Since
trace(A) = trace(U^T A U) = Q(u) + trace(B)
= trace(A)/n + trace(B)
we get trace(B)/(n-1) = trace(A)/n, and so
(U^T A U)_{jj} = trace(A)/n for all n.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Robert Israel wrote:
In article <1158591603.311291.165840@i42g2000cwa.googlegroups. com>,
pd <pannaga_ks@yahoo.comwrote:
>Hello,
>
>Consider the following ellipsoid in R^n
>
>x_1^2/a_1^2+x_2^2/a_2^2++x_n^2/a_n^2 = 1
>
>( can assume a_i <= a_j for i < j without loss of generality)
>
>I am interested in finding out a regular simplex (equilateral triangle
>in 3D, regular tetrahedron in 4D etc) on this surface. While I have an
>intutive idea as to how the points could be placed on the surface of
>the ellipsoid to form a regular simplex, I was wondering if there is a
>closed form formula for these points. Any pointers on related
>work in n
>dimensions will be appreciated. It turns out to be very trivial in 2D.
>
>Thanks
>
>
Represent your ellipsoid in R^n by Q(x) = x^T A x = 1
where A is a positive definite matrix. Thus for any nonzero
vector x, x/sqrt(Q(x)) is on the ellipsoid. If U is any
n x n orthogonal matrix, the column vectors u_1, , u_n
of U form a regular n-1-simplex. Thus if
Q(u_j) = (U^T A U)_{jj} all happen to be equal, the vectors
u_j/sqrt(Q(u_j)) form a regular n-1-simplex on the ellipsoid.
The question is whether such an orthogonal matrix always
exists. Note that sum_j Q(u_j) = trace(U^T A U) = trace(A),
so if Q(u_j) are all equal they are equal to trace(A)/n.
I will prove by induction on n that such a matrix does always
exist. The case n=1 is trivial. For the induction step,
suppose it's true for n-1. Let u be any unit vector in R^n
such that Q(u) = trace(A)/n. Such a vector will exist on any
curve on the unit sphere joining an eigenvector of A for
its least eigenvalue to an eigenvector for its greatest
eigenvalue. Let V be any orthogonal matrix whose last
column is u, and consider the orthogonal matrices
U = V W where W is a block matrix of the form

[ S 0 ]
[ 0 1 ]

where S is an (n-1) x (n-1) orthogonal matrix.
Let the (n-1) x (n-1) upper left submatrix of V^T A V be
B. This of course is a positive definite matrix.
Then (U^T A U)_{jj} = (S^T B S)_{jj} for 1 <= j <= n-1
and Q(u) for j=n. By the induction hypothesis we can
choose S so that (S^T B S)_{jj} for 1 <= j <= n-1 are
all equal to trace(B)/(n-1). Since
trace(A) = trace(U^T A U) = Q(u) + trace(B)
= trace(A)/n + trace(B)
we get trace(B)/(n-1) = trace(A)/n, and so
(U^T A U)_{jj} = trace(A)/n for all n.

I think it should be possible to do better, though: not just a
regular (n-1)-simplex but a regular n-simplex.
Thus for n=2 there is an equilateral triangle inscribed in any
ellipse.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

I think it should be possible to do better, though: not just a
regular (n-1)-simplex but a regular n-simplex.
Thus for n=2 there is an equilateral triangle inscribed in any
ellipse.

i think i was not clear enough in my initial post. I am interested in
placing a regular n-simplex in an n dimesional ellipsoid. Let me know
if you have any ideas on this.

In any case, your response was very insightful, thank you very much

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada