Intersection of a Hyperplane and an Ellipsoid

Date: 07/02/2007 at 17:34:34
From: Niels
Subject: intersection of a hyperplane and an ellipsoid

In a previous post Dr. Math gave an excellent description of the
intersection of a plane and a 3-D ellipsoid (which forms an ellipse).
I would like to know whether the intersection of an ellipsoid in R^p
with a plane in R^{p-1} is also an ellipsoid in R^{p-1}.  Any
references to a published proof would also be most helpful.  I am a
psychometrician by trade and I need to understand this for a
statistics problem that I am working on.

I don't have the slightest idea of how to generalize the R^3 problem 
to higher dimensions.  Many thanks for any and all tips!

Date: 07/03/2007 at 11:26:31
From: Doctor George
Subject: Re: intersection of a hyperplane and an ellipsoid

Hi Niels,

Thanks for writing to Doctor Math.

I think the answer has to be yes.  An ellipsoid in R^p space can be
written as an extension of the generalized second degree equation.  
All of the quadratic terms will have the same sign.

If you solve the equation of the hyperplane for one of the variables
and substitute it into the generalized second degree equation you get
a generalized second degree equation with one less degree of freedom.
The coefficients of the new quadratic terms will all still have the
same sign, so the reduced equation is still an ellipsoid.

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/03/2007 at 12:15:46
From: Niels
Subject: intersection of a hyperplane and an ellipsoid

Dr. George,

First, let me say "Thank you very much" for your help.  I would like 
to give you a little more information.

Let R equal a p x p symmetric, full rank matrix (its a correlation
matrix if you know statistics).  Also, let a_i = a p x 1 vector of
weights such that a_i' R a_i = c1 (where c1 is a constant).  Also, let
b denote a p x 1 vector (no subscript--there is only 1 b).  My two
constraints are

  a_i' R a_i = c1   and   a_i' R b = c2

where c2 is another constant.  As I understand this problem (and my
knowledge of geometry is almost non existent) then the 1st constraint
defines an ellipsoid in p space and the second constraint defines a
hyperplane.  The two constraints therefore define an ellipsoid in p-1
space.  Is that correct?  Also, do you know of any references for
this fact (I would like to cite a ref in a paper that I am writing--
I am prepared to purchase and read any book with the necessary
information so that I thoroughly understand this problem).

Thanks again for any and all help.

Niels

Date: 07/05/2007 at 07:52:14
From: Doctor George
Subject: Re: Thank you (intersection of a hyperplane and an ellipsoid)

Hi Niels,

I don't have a good reference for you, but you can just prove it in
your paper.  Here is what I think is a solid proof.  Let Q be an
orthogonal matrix such that

  Q' R b = (0,... 0, g)

for some constant g.  Note that g will be the length of R b since Q is
orthogonal.  The strategy is to simplify the problem by using Q to
rotate the ellipsoid and hyperplane so that the hyperplane is
"horizontal", so to speak.

Now let

  a_i = Q b_i

The original constraints become

  (Q b_i)' R (Q b_i) = c1 

  (Q b_i)' R b = c2

Simplifying the first, we get

  b_i' (Q' R Q) b_i = c1                (1)

which is now an ellipsoid in b_i.

Simplifying the second, we get

  b_i' (Q' R b) = c2

  b_i' (0,... 0, g) = c2

so the pth component of b_i = c2 / g.     (2)

If we substitute (2) into (1) we get a generalized 2nd order equation
in the other p-1 variables that is also an ellipsoid.

I think that should do it.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/16/2007 at 15:33:24
From: Niels
Subject: Thank you (intersection of a hyperplane and an ellipsoid)

Dr. George,

Thanks again for your help on my ellipsoid/plane intersection problem.

Best wishes,

Niels