Intersection of a Hyperplane and an EllipsoidDate: 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 |
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