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MathGroup Archive 1996

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Re: "Optimized" Plane

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4726] Re: "Optimized" Plane
  • From: rubin at msu.edu (Paul A. Rubin)
  • Date: Sat, 31 Aug 1996 03:57:43 -0400
  • Organization: Michigan State University
  • Sender: owner-wri-mathgroup at wolfram.com

In article <4uphl7$6b5 at dragonfly.wolfram.com>,
   TTCJ34A at prodigy.com (DR JOHN C ERB) wrote:
->August 09, 1996
->1:50 P.M. (CDT)
->
->Greetings;
->
->I have a 3D array of values -- that is, for the x,y,z rectangular 
->coordinate system I have an array
->of values (e.g., air densities) at each x,y,z coordinate in a given 
->volume of space.
->
->Question: How do I find a plane which, if one adds up all the values 
->in that plane (values in the plane
->are determined by the value assgined to the volume elements which the 
->plane intersects), 

Do you mean here that you have values at a discrete set of points?  Or are 
you saying that each of your finitely many values applies over a 
rectangular "bar" in 3-space (a rectangular parallelopiped)?

->such that
->the sum of all the values in the plane are greater than the sum of 
->values for all other planes?
->Stating the problem another way -- how does one find the "densest" 
->plane?

Do you really want a sum here?  I'm thinking of four distinct 
possibilities:

(1) density values apply at isolated points, and you literally sum the 
densities at those points which fall on the plane;

(2) density values are assumed constant through a rectangular zone, and you 
 sum the densities of each zone which intersects the plane, regardless of 
the size of the intersection (so tangency at one corner of the zone is as 
good as cutting across the zone);

(3) density values are constant through rectangular zones, but you really 
want to *integrate* the density along the plane (equivalently, sum across 
zones the products of density times are of intersection of zone and plane);

or

(4) density values are a sample from a continuous function of x, y and z, 
and you want to interpolate/approximate the true function and integrate the 
approximation along the plane.

Interpretations (3) and (4) lead you to integrals.  Interpretation (1) 
suggests an integer program to me (for which Mathematica is not well 
suited).  I believe (2) can also be done via a (somewhat more complicated) 
integer program.
->
->Thank you.
->John C. Erb

Paul Rubin

**************************************************************************
* Paul A. Rubin                                  Phone: (517) 432-3509   *
* Department of Management                       Fax:   (517) 432-1111   *
* Eli Broad Graduate School of Management        Net:   RUBIN at MSU.EDU    *
* Michigan State University                                              *
* East Lansing, MI  48824-1122  (USA)                                    *
**************************************************************************
Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE

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