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MathGroup Archive 2007

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Re: Ellipse equation simplification on Mathematica:

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76265] Re: Ellipse equation simplification on Mathematica:
  • From: CKWong <CKWong.P at gmail.com>
  • Date: Fri, 18 May 2007 06:29:12 -0400 (EDT)
  • References: <f2emof$35h$1@smc.vnet.net>

Errata on my last posting:
Let r, a, and b be vectors in R^3 and consider the surface S given by
r = { x, y, z } that satisfies
    || r - a || + || r - b || = constant                (1)
where || r || is the norm (length) of vector r.
We shall assume S to be a closed surface that bounds a 3-D volume V.

To begin, every cross section of V that passes through points a and b
is an ellipse with a and b as foci.
This means V is an ellipsoid of revolution with the line through a and
b as the axis of revolution.
For the case b = -a, the center of V is at the origin.  Setting z = 0
in eq(1) then gives the intersect C of S with the z = 0 plane.  C is
an ellipse since every cross section of an ellipsoid that passes
through its center is an ellipse.

The directions of the principal axes of C are along the vectors  ( a
cross z-axis ) and (projection of a onto the x-y plane ).  These can
be substituted into eq(1) with z=0 to give the semi-major & semi-minor
axes of C;  and hence an elliptic equation of the canonical form.



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