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.