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I'm trying to think of a 3D analogue to the ellipse, but not the ellipsoid.

A circle is constructed with a piece of string anchored at the radius. So the distance from center to the curve is always the same.

An ellipse is constructed with a piece of string ending at two foci. The sum of the two lines is always the same.

For 3D, I can think of 3 ways:

  • Three points define a plane. These are our 3 foci. Make lines from each of them that meet at some point on the curve of the shape. But the 3 lines must always sum to the same number. (I'm pretty sure this is not and ellipsoid, but possibly, if 2 foci coincide, it would be an ellipsoid?)

  • Three points define a plane. These are our 3 foci. Make lines from each of them that meet at some point on the curve of the shape. This also defines a tetrahedon. The surface area of the tetrahedron must be constant.

  • Same as above, except we want the volume of the tetrahedron to be constant.

What do those shapes look like? Maybe someone knows how to program a 3D model, or graph it somehow? If so please post pictures.

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    $\begingroup$ Did you try using Geogebra? geogebra.org $\endgroup$ – Kartik Feb 27 '17 at 6:05
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The set of points $M$ such that $MA+MB+MC=k$ in the case where $A=(0,0,0)$, $B=(1,0,0)$ and $C=(2,1,0)$ and $k=3.5$ to $6.5$ (with step $0.5$) look like this:

enter image description here

(points $A,B,C$ are materialized in blue).

Here is the Matlab program which has generated this figure:

      clear all;close all;axis equal;grid on
      Li=-4:0.1:4;L=length(Li);
      [X,Y,Z] = meshgrid(Li,Li,Li);
      V = sqrt(X.^2 + Y.^2 + Z.^2)+...
      sqrt((X-1).^2 + Y.^2 + Z.^2)+...
      sqrt(X.^2 + (Y+3).^2 + Z.^2);
      scatter3([0,1,0],[0,0,-3],[0,0,0],10,'b','filled');
      for k=3:0.5:6.5;
          p=patch(isosurface(X,Y,Z,V,k));
          set(p,'FaceColor','r','EdgeColor','none');lighting gouraud;
          alpha(0.1);
          view([112,18]);
          camlight right
      end
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  • $\begingroup$ It looks kinda like a clam. Is there an official name for this shape? When I have time, I'll try to find a free demo of Matlab and learn how to use it for the other shapes. $\endgroup$ – DrZ214 Feb 28 '17 at 1:50
  • $\begingroup$ Here is a reference with many pointers: (math.stackexchange.com/q/124333). They give the name "n-ellipse" to the curves; volumes are not mentionned. Another interesting reference: (math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/nellipse.pdf). $\endgroup$ – Jean Marie Feb 28 '17 at 7:26
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    $\begingroup$ For a tetrahedron MABC with fixed points $A,B,C$, if you impose a constant volume for this tetrahedron, the locus of M will be a plane (or two planes) parallel to plane ABC because the volume of a tetrahedron is (1/3) area of triangle ABC $\times$ its height. $\endgroup$ – Jean Marie Mar 1 '17 at 1:16
  • $\begingroup$ I see. Then I would hafta derive height from volume and base area. h = 3V/B. But yes this would be two parallel planes extending infinitely, so not a closed shape and not even a curved one. Fascinating. $\endgroup$ – DrZ214 Mar 1 '17 at 2:48

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