How Does Separating the Foci Affect the Shape of a Spheroid?

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Discussion Overview

The discussion revolves around the geometric properties of spheroids, specifically how separating the foci affects the shape of a prolate spheroid. Participants explore the implications of moving the foci apart, including the lengths of the semi-major and semi-minor axes, and the distance from the foci to the surface of the spheroid.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant assumes that the 2D geometry of circles and ellipses is analogous to the 3D geometry of spheres and ellipsoids, questioning what happens when the foci of a sphere are moved apart.
  • Another participant suggests that the resulting ellipsoid is formed by rotating an ellipse around its symmetry axis, noting that not all ellipsoids can be produced this way.
  • There is curiosity about the shape formed when the sum of distances to three points is constant, with references to egg-like shapes.
  • One participant expresses uncertainty about whether the focus-to-surface distance at the pole decreases to less than one unit as the foci separate.
  • Another participant questions the applicability of the eccentricity equation from Wikipedia, suggesting it may not provide the answer needed for their calculations.
  • One participant realizes that the assumption of constant radii leads to a limitation, as the major axis cannot exceed a certain length without degenerating.
  • There is an exploration of how to relate focal separation to the major and minor axes, with a need to specify parameters to maintain a constant volume.

Areas of Agreement / Disagreement

Participants express differing views on the implications of moving the foci apart and the resulting geometric properties. There is no consensus on the relationship between focal separation and the dimensions of the spheroid, nor on the applicability of certain equations.

Contextual Notes

Participants note limitations in their assumptions regarding constant radii and the relationship between volume and axis lengths, indicating that further specification is needed to resolve their questions.

DaveC426913
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A story I wrote depends on some geometry, and I want to get it straight.

Assumptions:
1] The 2D math that applies to circles and ellipses is analagous to the 3D math for spheres and ellipsoids in the ways relevant to the rest of my post.
2] An ellipse is a circle whose two foci are co-incident. Thus, a sphere is an ellipsoid whose two foci are co-incient.

What I want to figure out is what happens to the surface of an ellipse/ellipsoid when the two co-incident foci are moved apart.

Say I have a sphere of unit radius. I move its two foci two units apart. What happens to the surface?
What is the length of its semi-minor axis?
What is the length of its semi-major axis?

And, more specifically, what happens to the distance from focus-to-surface near the "ends"?
Does the distance from focus to surface decrease as the ellipsoid "stretches" thinner? (i.e. is the focus now closer to the surface than one unit?)[ EDIT ] The correct term for my shape is a prolate spheroid - a cigar shape**. Equatorial radii a and b are the same. Polar radius c is longer.

** by magical coincidence, I am writing this while smoking a cigar.
 
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DaveC426913 said:
Say I have a sphere of unit radius. I move its two foci two units apart. What happens to the surface?
What is the length of its semi-minor axis?
What is the length of its semi-major axis?
Same as in the two-dimensional case. The ellipsoid you get in that way is the ellipse rotated around the symmetry axis. Note that not all ellipsoids can be produced that way - you are limited to those with one large axis and two identical smaller axes.
I wonder what shape you get if you require the sum of distances to three (instead of two) points to be constant. The two-dimensional shape can be egg-like. Another egg.
 
mfb said:
Same as in the two-dimensional case. The ellipsoid you get in that way is the ellipse rotated around the symmetry axis.
Right. Which is what?

(The only reason I mentioned the 2D case is so that, when I set up some details in my story, I can sketch the simpler 2D geometry. I don't have to wory about whether 3D geometry has different behavior).
 
The end-result of my question is this: as the two foci separate, does the focus-to-surface distance (what was a moment ago, the radius) at the pole decrease to less than one unit?
 
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DaveC426913 said:
** by magical coincidence, I am writing this while smoking a cigar.
... as long as the NFL is off season ...
DaveC426913 said:
The end-result of my question is this: as the two foci separate, does the focus-to-surface distance (what was a moment ago, the radius) at the pole decrease to less than one unit?
I have a question. Why isn't the equation for the eccentricity ##e^2 = 1 - \frac{a^2}{c^2}## from Wiki (https://en.wikipedia.org/wiki/Spheroid) not the answer because the planar foci are ##(0,±e)##?
 
Ah. OK. Once I drew it out, it was pretty simple. Should have started with that.

elliptoid.png

So, the distance I was looking for was BP = .5
 
Shoot. So THAT's a problem. The above ellipse assumes that it is the radii that remain constant. It can not grow larger than a major axis of 2. At that point, it degenerates to a line. That's useless to me.

I'm going to need to redo it using 3D volume as a constant.

So, a sphere of one unit radius has a volume of 4/3π. I need to find the major and minor axes of a prolate elliptoid with a volume of 4/3π whose foci are one unit apart.Volume of sphere s = 4/3π*r3
Volume of prolate elliptoid e = 4/3π*a2c

e=s
4/3π*r3 = 4/3π*a2c
r=a2c
a2c = 1

Hm. Missing something here...
I need to specify a as a ratio of c.

How does the focal separation relate to the major and minor axes?
More specifically, how can I specify a and c so that the foci are one unit apart?
 
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fresh_42 said:
I have a question. Why isn't the equation for the eccentricity ##e^2 = 1 - \frac{a^2}{c^2}## from Wiki (https://en.wikipedia.org/wiki/Spheroid) not the answer
No idea, sorry.
 

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