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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.
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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