# Ellipsoid

Discussion in 'Math' started by sytem_recon, Aug 26, 2009.

1. ### sytem_recon Thread Starter Active Member

Apr 21, 2009
52
0
If I have an ellipsoid. If light emmitting in spherical or circular shape from its one focus (focal point), is it possible to reflect maximum of the rays on its other focus (focal point)? what will be the method and how to check/verify it? Is there any other geometry to focus light from one point to another.

2. ### rspuzio Active Member

Jan 19, 2009
77
0
Yes, for an ellipsoid of revolution, every ray emerging from one focal
point will be reflected to the other focal point. In practice since your
source is not an ideal point and there will be some supports or power
connections to the source, there will be some rays which reflect back
to the source or land on the supports.

Are you interested in a mathematical proof or how to check it
physically using an ellipsoidal mirror?

I feel quite confident that the answer is no, but I'd want to check
through the details and write out the proof before saying for sure.
If you are interested, let me know and I will gladly explain my line
of reasoning --- it is based on Expressing the question as a Pfaff
problem, arguing that the solution must be spherically symmetric,
reducing to one dimension, where the problem reduces to finding
integral curves of a vector field, then noting that the ellipses are
all the integral curves.

3. ### sytem_recon Thread Starter Active Member

Apr 21, 2009
52
0
Thanks a lot for reply. I need not only mathematical proof and also physical(through some computer simulation). If i have a source on the focal point emitting light in spherical shape and the mirror will be ellisoid.

4. ### rspuzio Active Member

Jan 19, 2009
77
0

Jul 7, 2009
1,585
141
Also consider that the paraboloid is in some sense a degenerate ellipsoid; one focus is "off at infinity".

6. ### sytem_recon Thread Starter Active Member

Apr 21, 2009
52
0
This proof is only for a 2D ellipse. I need a proof of 3D ellipsoid. Also you discuss reasoning for not other geometry. How i can simulate/program through a computer that when a light ray emit from a focus of ellipsoid will reflect to the other focus of it.

7. ### rspuzio Active Member

Jan 19, 2009
77
0
By rotational symmetry, the 3D case reduces to 2D.

8. ### sytem_recon Thread Starter Active Member

Apr 21, 2009
52
0
I am very good in maths. What do you mean by rotatinal symmetry. I don't understand it.

9. ### sytem_recon Thread Starter Active Member

Apr 21, 2009
52
0
I am not very good in maths. What do you mean by rotatinal symmetry. I don't understand it.