Understanding Elliptical Formulas: Focal Point Discrepancy Explained

[nautica]

Confused as why these two formulas do not match up. Or do they?

http://www.geocities.com/thesciencefiles/ellipse/facts.html

c^2=a^2-b^2

This does not seem to match up with the fact that the focal point should be 1/2 the distance of the radius of curvature.

If you make a = b in the elliptical formula, you will essentially be making a circle, and in that case the focal point would be zero or at the center according to the formula.

But this is not true, the focal point should be 1/2 the radius.

Thanks
Nautica

 

[robphy]

This does not seem to match up with the fact that the focal point should be 1/2 the distance of the radius of curvature.

Are you referring to the focal point of a spherical mirror, where a small pencil of rays parallel to the principal axis (i.e. the rays from an "object at infinity") focuses?

That situation is different from an ellipse where [nonparallel] rays leaving the one focus of an ellipse reflect off the ellipse and focus (converge) on the other focus. For a circle, of course, rays leaving the center of a circle reflect off the circle and converge back on the center.

 

[nautica]

Yes, that is what I am referring too. But an ellipse is just a special case of a circle and as you close the sides, the ellipse becomes a cirlce. So the formula I stated should work. Right?

I don't guess I completely understand what you are saying about the ellipse. Are these not "focal points" in relation to optics.

Thanks
Nautica

 

[robphy]

Here is the elliptical mirror (which has the spherical mirror as a special case)
http://cage.rug.ac.be/~hs/billiards/billiards.html
http://www.math.ubc.ca/~cass/courses/m309-01a/dawson/
In this case, ALL rays from one focus reflect off the mirror and converge at the other focus. (The object at one focus has its image at the other focus. In the circular case, the object at the center has its image at the center.)

Here is the spherical mirror
http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l3a.html
where rays "close to and parallel to the principal axis" reflect off the mirror and converge at point F (where F=R/2). (The object at infinity along the principal axis has its image at F.)

The connection between two constructions is a little subtle.

First, recall from optics that a spherical mirror has "spherical aberration" in the sense that parallel rays "far from the principal axis" do not focus at F.
http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l3g.html
The required shape so that ALL parallel rays focus at F is a parabola.
Look at page 3 (figure 1) of this pdf file
http://www.math.technion.ac.il/~rl/docs/parabola.pdf
The osculating (best fitting, best approximating) circle through the vertex has radius R=2F (that is F=R/2).

Second, take the elliptical mirror above and make it more oblong (effectively moving one focus out to infinity). You'll end up with a parabola. [For an example, consult
http://www.math.unifi.it/archimede/archimede_inglese/curve/curve_giusti/curve5.html ]

Hopefully, you have enough to put the pieces together.
I apologize if this presentation is a little unclear. Maybe someone else can clarify.