Originally Posted by **MichaelChan** Hello, everybody! Good morning!
Look at the picture, in the oval, F and F' is the focal points. And the light FM travels from F to M then M relects the light to F'. It's easily known that:
Optical Length (FMF') = FM + MF' = Const
In the picture, there are two surface SMT and PMQ that is tangent to the the oval but of which the curvature are different from the oval's:
Suface PMQ ∵ Curvature : PMQ < Oval
∴ (FMF') is the Maximum
Suface PMQ ∵ Curvature : SMT > Oval
∴ (FMF') is the Minmum
This conclusion is drawn based on Fermat's Principle. I konw what is Fermat's Principle but I understand why the optical length is different from each other because their curvatures are not the same. Why? Is there anybody explaining the reason to me? Thank you very much. |

When the reflecting surface is an ellipse, then a diverging bundle of light rays from point F all focus to F'. If the other surfaces have the same slope at M (that is, they are tangent), then the direction MF' will be the same, BUT the focal point will be different.

For a smaller radius of curvature such as PMQ, the focal length will be shorter than MF'. By Fermat's Principle, neighboring rays at slightly different angles will converge with a smaller optical length.

For a longer radius of curvature such as SMT, the focal point will be extended beyond point F'. The optical length is longer than FMF'.

Language note: "elipse" is a more precise word than "oval" to describe the geometry in English. Also, "curvature" is the

*reciprocal *of the

*radius *of curvature, so the curvature of PMQ is greater then the ellipse, and the curvature of SMT is less.