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Old Mar 8th 2013, 09:51 PM   #1
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Smile Fermat's Principle

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.
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Old Mar 9th 2013, 10:07 AM   #2
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Originally Posted by MichaelChan View Post
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.
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