How to derive the principal radii of an ellipse

[fruitkiwi]

Hi, dear all,

hope your guys allow me to ask this tricky question.
Refer to the attachment, i would like to derive the principal radii of an ellipse, the final equation and figure is provided in Stephen Timoshenko Theory of plates and shells 2nd edition.

consider the ellipse has form b^2*x^2+a^2*y^2=(a^2)*(b^2)
may i know how can i get the correct variable for the red color highlighted length?
because the final equation is so complex, i expect it is not y=2b+unknown.

I use the center as focus origin, the final equation i derive is not correct.as the focus origin move.

In others words , how to get the length for the focus origin as highlighted in red color?

 

[fruitkiwi]

or it is just r1=2b+a?

 

[HallsofIvy]

Your title talks about "principle radii" as if it were a basic property of an ellipse. But the article you show talks about the "principle radii of curvature". That is different at every point and is given by the formula in the article. There is no such thing as a single "principle radius" of an ellipse.

 

[fruitkiwi]

Your title talks about "principle radii" as if it were a basic property of an ellipse. But the article you show talks about the "principle radii of curvature". That is different at every point and is given by the formula in the article. There is no such thing as a single "principle radius" of an ellipse.

Dear HallsofIvy,

Thanks for your reply.
as i read through the book, i do not know how to start to derive, therefore i start with ellipse form equation.

based on your experience, if i am going to derive that equation "principle radii of curvature" of the ellipse, what is the first equation i should use?

i need a starting point to crack through it.any help?