How to derive the principal radii of an ellipse

In summary, HallsofIvy asks how to get the length for the focus origin highlighted in red color in Stephen Timoshenko Theory of plates and shells 2nd edition. However, the article HallsofIvy shows talks about the principle radii of curvature which 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.
  • #1
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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?
 

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  • #2
or it is just r1=2b+a?
 
  • #3
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.
 
  • #4
HallsofIvy said:
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?
 

1. What is the formula for finding the principal radii of an ellipse?

The formula for finding the principal radii of an ellipse is: r1 = a and r2 = b, where a is the semi-major axis and b is the semi-minor axis of the ellipse.

2. How do you determine the semi-major and semi-minor axes of an ellipse?

The semi-major and semi-minor axes can be determined by measuring the longest and shortest distances respectively from the center of the ellipse to its edge. Alternatively, you can also use the equation x^2/a^2 + y^2/b^2 = 1 to find the values of a and b.

3. Can the principal radii of an ellipse be negative?

No, the principal radii of an ellipse cannot be negative. They are always positive values that represent the distances from the center of the ellipse to its edge.

4. How are the principal radii of an ellipse related to its foci?

The principal radii of an ellipse are directly related to its foci. The sum of the distances from any point on the ellipse to its two foci is always equal to the length of the major axis, which is equal to 2a. This relationship can be expressed as r1 + r2 = 2a.

5. Can the principal radii of an ellipse be equal?

Yes, the principal radii of an ellipse can be equal only if the ellipse is a circle. In a circle, both the semi-major and semi-minor axes are equal, making the principal radii r1 and r2 equal as well.

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