Elipse Perimeter Equation

[Techclerk]

A room is shaped like a circle. All walls (or, actually, the wall) measures at a constant radius around the center point.

A curved window is in the wall, and it shares the same curvature and radius focal point as the round wall of the room.

The window appears to be a flat, 2D circle when viewed from the focal point of the room. It looks like a round window in a flat wall would look.

Using the variables of the window diameter and room radius, what is the equation for the perimeter (length) of the concave elipse that describes the shape of the window?

 

[soroban]

Hello, TechClerk!

I have a start on this problem . . .

                      A 
                      *            O is the center of the room;
           R     *    | *          Its radius is OA = OB = OC = R.
            *        r|  * 
       * t            |   *        The window is on arc ACB.               
O * - - - - - - - - - * - *C       Its center is P.
       *              |P  *        Its radius is PA = PB = r.
            *         |  *
            R    *    | *          Let angle AOC = t (theta).
                      *
                      B

The semimajor axis of the ellipse (a) is arc AC.
The semiminor axis of the ellipse (b) is r.

In right triangle APO: . sin θ .= .r/R

Then: . a .= .Arc AC .= .R·θ .= .R·arcsin(r/R)

Therefore, the equation of the ellipse is:

. . . . . . . . .x² . . . . . . . y²
. . . ------------------- + ----- . = . 1
. . . [R·arcsin(r/R)]² . . .r²

I'll let <u>you</u> work out its perimeter . . .

 

[Techclerk]

Thank You

That is a brilliant text diagram. Thank you very much. I'll try it.
You made it look easy!