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Look at picture below :

enter image description here

Q: I wonder what's the equation to find $L$ (the red line on {x',y'}, which x' will always be $0$), assuming if you don't know about the ellipse size.

Btw, I want to use the equation to programatically move the red line, on Y axis (bot->top), while also updating it's length ($L$) to match the ellipse on current Y position. Unfortunately, I'm not a math guy, so... Hahaha.

Thank you very much!

User8976
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Moses Aprico
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1 Answers1

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We assume that the ellipse is placed symmetrically about the $x$-axis. We can without loss of generality assume that it is also symmetrical about the $y$-axis. Then the equation of the ellipse is $$\frac{x^2}{r^2}+\frac{y^2}{R^2}=1.$$ It follows that at height $y$ above the $x$-axis, we have $$L=2r\sqrt{1-\frac{y^2}{R^2}}.$$ You may prefer the equivalent $$L=\frac{2r}{R}\sqrt{R^2-y^2}.$$

André Nicolas
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