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Given an ellipse at (0, 0), with height "h" and width "w", what's the "x" coordinate along the perimeter for a given "y" coordinate?

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up vote 6 down vote accepted

An ellipse is in the standard form if its major and minor axes are co-ordinate axes and intersect at origin. This point of intersection is usually called the center of the ellipse.

This is a standard ellipse whose equation is $$ \dfrac{4x^2}{w^2}+\dfrac{4y^2}{h^2}=1$$

Also, notice that, since the curve is symmetric about origin, there are always $2$ $x$- coordinates that satisfy a given $y$-coordinate and vice versa.

$$x=\pm \dfrac{w}{2}\sqrt{1-\dfrac{4y^2}{h^2}}$$ $$y=\pm \dfrac{h}{2}\sqrt{1-\dfrac{4x^2}{w^2}}$$

Now, that you have pointed out to a positive $x$, we can resolve this sign ambiguity and we'll have that, $$\boxed{x=+\dfrac{w}{2}\sqrt{1-\dfrac{4y^2}{h^2}}}$$

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This raises a question about the factor of $2$ you have included: In his diagram, does $h$ refer to the distance from the top of the ellipse to the bottom, or from the origin to the top? (I really can't tell from the picture) –  Eric Naslund Feb 2 '12 at 14:58
@Eric, OP says his/her ellipse is of height "h", and not "2h". So, I thought this was a reasonable interpretation. –  user21436 Feb 2 '12 at 15:00
Just tested it. That's exactly what I needed. Thanks! –  ElephantHunter Feb 2 '12 at 15:19
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The equation for the ellipse will be $$\frac{x^2}{w^2}+\frac{y^2}{h^2}=1.$$

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