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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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 02 '12 at 14:58
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@Eric, OP says his/her ellipse is of height "h", and not "2h". So, I thought this was a reasonable interpretation. – Feb 02 '12 at 15:00
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Just tested it. That's exactly what I needed. Thanks! – ElephantHunter Feb 02 '12 at 15:19
