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?
Finding Coordinate along Ellipse Perimeter
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conic-sections
2 Answers
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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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0Just tested it. That's exactly what I needed. Thanks! – 2012-02-02
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The equation for the ellipse will be $\frac{x^2}{w^2}+\frac{y^2}{h^2}=1.$