Finding Coordinate along Ellipse Perimeter

Question

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?

Answer 1

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

Answer 2

The equation for the ellipse will be $$\frac{x^2}{w^2}+\frac{y^2}{h^2}=1.$$