I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format –
Where $2a$ is the total height of the ellipse and $2b$ being the total width.
You can then find the radial length, $r$, at any angle $\theta$ to major axis as…
$$r(\theta) = \sqrt{(b \sin(\theta))^2 + (a \cos(\theta))^2}$$
…by just following the Pythagorean theorem. Yet Wikipedia's equation for the polar coordinate ellipse is as follows:
$$r(\theta) = \frac{ab}{\sqrt{(b \cos(\theta))^2 + (a \sin(\theta))^2}}$$
Here is the link to the Wikipedia page:
Can someone explain this, please? Why divide by the hypotenuse? Why the $ab$? Thank you!
Best Answer
It's easiest to start with the equation for the ellipse in rectangular coordinates:
$$(x/a)^2 + (y/b)^2 = 1$$
Then substitute $x = r(\theta)\cos\theta$ and $y = r(\theta)\sin\theta$ and solve for $r(\theta)$.
That will give you the equation you found on Wikipedia.