Okay, so I have just broken into the polar coordinate system, and I like to derive things on my own to strengthen my intuition. I decided to try and derive the equation of an ellipse swiftly on my own, and had the high ambitions of eventually deriving the area of an ellipse with polar integration. When I think of a radius as a function of theta, especially in regards to circles/ellipses, I think of distance with regards to a point, so when I think of finding the equation for the radius of an ellipse as a function of theta, I immediately go to Pythagoreas.(adsbygoogle = window.adsbygoogle || []).push({});

My exact line of reasoning is this:

Pythagorean theorem holds for any point on an ellipse.

c^2 = a^2 + b^2

Except c changes, of course.

If I think of a as x and b as y, I can easily find them independently in terms of theta.

X would be some constant "a" (max x value) times cos(theta): [itex]a*cos(\theta)[/itex].

Y would be some other constant "b" (max y value) times sin(theta): [itex]b*cos(\theta)[/itex].

R would therefore be c, making it: [itex]r^{2} = (a*cos(\theta))^{2} + (b*sin(\theta))^{2}[/itex]

(Or r = sqrt of that)

This makes perfect sense to me, but then I read that r is officially defined as [itex]\frac{ab}{\sqrt{(b*cos(\theta))^{2} + (a*sin(\theta))^{2}}}[/itex]

Plugging in a value for theta shows that these are not equivalent, and I have no doubt that I went wrong somewhere, but how? If I were to pick any point on an ellipse, and were given theta/a/b, that is how I would solve for r. I have also established that my definition of a and b matches the official equation's definition. It goes without saying that my integration attempts later did not turn up sensible results. I am utterly confused.

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# Troubles with Ellipses (Cartesian -> Polar)

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