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I'm not awfully familiar with polar coordinates, and I'm trying to prove Kepler's second law via observations. I found plenty of articles regarding proof of Kepler's second law, and the following seemed the easiest to understand: https://www.math.ksu.edu/~dbski/writings/planetary.pdf (third page).

My question is, how can I move $r$ so that it will stretch from a focal point $(c, 0)$ instead of $(0, 0)$? And when doing so, will/can it change the function in the following integral?

$$\text{Area swept between times}\ t_1\ \text{and}\ t_2=\frac12\int_{\theta_1}^{\theta_2}r^2\ \mathrm d\theta$$

Since $r=a-\dfrac{cx}a$, I figured moving the starting point of $r$ to $(c, 0)$ and rewriting $x$ to $\theta$ through trig would be the best way to implement my given information into the integral.

I have $a$, $b$ and $c$, which is why I want to hold onto that function.

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    Aren’t your radii measured from a focus (the planet that the moons are orbiting) in the first place? (Actually, the planet is slightly offset from the focus, but it’s so massive compared to the moons that you may as well assume that it’s not.) – 2017-02-28
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    @amd They are measured from the focus, but in the article I found it appears that they measure r from (0 , 0), so I wanted to know how I can move their function to start at (c , 0) instead. – 2017-02-28
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    In the article you cite, the focus is at the origin. There’s no need to move anything. – 2017-02-28

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