What would be the average distance from the point $P=(a,b)$ (outside the circle), to any point on the circumference with center at $(0,0)$ and radius $r$ be?
Asked
Active
Viewed 92 times
0
-
what are your thoughts, trials, and sticking points ? – Dec 03 '19 at 20:17
-
Averaged over what? Over all points $P$ outside of the fixed circle $C_{(0,0),r}$? It isn't clear to me what this is asking... – MPW Dec 03 '19 at 20:17
-
I think it's asking for a generic formula for any given point. – Dec 03 '19 at 20:20
-
Intuitively is seems to be infinite if each point is equally likely? – Goldbug Dec 03 '19 at 20:22
1 Answers
0
Use the SAS (side-angle-side) formula for the green distance, $d = \sqrt{1^2 + r^2 - 2 r \cos \theta}$, and integrate over $\theta$ (using symmetry)
$$2 \int\limits_{\theta = 0}^\pi \sqrt{1^2 + r^2 - 2 r \cos \theta}\ d \theta = 4 (r+1) E\left(\frac{4 r}{(r+1)^2}\right)$$
where the $E$ is the elliptic integral function.
Can you polish this ensuring normalization? (Think about the length of the arc integrated over.)
David G. Stork
- 29,774