Distance Between Two Points Travelling on a Polar Graph Chart
Haskell, 49 48 bytes

(a!q)c r=min(q+r)$abs(q-r)+acos(cos$a-c)*min q r


Usage:

&gt; let rect2polar (x,y)=(atan2 y x, sqrt(x^2+y^2))
&gt; let test c1 c2=let [(a1,r1),(a2,r2)]=rect2polar&lt;$&gt;[c1,c2] in (a1!r1)a2 r2
&gt; test (1,0) (-0.4161, 0.90929)
1.9999342590038496


Thanks to @Zgarb for saving a byte