So I've been trying to write a hard sci-fi story, and I need some help with figuring out some of the math behind it. I remember reading somewhere that if you made an arc and drew another 2 arcs inside and outside of that curve with the same angle and the same focus, the arc outside the curve would be longer than the arc inside the curve.
So, I'm trying to design an FTL drive for the story that applies this theorem to space. Can someone please tell me how I could calculate the arc length of these 2 arcs and the scaling factor between them? I want to know so I could get accurate distances and travel times for ships traveling between stars, but I only know math up to calculus 1.
Angle $\theta$ subtended at center is same. Concentric arc lengths are basically
$$ s_1 = r \theta,\, s_2 = R \theta,\, \frac{s_1}{s_2}=\frac{r}{R} = Scaling Factor $$