One day I thought about chasing curve on a sphere. I wanted to find parametric equations such as θ(t) and ϕ(t) of running dog always towards a rushing cat on a big spherical world. Orange point is running along the bigest great circle of a sphere but red one is chasing orange point.
This problem led me to very complicated set of 2 equations:
$$r \cos^2(\theta) \dot \phi = u t \sin(\frac{\pi ^2 r - ut}{2 \pi r}) (\cos(\theta) \sin(\phi) \dot \phi + \sin(\theta) \cos(\phi) \dot \theta)$$
$$v^2 = r^2 \dot \theta ^2 + r^2 \dot \phi ^2 \cos^2(\theta)$$
Do anyone can derive equations of motion? I think I made some mistakes here. I will be very appreciate.