This is a question I thought about while crossing the street.
Suppose you're standing at the bottom-left corner of a rectangle.
Your goal is moving to the the top-right corner, efficiently, Considering the following rules:
- As you get closer to the center it's becoming more dangerous.
- You want to get as fast as possible to the top-right corner.
Let $a \in (0,1)$ the importance of rule #1.
Let $b \in (0,1)$ the importance of rule #2.
So, if $a=0.5$ and $b=0.5$ then your path would be a straight line between the two corners.
What would be the general function for the curve? (the path) for any $a,b$
This is a calculus of variations problem, so there is no really simple solution. Some suggestions to start: since your rule $a$ concerns distance to the origin, you want to write your path in polar coordinates; you only need one parameter (since you can normalize by $a+b=1;$ but you also want to say how MUCH more dangerous it becomes as you get closer to the center. Once you state your problem properly, the techniques are pretty standard (calculus of variations has been around about as long as calculus itself).