Modeling social distance on a walk

Question

I’ve been thinking about this on my walks for the last year: I’m walking down the sidewalk and someone is coming towards me in the opposite direction. I step of the sidewalk to make sure that we maintain 6' of separation at all times. How would I describe the optimal path that minimizes my time off the sidewalk? I can envision roughly what it looks like, and the boundary cases are simple enough. We can define $v_r$ to be the ratio between the sidewalk walker’s velocity and the grass walker’s velocity. if $v_r=0$ (i.e., someone stands on the sidewalk and the other person walks around them), the path is to follow the sidewalk and then form a semi-circle of radius $a$ around them before continuing on the sidewalk. And in the case of $\lim_{v_r\to\infty} v_r$, then the grass walker is walking parallel to the sidewalk at a distance of $a$ until the sidewalk walker passes then she moves straight to the sidewalk. Can we describe the curves of the path otherwise as a function of $v_r$?