Track optimisation over waypoints radius

Question

I can't seem to solve the following problem :

Using Google map (and WSG84) I have a set of waypoints defining an air/sea route. (There is no roads : Just straight lines !). I can draw and estimate a route length easily because I have the coordinates of all waypoints but I'm trying to take into account any existing radius around each waypoints to optimize the fastest available route and get the shortest distance. A picture may illustrate this better.

_{(source: ffvl.fr)}

The blue line is easy to sort out. How can I get all intermediates points belonging to each waypoint's radius on the red line ?

Answer 1

For the record, Téo Bouvard's solution described Here with quasi Newtons methods was what I was looking for for such a long time.

His question here look similar than mine.