Path from a start point at a certain heading to an end point at a certain heading while obeying a minimum turn radius

Question

So not sure if my title is clear (also no idea what to tag, because you need 1000 rep to add tags) so I will try my best to explain the problem.

I'm working in 2D space and to simplify the problem, I'm normalizing headings and locations such that start heading = (1,0) and start position = (0,0).

Given an end point (x,y), a final heading $v$, and a minimum turn radius $r$ is it possible to generate a curved path from start to finish?

What I mean by minimum turn radius is the path cannot turn tighter than the given radius but larger radius turns are possible.

I haven't gotten very far, if at all. Any help or ideas would be greatly appreciated.

Answer 1

You draw a circle of min. radius $r$ tangent to the $x$-axis in the origin, and another passing by $P=(x,y)$ and tangent to $v$. Then you draw a circle, of radius $\ge r$ tangent to both. Choosing the appropriate side when drawing the circles, you can always arrive to a path having the required start and end points and headings.