I am thinking here that geometrically, the solution to this problem is three lofts of arcs where the arcs are tangent.
The top set of arcs starts with zero radius, then the center moves inwards and downwards until the center matches the end circle and its radius.
The bottom set of arcs does the same except that it goes upwards and inwards.
The third set of arcs starts with radius infinity (line) then the center progresses to the center of the circle.
The trick is to figure out a set of arcs that produces a smooth surface while maintaining tangency.
