[1] Lester Eli Dubins (1920–2010)[2] proved using tools from analysis[3] that any such path will consist of maximum curvature and/or straight line segments.
In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines.
In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle.
[4] In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length.
[5] More recently, a geometric curve-theoretic proof has been provided by J. Ayala, D. Kirszenblat and J. Hyam Rubinstein.
There are simple geometric[8] and analytical methods[9] to compute the optimal path.
An optimal path will always be at least one of the six types: RSR, RSL, LSR, LSL, RLR, LRL.
For example, consider that for some given initial and final positions and tangents, the optimal path is shown to be of the type 'RSR.'
The tangent direction of the path at initial and final points are constrained to lie within the specified intervals.