In 1957, Lester Eli Dubins proved that the shortest path between two

coordinates for a forward-moving vehicle with a minimum turning radius

is composed entirely of straight lines or no more than three circular arcs of radius

[1].

This Demonstration gives the reachable set of

,

locations from a given starting

coordinate. The boundary of this set is reachable by a circular arc of radius

followed by either a straight path or a circular arc of radius

in the opposite direction. Label a turn to the right at the maximum rate by the letter

, left as

and straight as

; then the optimal paths to the boundary are

,

,

,

.

The Dubins car is a simplified mathematical model of a car that moves on the

,

plane [1]. The car's location is specified by the

location of the center of the car's rear axle and the orientation

of the car. The car cannot move sideways because the rear wheels would have to slide rather than roll. The Dubins car model stipulates that the car be moving forward at a constant speed and have a maximum steering angle that translates into a minimum turning radius

. The minimum turning radius circles are drawn tangent to the starting and ending positions with gray dashed circles.

If the car has forward velocity of 1 unit per second, the system equations are

*,**,*,

where

is chosen from the interval

.

For a car starting at

, define the switching time as

and arc lengths traveled by the car as

and

.

The car position for an

path is

.

For

the ending position is

.

For

the ending position is

,

and for

the ending position is

.

See [2] for more details.

[1] L. E. Dubins, "On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents,"

*American Journal of Mathematics*,

**79**(3), 1957 pp. 497–516.

doi:10.2307/2372560.

[2] E. J. Cockayne and G. W. C. Hall, "Plane Motion of a Particle Subject to Curvature Constraints,"

*SIAM Journal on Control*,

**13**(1), 1975 pp. 197–220.

doi:10.1137/0313012.