# Approximation of Curvature-Constrained Shortest Paths through a Sequence of Points

## Abstract

Let *B* be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most 1, and let *X* denote a sequence of *n* points. Let *s* be the length of the shortest curvature-constrained path for *B* that visits the points of *X* in the given order. We show that if the points of *X* are given *on-line* and the robot has to respond to each point immediately, there is no strategy that guarantees apath whose length is at most *f(n)s*, for any finite function *f(n).* On the other hand, if all points are given at once, a path with length at most 5.03s can be computed in linear time. In the *semi-online* case, where the robot not only knows the next input point but is able to “see” the future input points included in the disk with radius *R* around the robot, a path of length (5.03 + *O(1/R))s* can be computed.

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