On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
An efficient and simple motion planning algorithm for a ladder amidst polygonal barriers
Journal of Algorithms
Moving a ladder in three dimensions: upper and lower bounds
SCG '87 Proceedings of the third annual symposium on Computational geometry
The complexity of robot motion planning
The complexity of robot motion planning
On the general motion-planning problem with two degrees of freedom
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Planning algorithm for a convex polygonal object in two-dimensional polygonal space
Discrete & Computational Geometry
Efficient motion planning for an L-shaped object
SIAM Journal on Computing
Ray shooting and parametric search
SIAM Journal on Computing
A convex polygon among polygonal obstacles: placement and high-clearance motion
Computational Geometry: Theory and Applications
Motion planning amidst fat obstacles (extended abstract)
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Extremal polygon containment problems
Computational Geometry: Theory and Applications
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Computing roadmaps of semi-algebraic sets (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On Translational Motion Planning of a Convex Polyhedron in 3-Space
SIAM Journal on Computing
Ray shooting and lines in space
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Combinatorial complexity of translating a box in polyhedral 3-space
Computational Geometry: Theory and Applications
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
An algorithm for planning collision-free paths among polyhedral obstacles
Communications of the ACM
Polygon decomposition for efficient construction of Minkowski sums
Computational Geometry: Theory and Applications - Special issue on: Sixteenth European Workshop on Computational Geometry (EUROCG-2000)
The Visibility Diagram: a Data Structure for Visibility Problems and Motion Planning
SWAT '90 Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory
A practical exact motion planning algorithm for polygonal object amidst polygonal obstacles
Proceedings of the Workshop on Geometry and Robotics
Almost tight upper bounds for vertical decompositions in four dimensions
Journal of the ACM (JACM)
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Consider a robot R that is either a line segment or the Minkowski sum of a line segment and a 3-ball, and a set S of polyhedral obstacles with a total of n vertices in R3. We design near-optimal exact algorithms for planning the motion of R among S when R is allowed to translate and rotate. Specifically, we can preprocess S in time O(n 4+ε) for any ε 0 into a data structure that given two placements α and β of R, can decide in time O(log n) whether a collision-free rigid motion of R between α and β exists and if so, output such a motion in time asymptotically proportional to its complexity. Furthermore, we can find in time O(n4+ε) for any ε 0 the largest placement of a similar (translated, rotated and scaled) copy of R that does not intersect S. A number of additional stronger results are provided. Our line segment motion planning algorithm improves the result of Ke and O'Rourke by two orders of magnitude and almost matches their lower bound, thus settling a classical motion planning problem first considered by Schwartz and Sharir in 1984. This implies a number of natural directions for future work concerning rigid motion planning in three dimensions.