Improved combinatorial bounds and efficient techniques for certain motion planning problems with three degrees of freedom

Dan Halperin*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study motion planning problems for several systems with three degrees of freedom. These problems can be rephrased as the problems of analyzing the combinatorial complexity of a single cell in arrangements of certain types of surfaces (actually, surface patches) in 3-dimensional space. The combinatorial complexity of the entire arrangement in each case that we study can be Θ(n3) in the worst case and for each such arrangement we obtain a subcubic bound on the total combinatorial complexity of all the 3D cells in the arrangement that contain a portion of the 1D boundary of a surface patch in their closure (these are called the interesting cells); the bound is O(n 7 3) in the case of arrangements related to the motion planning problem of a so-called telescopic arm moving in the plane among polygonal obstacles with n corners, and O(n 5 2) in the case of arrangements resulting from the motion planning problem for an L-shaped object in the plane amidst n point obstacles. We also devise an algorithm to compute the interesting cells in the second type of arrangements, whose time complexity is O(n 5 2log2n), and an algorithm with running time O(n 7 3) for the case of a telescopic arm moving among point obstacles, in both cases improving over the best previously known algorithms for these problems, whose time complexity is O(n3logn). Our approach reduces each three-dimensional problem into a collection of problems involving two-dimensional arrangements. To solve these two-dimensional problems we obtain two combinatorial results of independent interest for arrangements in the plane: (i) a tight bound Θ(nm 1 2) on the maximum joint combinatorial complexity of m 'concave chains' in an arrangement of n pseudo lines, and (ii) an upper bound O(m 2 3n 2 3+nα(n)) on the maximum number of edges of m distinct faces in certain types of arrangements of n pseudo segments, which is within an α(·) factor off the lower bound for this quantity.

Original languageEnglish
Pages (from-to)269-303
Number of pages35
JournalComputational Geometry: Theory and Applications
Issue number5
StatePublished - May 1992


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