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

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 2}log²n), 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 3}n ^{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.