Vertical Decomposition of a Single Cell in a Three-Dimensional Arrangement of Surfaces

Abstract. Let Σ be a collection of n algebraic surface patches in ${\Bbb R}^3$ of constant maximum degree b, such that the boundary of each surface consists of a constant number of algebraic arcs, each of degree at most b as well. We show that the combinatorial complexity of the vertical decomposition of a single cell in the arrangement ${\cal A}(\Sigma)$ is O(n^{2+ɛ}), for any ɛ > 0, where the constant of proportionality depends on ɛ and on the maximum degree of the surfaces and of their boundaries. As an application, we obtain a near-quadratic motion-planning algorithm for general systems with three degrees of freedom.

[1]  Micha Sharir.  Almost tight upper bounds for lower envelopes in higher dimensions , 1994, Discret. Comput. Geom..

[2]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 2015, J. Comb. Theory, Ser. A.

[3]  Micha Sharir,et al.  Arrangements and their applications in robotics: recent developments , 1995 .

[4]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[5]  Leonidas J. Guibas,et al.  A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications , 1989, ICALP.

[6]  Boaz Tagansky,et al.  A new technique for analyzing substructures in arrangements , 1995, SCG '95.

[7]  Mariette Yvinec,et al.  Applications of random sampling to on-line algorithms in computational geometry , 1992, Discret. Comput. Geom..

[8]  Micha Sharir,et al.  A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment , 1996, Discret. Comput. Geom..

[9]  Micha Sharir,et al.  On translational motion planning in three dimensions , 1994 .

[10]  Micha Sharir,et al.  Castles in the air revisited , 1994, Discret. Comput. Geom..

[11]  Micha Sharir,et al.  Arrangements in Higher Dimensions: Voronoi Diagrams, Motion Planning, and Other Applications , 1995, WADS.

[12]  Micha Sharir,et al.  Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes , 1986, FOCS.

[13]  Micha Sharir,et al.  The overlay of lower envelopes in three dimensions and its applications , 1995, SCG '95.

[14]  Mark de Berg,et al.  On lazy randomized incremental construction , 1995, Discret. Comput. Geom..

[15]  M. Sharir,et al.  New bounds for lower envelopes in three dimensions, with applications to visibility in terrains , 1993, SCG '93.

[16]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[17]  Leonidas J. Guibas,et al.  A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications , 1991, Theor. Comput. Sci..

[18]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[19]  Micha Sharir,et al.  Almost tight upper bounds for the single cell and zone problems in three dimensions , 1995, Discret. Comput. Geom..