Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications

Let X be a collection of n algebraic surface patches of constant maximum degree in IR3. We show that the combinatorial complexity of the vertical decomposition of a single cell in the arrangement A(Z) is 0(n2+E ), for my E > CI, where the constant of proportionality de pends on E and on the maximum degree of the surfaces and of their boundaries. As an application, we obt~”n a near-quadratic motion planning algorithm for general systems with three degrees of freedom.

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

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

[3]  Mark de Berg,et al.  On lazy randomized incremental construction , 1994, STOC '94.

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

[5]  Micha Sharir,et al.  Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

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

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

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

[9]  Micha Sharir,et al.  Castles in the air revisited , 1992, SCG '92.

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

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

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

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

[14]  Micha Sharir,et al.  Almost tight upper bounds for the single cell and zone problems in three dimensions , 1994, SCG '94.

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

[16]  Micha Sharir,et al.  Triangles in space or building (and analyzing) castles in the air , 1990, Comb..

[17]  Micha Sharir,et al.  On translational motion planning in 3-space , 1994, SCG '94.

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

[19]  Vladlen Koltun.  Almost tight upper bounds for lower envelopes in higher dimensions , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

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