Finding Specified Sections of Arrangements: 2D Results

Given a configuration C of geometric objects in R2 (called the input configuration), a target configurationT of geometric objects in R1, and a class S of allowable sectioning lines we consider in this paper many variations on the following problem: ‘Is there a line S∈S such that the section S∩C is equivalent by rigid motion to the target T?’

[1]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .

[2]  P. McMullen,et al.  On Hammer's X-Ray Problem , 1980 .

[3]  Gabor T. Herman,et al.  Mathematical Aspects of Computerized Tomography , 1981, Lecture Notes in Medical Informatics.

[4]  S. Deans.  The Radon Transform and Some of Its Applications , 1983 .

[5]  Raimund Seidel,et al.  Finding the optimal shadows of a convex polytope , 1985, SCG '85.

[6]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[7]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[8]  Gabor T. Herman,et al.  Basic methods of tomography and inverse problems , 1987 .

[9]  Steven Skiena,et al.  Probing Convex Polygons with X-Rays , 1988, SIAM J. Comput..

[10]  Jean-Daniel Boissonnat,et al.  Shape reconstruction from planar cross sections , 1988, Comput. Vis. Graph. Image Process..

[11]  R. L. Shuo-Yen.  Reconstruction of polygons from projections , 1988 .

[12]  Shuo-Yen Robert Li,et al.  Reconstruction of Polygons from Projections , 1988, Information Processing Letters.

[13]  Leonidas J. Guibas,et al.  The complexity of cutting complexes , 1989, Discret. Comput. Geom..

[14]  David P. Dobkin,et al.  Probing Convex Polytopes , 1990, Autonomous Robot Vehicles.

[15]  B. Faverjon,et al.  On computing three-finger force-closure grasps of polygonal objects , 1991 .

[16]  Peter Gritzmann,et al.  Successive Determination and Verification of Polytopes by their X-Rays , 1992, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[17]  S. Skiena.  Interactive reconstruction via geometric probing , 1992, Proc. IEEE.

[18]  Steven Skiena,et al.  Reconstructing polygons from x-rays , 1993, CCCG.

[19]  Peter Gritzmann,et al.  Polytope Projection and Projection Polytopes , 1996, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[20]  R. Gardner.  Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[21]  Peter Gritzmann,et al.  Discrete Tomography: Determination of Finite Sets by X-Rays , 1995, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[22]  N. Amenta,et al.  Deformed products and maximal shadows of polytopes , 1996 .

[23]  Joseph O'Rourke,et al.  On reconstructing polyhedra from parallel slices , 1996, Int. J. Comput. Geom. Appl..

[24]  Micha Sharir,et al.  Piecewise-Linear Interpolation between Polygonal Slices , 1996, Comput. Vis. Image Underst..

[25]  Prosenjit Bose,et al.  On the sectional area of convex polytopes , 1996, SCG '96.

[26]  Nina Amenta,et al.  Shadows and slices of polytopes , 1996, SCG '96.

[27]  Jean-Daniel Boissonnat,et al.  On Computing Four-Finger Equilibrium and Force-Closure Grasps of Polyhedral Objects , 1997, Int. J. Robotics Res..

[28]  P. McMullen.  GEOMETRIC TOMOGRAPHY (Encyclopedia of Mathematics and its Applications 58) , 1997 .

[29]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[30]  Prosenjit Bose,et al.  Drawing Nice Projections of Objects in Space , 1999, J. Vis. Commun. Image Represent..