### Seminar

**Lecture**: Wednesday, 17:00–19:00, Schreiber 7 note change of room

**Instructor**: Dan Halperin

**Office hours**: Wednesday, 16:00–17:00, Schreiber 114

An * arrangement * of geometric objects is the decomposition of space induced by these objects. An arrangement of a finite collection \(L\) of lines in the plane, for example, is the subdivision of the plane into vertices, edges, and faces induced by the lines in \(L\). Arrangements of curves and surfaces are fundamental constructs in computational geometry, and they are useful for solving problems in a variety of domains such as computer vision, robot motion planning, molecular modeling, mechanical assembly planning, and more. For a survey of definitions, problems, results and applications of arrangements [psz]

In the seminar we will cover basic techniques for manipulating arrangements, with an emphasis on applications.

After two introductory meetings, we will follow the schedule below of students presentations (temporary list, stay tuned).

by now all the dates are of course outdated; they will be revised after the strike ends

### Aspect Graphs (Maintaining Invariants, Sparseness)

**11.11****Efficiently computing and representing aspect graphs of polyhedral objects**

Z. Gigus, J. Canny and R. Seidel,*IEEE Trans. Pattern Anal. Mach. Intell.*, Vol. 13. no. 6 (1991), 542–551

**18.11****Sparse arrangements and the number of views of polyhedral scenes**

M. de Berg, D. Halperin, M.H. Overmars and M. van Kreveld,*International Journal of Computational Geometry and Applications,*7 (1997), 175–195

### Geometric Modeling of Molecules (“Fatness”, Robustness I)

**25.11****Spheres, molecules, and hidden surface removal**

D. Halperin and M.H. Overmars, In*Proceedings 10*, Stony Brook, 1994, pp. 113–122^{th}ACM Symposium on Computational Geometry

**02.12****A perturbation scheme for spherical arrangements with application to molecular modeling**

D. Halperin and C.R. Shelton,*Proceedings of the 13*, Nice, 1997, pp. 183–192^{th}ACM Symposium on Computational Geometry

### Robot Motion Planning (Configuration Space)

**9.12**- Chapter 13,
**Robot motion planning in**M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Springer, 1997*Computational Geometry: Algorithms and Applications*

**On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles**

K. Kedem, R. Livne, J. Pach and M. Sharir,*Discrete Computational Geometry*1 (1986), pp. 59–71

- Chapter 13,
**16.12****Robot motion planning and the single cell problem in arrangements**

D. Halperin,*Journal of Intelligent and Robotic Systems*, 11 (1994) pp. 45-–65**A near-quadratic algorithm for planning the motion of a polygon in a polygonal environment**

D. Halperin and M. Sharir,*Discrete and Computational Geometry*, 16 (1996), 121–134

### Assembly Planning (Motion Space)

**23.12****A general framework for assembly planning: The motion space approach**

D. Halperin, J.-C. Latombe and R.H. Wilson,*Algorithmica*, to appear.

**30.12****Polyhedral assembly partitioning using maximally covered cells in arrangements of convex polytope**

L. Guibas, D. Halperin, H. Hirukawa, J.-C. Latombe and R.H. Wilson,*International Journal of Computational Geometry and Applications*, 8(2), 1998, 179–199

### Part Manipulation (Duality, Zone)

**6.1.99****Orienting polygonal parts without sensors**

K.Y. Goldberg,*Algorithmica*, 10 (1993), pp. 201–225**The complexity of oblivious plans for orienting and distinguishing polygonal parts**

Y.-B. Chen and D. Ierardi,*Algorithmica*, 14 (1995), pp. 367–397

**13.1****The area bisectors of a polygon and force equilibria in programmable vector fields**

Karl F Böhringer, B.R. Donald and D. Halperin,*Proceedings of the 13*, Nice, 1997, pp. 457–459^{th}ACM Symposium on Computational Geometry

The full version [psz]

### Miscellaneous (3D Vertical Decomposition, Robustness II)

**20.01****Vertical decompositions for triangles in 3-space**

M. de Berg, L. Giubas abd D. Halperin,*Discrete and Computational Geometry*, 15 (1996), 36–61

**20.01****Snap rounding line segments efficiently in two and three dimensions**

M. Goodrich, L. Guibas, J. Hershberger, and P. Tanenbaum, In*Proceedings of the 13*, Nice, 1997, pp. 284–293^{th}ACM Symposium on Computational Geometry

**27.01****Robust plane sweep for intersecting segments**

J.-D. Boissonnat and F.P. Preparata, Technical report No. 3270, INRIA, September 1997