Geometric computation software tends to be fragile and fails occasionally. This robustness problem is rooted in the difficulty of making unambiguous decisions about incidence and nonincidence, fundamentally impairing layering the geometry software reliably. Additionally, geometric operations tend to have a large number of special and singular cases, further adding to the difficulty of creating dependable geometric software. We review the problem origins and ways to address it.

1.
Lakos, J., 1996, Large-Scale C++ Software Design, Addison-Wesley, Reading, MA.
2.
Hoffmann, C. M., Hopcroft, J., and Karasick, M., 1988, “Towards Implementing Robust Geometric Computations,” Proc. 4th ACM Symp. on Comp. Geometry, pp. 106–117.
3.
Hoffmann, C. M., 1989, Geometric and Solid Modeling, An Introduction, Morgan Kaufman, San Mateo, CA.
4.
Gavrilova, M., and Rokne, J. G., 2000, “Reliable Line Segment Intersection Testing,” CAD 32, pp. 737–746.
5.
Ratschek
,
H.
, and
Rokne
,
J.
,
1999
, “
Exact Computation of the Sign of a Finite Sum
,”
Appl. Math. Comp.
, pp.
99
127
.
6.
de Berg, M., van Kreveld, M., Overmars, M., and Schwarzkopf, O., 1997, Computational Geometry, Algorithms and Applications, Springer-Verlag, New York.
7.
Sugihara
,
K.
, and
Iri
,
M.
,
1989
, “
A Solid Modeling System Free From Topological Inconsistency
,”
J. of Inf. Proc.
,
12
, pp.
380
393
.
8.
Fortune, S., 1995, “Polyhedral Modeling with Exact Arithmetic,” Proc. 3rd Symp. Solid Modeling, ACM Press, NY, pp. 225–234.
9.
Fortune
,
S.
, and
Van Wyk
,
C.
,
1993
, “
Efficient Exact Arithmetic for Computational Geometry
,”
Proc. 9th Symp. Comp. Geometry
, ACM Press, NY, pp. 163–172.
10.
Yu, J., 1991, “Exact Arithmetic Solid Modeling,” Ph.D Thesis, CS, Purdue University.
11.
Sugihara
,
K.
,
1992
, “
A Simple Method for Avoiding Numerical Error and Degeneracy in Voronoi Diagram Construction
,”
IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
,
75-A
, pp.
468
477
.
12.
Hopcroft
,
J. E.
, and
Kahn
,
P. J.
,
1992
, “
A Paradigm for Robust Geometric Algorithms
,”
Algorithmica
,
7
, pp.
339
380
.
13.
Golub, G., and van Loan, C., 1983, Matrix Computations, Johns Hopkins University Press.
14.
Hammer, R., Hocks, M., Kulisch, U., and Ratz, D., 1995, C++ Toolbox for Verified Computing, Basic Numerical Problems, Springer Verlag, New York.
15.
Jenkins
,
M.
, and
Traub
,
J.
,
1970
, “
A Three Stage Variable-Shift Iteration for Polynomial Zeros and Its Relation to Generalized Rayleigh Iteration
,”
Numer. Math.
,
14
, pp.
252
263
.
16.
Press, W., Teukolsky, S., Wetterling, W., and Flannery, B., 1992, Numerical Recipes in C, 2nd edition, Cambridge University Press.
17.
Neumaier, A., 1990, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, England.
18.
Sederberg
,
T.
, and
Farouki
,
R.
,
1992
, “
Approximation by Interval Be´zier Curves
,”
IEEE Comput. Graphics Appl.
,
87
95
.
19.
Hu
,
C-Y.
,
Patrikalakis
,
N.
, and
Ye
,
X.
,
1996
, “
Robust Interval Solid Modeling
,”
CAD
, pp. 807–817 and 819–830.
20.
Wallner
,
J.
,
Krasauskas
,
R.
, and
Pottmann
,
H.
,
2000
, “
Error Propagation in Geometric Computations
,”
CAD 32
, pp. 631–641.
21.
Farin, G., 1992, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Boston; 3rd ed.
22.
Farouki
,
R.
, and
Rajan
,
V.
,
1987
, “
On the Numerical Condition of Polynomials in Bernstein Form
,”
Comp. Aided Geom., Design
,
4
, pp.
191
216
.
23.
Keyser
,
J.
,
Culver
,
T.
,
Manocha
,
D.
, and
Krishnan
,
S.
,
2000
, “
Efficient and Exact Manipulation of Algebraic Points and Curves
,”
CAD
, pp.
649
662
.
24.
Kortenkamp, U., 1999, “Foundations of Dynamic Geometry,” Ph.D Thesis, Informatik, Swiss Fed. Inst. of Technology.
25.
Agrawal, A., 1995, “A General Approach to the Design of Robust Algorithms for Geometric Modeling,” Ph.D Thesis, Comp. Science, University of Southern California.
You do not currently have access to this content.