Page 177 - Geometric Modeling and Algebraic Geometry

9 Intersecting Biquadratic Patches 179
hand, the use of an approximate technique introduces inaccuracies, which may cause
problems with singular and almost singular situations. We feel that this price for
using a lower degree implicit representation is too high.
The parameter–line based approach adds some geometric interpretations to the
process of eliminating variables from the problem. As an advantage, it is possible
to correctly establish the region(s) of interest. This avoids problems with unwanted
branches of the (self–) intersection curves. In the case of two touching surfaces,
using this approach becomes more expensive, since standard techniques for tracing
the intersection cannot be applied.

Acknowledgment
This research was supported by the European Union through project IST 2001–
35512 ‘Intersection algorithms for geometry based IT applications using approxi-
mate algebraic methods’ (GAIA II), by the Austrian Science Fund through the Joint
Research Programme FSP S92 ‘Industrial Geometry’, and by Aim@Shape (IST NoE
506766).

References

1. L. Andersson, J. Peters, and N. Stewart, Self-intersection of composite curves and sur-
faces, Computer Aided Geometric Design, 15 (1998), pp. 507–527.
2. L. Bus´ e, Etude du r´ esultant sur une vari´ et´ e alg´ ebrique, PhD thesis, University of Nice,
December 2001.
3. L. Bus´ e and C. D’Andrea, Inversion of parameterized hypersurfaces by means of subre-
sultants, Proceedings ACM of the ISSAC 2004, pp. 65–71.
4. L. Bus´ e, M. Elkadi, and B. Mourrain, Using projection operators in Computer Aided
Geometric Design, In Topics in Algebraic Geometry and Geometric Modeling, pp. 321–
342, Contemporary Mathematics, AMS, 2003.
5. L. Bus´ e, I.Z. Emiris and B. Mourrain, MULTIRES, http://www-sop.inria.
fr/galaad/logiciels/multires.
6. L. Bus´ e and A. Galligo, Using semi-implicit representation of algebraic surfaces, Pro-
ceedings of the SMI 2004 conference, IEEE Computer Society, pp. 342–345.
7. E.W. Chionh and R.N. Goldman, Using multivariate resultants to find the implicit equa-
tion of a rational surface, The Visual Computer 8 (1992), pp. 171–180.
8. D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms, Springer-Verlag,New
York, 1992 and 1997.
9. D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Springer-Verlag, New York,
1998.
10. C. D’Andrea, Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc.,
354(7) (2002), pp. 2595–2629.
11. T. Dokken, Aspects of Intersection Algorithms and Approximation, Thesis for the doctor
philosophias degree, University of Oslo, Norway 1997.
12. T. Dokken, Approximate implicitization, Mathematical Methods for Curves and Surfaces,
T. Lyche and L.L. Schumaker (eds.), Vanderbilt University Press, 2001, pp. 81–102.