Degree, multiplicity, and inversion formulas for rational surfaces using \(u\)-resultants. (English) Zbl 0756.65011
Given an algebraic surface, the basic problems of identification of base points, minimal degree, generic representation, etc. are solved in algebraic geometry by intersecting the surface with a linear pencil. The authors note that if the surface is given parametrically in projective (3-)space by \(x_ i=f_ i(s_ k)\) then the intersection of the surface with a line given as intersection of two planes gives two equations \(\sum a^{(j)}_ if(s_ 1,s_ 2,s_ 3)=0\), (\(j=1,2\)).
The adjunction of a third \(\sum u_ ks_ k=0\) with indeterminates \(u_ k\) allows one to compute the resultant which can be used to solve problems one is interested in {unless one runs into a surface with infinitely close singularities, the problem that is unsolvable with methods of enumerative algebraic geometry}. However, in the untractable cases the resultant will vanish and therefore it is a very safe indicator of trouble and that is really the most important aspect for computational geometry.
The adjunction of a third \(\sum u_ ks_ k=0\) with indeterminates \(u_ k\) allows one to compute the resultant which can be used to solve problems one is interested in {unless one runs into a surface with infinitely close singularities, the problem that is unsolvable with methods of enumerative algebraic geometry}. However, in the untractable cases the resultant will vanish and therefore it is a very safe indicator of trouble and that is really the most important aspect for computational geometry.
Reviewer: H.Guggenheimer (West Hempstead)
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 14Q10 | Computational aspects of algebraic surfaces |
Keywords:
degree; multiplicity; inversion formulas; rational surfaces; algebraic surface; algebraic geometry; intersection; resultant; computational geometrySoftware:
MapleReferences:
| [1] | Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer Aided Geometric Design, 1, 1-60 (1984) · Zbl 0604.65005 |
| [2] | Canny, J., The complexity of Robot Motion Planning, PhD Thesis (1987), Department of Electrical Engineering and Computer Science, MIT |
| [3] | Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Monagan, M. B.; Watt, S. M., MAPLE Reference Manual (1988), WATCOM Publ. Ltd: WATCOM Publ. Ltd Waterloo, Ontario, Canada · Zbl 0758.68038 |
| [4] | Chionh, E. W., Base Points, Resultants, and the Implicit Representation of Rational Surfaces, PhD Thesis (1990), University of Waterloo |
| [5] | Chionh, E. W.; Goldman, R. N., A tutorial on resultants and elimination theory (1989), submitted for publication |
| [6] | Chionh, E. W.; Goldman, R. N., On the existence and the coefficient field of the implicit equation for a rational surface (1990), submitted for publication |
| [7] | Kaltofen, E., Computing the irreducible real factors and components of an algebraic curve, (Proc. 5th Annual Symp. on Computational Geometry (1989), ACM Press: ACM Press New York), 79-87 |
| [8] | Knuth, D. E., The Art of Computer Programming, Vol. 2: Seminumerical Algorithms (1969), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0191.17903 |
| [9] | Lefschetz, S., Algebraic Geometry (1953), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0052.37504 |
| [10] | Macaulay, F. S., On some formula in elimination, Proc. London Math. Soc., 3-27 (1902) · JFM 34.0195.01 |
| [11] | Macaulay, F. S., Note on the resultant of a number of polynomials of the same degree, Proc. London Math. Soc., 14-21 (1921) · JFM 48.0108.02 |
| [12] | Salmon, G., Lessons Introductory to the Modern Higher Algebra (1924), G.E. Stechert & Co: G.E. Stechert & Co New York |
| [13] | Shafarevich, I. R., Basic Algebraic Geometry (1974), Springer: Springer Berlin · Zbl 0284.14001 |
| [14] | Sharpe, F. R., The mapping of a rational surface on a plane, Selected Topics in Algebraic Geometry, 275-290 (1970), Ch. XII |
| [15] | van der Waerden, B. L., Modern Algebra (1950), Frederick Ungar: Frederick Ungar New York · Zbl 0037.01903 |
| [16] | Walker, R. J., Algebraic Curves (1950), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0039.37701 |
| [17] | Warren, J., The effect of base points on rational Bézier surfaces, Rice COMP TR90-12 (1990) |
| [18] | Zariski, O., Algebraic Surfaces (1971), Springer · Zbl 0219.14020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
