Automatic parameterization of rational curves and surfaces IV: Algebraic space curves

Abhyankar, S. S. ; Bajaj, C. J. (1989) Automatic parameterization of rational curves and surfaces IV: Algebraic space curves ACM Transactions on Graphics, 8 (4). No pp. given. ISSN 0730-0301

[img]
Preview
PDF - Author Version
290kB

Official URL: http://portal.acm.org/citation.cfm?id=77269.77273

Related URL: http://dx.doi.org/10.1145/77269.77273

Abstract

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f (x, y, z) = 0 and g (x, y, z) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P, whose genus is the same as that of C. Thus C is rational if the genus of P is zero. Given an irreducible space curve C = (f ∩ g), with f and g not tangent along C, we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C. Together with [4], this method yields an algorithm to compute the genus of C, and if the genus is zero, the rational parametric equations for C. As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C. The birational mappings of implicitly defined space curves find numerous applications in geometric modeling and computer graphics since they provide an efficient way of manipulating curves in space by processing curves in the plane. Additionally, having rational surfaces containing C yields a simple way of generating related families of rational space curves.

Item Type:Article
Source:Copyright of this article belongs to Association for Computing Machinery.
ID Code:70874
Deposited On:22 Nov 2011 12:56
Last Modified:18 May 2016 16:48

Repository Staff Only: item control page