Approximate implicitization of space curves and of surfaces of revolution. (English) Zbl 1147.53006

Shalaby, Mohamed; Jüttler, Bert

Jüttler, Bert (ed.) et al., Geometric modeling and algebraic geometry. Berlin: Springer (ISBN 978-3-540-72184-0/hbk). 215-227 (2008).

The present article discusses two algorithms to determine an exact or approximate implicit representation of a non-planar space curve, given in parametric form or by a set of data points. These methods are based on the approximate implicitization of plane curves. Additionally, approximate implicit representations of surfaces of revolution are determined.
The first algorithm represents a space curve \(C\) as intersection of two implicitly given cylinders \(f(x,y)=0\) and \(g(x,z)=0\). These cylinders serve as an input for the second algorithm determining two surfaces \(F=0\) and \(G=0\) intersecting approximately orthogonally along \(C\). This pair of surfaces also determines a good local approximation of the distance function of the space curve \(C\).
The implicit representations of surfaces of revolution \(S\) are based on an implicit description of a meridian curve, containing the radius function of \(S\). The equation of the surface is eventually found by elimination or substitution of the radius function from an implicit representation of an meridian curve. Some examples and problems are discussed.
For the entire collection see [Zbl 1123.53003].

Reviewer: Martin Peternell (Wien)

MSC:

53A05	Surfaces in Euclidean and related spaces
53A04	Curves in Euclidean and related spaces
65D17	Computer-aided design (modeling of curves and surfaces)

Keywords:

space curve; approximate implicitization; intersection of surfaces; surface of revolution