Chanderjit L. Bajaj
Insung Ihm
Abstract
This paper presents an efficient algorithm called Hermite interpolation, for constructing low-degree algebraic surfaces, which contain, with C1 or tangent plane continuity, any given collection of points and algebraic space curves having derivative information. Positional as well as derivative constraints on an implicitly defined algebraic surface are translated into a homogeneous linear system, where the unknowns are the coefficients of the polynomial defining the algebraic surface. Computaional details of the Hermite interpolation algorithm are presented along with several illustrative applications of the interpolation technique to construction of joining or blending surfaces for solid models as well as fleshing surfaces for curved wire frame models. A heuristic approach to interactive shape control of implicit algebraic surfaces is also given, and open problems in algebraic surface design are discussed.
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Index Terms
- Algebraic surface design with Hermite interpolation
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Reviews
Nickolas S. Sapidis
The problem of constructing an algebraic surface of the lowest degree that interpolates points of curves with normals ( C 1 interpolation) is addressed. The authors present a complete discussion of the problem and establish that the interpolants correspond to nontrivial solutions of a homogeneous linear system of equations. This result is translated into an algorithm, which is applied in a large number of test cases (examples) that clearly show the advantages of the proposed technique. The authors discuss, and demonstrate with an example, one of the major open problems in algebraic-surface modeling: often, the constructed interpolating surface is unsuitable for design applications because the interpolated points or curves lie on separate components of the same algebraic surface. This problem is only one instance of the more general issue of topological unpredictability that one faces when dealing with algebraic surfaces. Other instances of this phenomenon are extra branches that an interpolating algebraic surface may have (see Warren [1] and the references therein) and self-intersections. Another issue in algebraic-surface interpolation is that the result of C 1 interpolation is generally a q -parameter family of surfaces that may have drastically varying topological characteristics, as some of the examples in this paper demonstrate. Thus, one faces the problem of selecting from the above q -parameter family of interpolants a bounded, connected piece of a surface [1] that is appropriate for geometric modeling applications. For this purpose, the authors convert the interpolating surfaces to the Bernstein basis and attempt to control their shapes with NURBS-like control points and weights. This conversion requires the specification of a tetrahedron T enclosing the portion of the surface that is of interest. (The fact that this surface does not yet exist makes this part of the methodology vague.) The vertices of T directly determine the surface's control points. Finally, the weights are determined by various heuristics, user interaction, or both. This paper is an important contribution in the area of algebraic-surface interpolation. It will be useful to researchers as well as software developers considering implicit surfaces for geometric modeling applications.
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