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# Hagen H. (ed.) Curve and Surface Design

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Society for Industrial and Applied Mathematics, 1992, -216 pp.
Curve and surface design in industrial applications still has many unresolved issues, even though research in this field has been ongoing for more than 30 years. This book is a collection of new ideas and results on topics of curve and surface design. It is intended not only for research in the academic environment, but for practical use in industrial applications.
The starting point for this book was the SIAM Conference on Geometric Design held in Tempe, Arizona, from November 6-10, 1989. Many of the ideas contained in this volume were presented there. Additionally, some of the leading experts in the field were invited to contribute. About60% of the submissions were finally accepted for publication.
The book is organized in two sections: curve design and nontensor product surfaces.
Part 1 . Curve Design. This section contains the newest developments in curve design. The first two contributions are concerned with minimal-energy splines. Smoothing algorithms are frequently based on the idea of simulating the behavior of elastic materials that tend to minimize their elastic energy. The energy stored in a thin elastic beam is proportional to the integral of the squared curvature of the beam with the length of the beam as the variable of integration. Curves minimizing this energy functional are therefore used to define a model of a smooth curve shape.
The article by Brunnett presents an analysis of the criterion of minimal energy, giving new insight into the properties of minimal-energy splines and their behavior. The second paper on this topic, by Jou and Han, shows how to introduce various end constraints in the concept of minimal-energy splines.
A milestone in free-form curve design was Nielsen's development of a piecewise polynomial alternative to splines under tension, the so-called v-spline, a piecewise cubic, curvature continuous spline. Hagen generalized Nielsen's approach resulting in polynomials of degree n=2L-1, L>3 . This concept of geometric spline curves includes Nielsen's v-splines for L=2 and curvature and torsion continuous quintics for L=3, the so-called T-splines. Geometric spline curves provide shape parameters having the characteristics of point weights.
The idea of interval weights is attributed to Salkauskas. In many applications there is a need to combine both concepts. Just recently, Foley generalized Nielsen's v-spline to an interval-weighted v-spline. The gap to T-splines is bridged in this volume. Lasser and Hagen present interval-weighted T-splines, and Neuser presents curve and surface interpolating techniques using quintic weighted T-splines.
The latest progress in weighted spline methods is presented by Bos and Salkauskas. The topics are completed by Eck's algorithms for geometric spline curves. The contribution of Fritsch and Nielson is concerned with the problem of determining the distance between parametric curves. They present an elegant, application-oriented solution.
Part 2 . Nontensor Product Surfaces. The problem of fitting a surface through a set of three-dimensional data points is a key problem in the field of computer aided geometric design. Tensor product schemes work quite well for modeling surfaces based on strict rectangular networks, but very often applications need more general topologies.
In this section, we present three different approaches dealing with this problem. DeRose et al. give a critical survey on triangular surface interpolating methods. The paper by Bloor and Wilson describes a method for surface generation, whereby a smooth surface is generated as a solution to a boundary value problem. In his contribution, Daehlen considers bivariate box splines with special emphasis on three-, five-, and six-sided surface design.
Part 1 Curve Design
Properties of Minimal-Energy Splines
Minimal-Energy Splines with Various End Constraints
Interval Weighted Tau-Splines
Curve and Surface Interpolation Using Quintic Weighted Tau-Splines
Weighted Splines Based on Piecewise Polynomial Weight Functions
Algorithms for Geometric Spline Curves
On the Problem of Determining the Distance Between Parametric Curves
Part 2 Nontensor Product Surfaces
A Survey of Parametric Scattered Data Fitting Using Triangular Interpolants
Free-Form Surfaces from Partial Differential Equations
Modeling with Box Spline Surfaces
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