Robert E. Barnhill

Fat surfaces: a trivariate approach to triangle-based interpolation on surfaces

Abstract We present a new technique for interpolation of scattered data on arbitrary surfaces. The interpolant is obtained as the restriction of a trivariate function and is piecewise defined over triangular surface patches. The smoothness of the resulting function over the domain surface is visualized by application of interrogation tools for surfaces on surfaces.

Constant-radius blending of parametric surfaces

A method for blending two parametric surfaces is presented. It is based an an algorithm which calculates the intersection of two offset surfaces using only the first-order derivatives of the progenitors. The method converges quadratically in non-singular cases.

Methods for Constructing Surfaces on Surfaces

Given data defined on a (domain) surface, we construct an interpolant, which is a “surface defined on a surface.” we provide four different solutions to this multidimensional problem.

Curves with quadric boundary precision

Abstract We describe a method for constructing rational quadratic patch boundary curves for scattered data in R 3 . The method has quadric boundary precision; if the given point and normal data are extracted from a quadric, then the boundary curves will lie on this quadric. Each boundary curve is a conic section represented in the rational Bezier representation.

Geometry processing: curvature analysis and surface-surface intersection

Abstract Geometry Processing is the extraction of geometric features from an already constructed curve or surface. This paper concentrates on two aspects of geometry processing: curvature analysis and surface-surface intersections. Curvature analysis is a means of interrogating the higher order smoothness of curves and surfaces. A curve fairing method utilizing curvature plots is discussed. Curvature analysis for surfaces is used to determine the fairness of surfaces and to measure the effects of different choices of twists for bicubic patches. Surface-surface intersections of parametric patches is an important topic in geometric modelling. Our algorithm for surface-surface intersection is presented, including its application to offset surfaces.

Parametric Offset Surface Approximation

Offset surfaces are of interest in a variety of engineering applications. The formulation of a parametric offset surface involves division by the square root of a parametric equation, therefore, the offset surface is typically non-polynomial. Because of this complexity, offset surfaces cannot, in general, be written as members of the same class of functions or their generating or progenitor surface. Approximation of the offset surface is therefore desirable. Three contemporary methods of offset surface approximation are described which serve as models for the development of a new approximation algorithm. An adaptive offset surface approximation method based on a visually smooth triangular interpolant to position and tangent plane data defined in a triangular mesh is then developed. The criteria used to develop the approximation method are discussed, the components of the algorithm are described and the results of an implementation are illustrated. A conclusion about the success and possible refinement of the triangular offset surface approximation method is drawn and ideas for further research are outlined.

Interpolating scattered multivariate data as a function of time

Abstract Two classes of methods are presented for interpolating scattered data sampled in a spatial domain at different times. Instead of treating time as another Euclidean variable, time is treated as a special variable in our two approaches. These methods make use of scattered data interpolants over the spatial domain and univariate interpolants over the time domain. When compared with existing scattered data interpolation methods, the new methods are more effective.

Nurbs and grid generation

This paper provides a basic overview of NURBS and their application to numerical grid generation. Curve/surface smoothing, accelerated grid generation, and the use of NURBS in a practical grid generation system are discussed.