Dianne Hansford

Discrete Coons patches

We investigate surfaces which interpolate given boundary curves. We show that the discrete bilinearly blended Coons patch can be defined as the solution of a linear system. With the goal of producing better shape than the Coons patch, this idea is generalized, resulting in a new method based on a blend of variational principles. We show that no single blend of variational principles can produce “good” shape for all boundary curve geometries. We also discuss triangular Coons patches and point out the connections to the rectangular case.

Dinus: Double insertion, nonuniform, stationary subdivision surfaces

The Double Insertion, Nonuniform, Stationary subdivision surface (DINUS) generalizes both the nonuniform, bicubic spline surface and the Catmull-Clark subdivision surface. DINUS allows arbitrary knot intervals on the edges, allows incorporation of special features, and provides limit point as well as limit normal rules. It is the first subdivision scheme that gives the user all this flexibility and at the same time all essential limit information, which is important for applications in modeling and adaptive rendering. DINUS is also amenable to analysis techniques for stationary schemes. We implemented DINUS as an Autodesk Maya plugin to show several modeling and rendering examples.

Natural neighbor extrapolation using ghost points

Among locally supported scattered data schemes, natural neighbor interpolation has some unique features that makes it interesting for a range of applications. However, its restriction to the convex hull of the data sites is a limitation that has not yet been satisfyingly overcome. We use this setting to discuss some aspects of scattered data extrapolation in general, compare existing methods, and propose a framework for the extrapolation of natural neighbor interpolants on the basis of dynamic ghost points.

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.

Agnostic G1 Gregory surfaces

Graphical abstractAgnostic G1 Gregory surfaces: input data, G0 intermediate surfaces, and final G1 surface.Display Omitted Highlights? G1 smoothness conditions for rectangular and triangular Gregory patches, which are rational surfaces, are presented. ? A G1 surface fitting algorithm for rectangular and triangular Gregory patches is outlined. ? The surface fitting method incorporates point-normal interpolation and the concept of tangent ribbons. We discuss G1 smoothness conditions for rectangular and triangular Gregory patches. We then incorporate these G1 conditions into a surface fitting algorithm. Knowledge of the patch type is inconsequential to the formulation of the G1 conditions, hence the term agnostic G1Gregory surfaces.

The neutral case for the min-max triangulation

Abstract Choosing the best triangulation of a point set is a question that has been debated for many years. Two of the most well known choices are the min-max criterion and the max-min criterion. The max-min triangulation criterion has received the most attention over the years because efficient algorithms have been developed for determining this triangulation. The ability to construct such efficient algorithms has been shown to be a result of the geometry of the neutral set for the max-min criterion. A point from the neutral set is formed from the special instance when the criterion is satisfied by more than one triangulation. For the max-min criterion, the neutral set is a circle. In this paper, we construct the neutral set for the min-max criterion. This construction is compared to that of the max-min triangulation and the results are analyzed in order to attain a better understanding of the nature of the min-max criterion.

On the approximation order of tangent estimators

A classic problem in geometric modelling is curve interpolation to data points. Some of the existing interpolation schemes only require point data, whereas others, require higher order information, such as tangents or curvature values, in the data points. Since measured data usually lack this information, estimation of these quantities becomes necessary. Several tangent estimation methods for planar data points exist, usually yielding different results for the same given point data. The present paper thoroughly analyses some of these methods with respect to their approximation order. Among the considered methods are the classical schemes FMILL, Bessel, and Akima as well as a recently presented conic precision tangent estimator. The approximation order for each of the methods is theoretically derived by distinguishing purely convex point configurations and configurations with inflections. The approximation orders vary between one and four for the different methods. Numerical examples illustrate the theoretical results.

Chapter 7 – Curve and Surface Constructions

This chapter introduces algorithms for the generation of Curves and Surfaces. It discusses some of the most fundamental interpolation and approximation methods in computer aided geometric design (CAGD). The developments emphasizes on Bezier and B-spline techniques because of their intuitive geometric definitions. The chapter focusses on polynomial curve methods, including Lagrange interpolation, point approximation, and Hermite interpolation. Next, a piecewise polynomial scheme, C 2 cubic spline interpolation is presented. For C 2 cubic spline interpolation, the choice of end conditions is important for the shape of the interpolant near the endpoints. The focus then moves to surface methods. The chapter describes interpolation to boundary curve data with Coons patches, interpolation to rectangular data with tensor product surfaces, approximation to large sets of data, and interpolation to point and derivative data. Mirroring the curve presentation, a piecewise polynomial surface scheme, C 2 bicubic spline interpolation, is discussed. The chapter concludes with a discussion on volume deformations.

Mathematical Principles for Scientific Computing and Visualization

This non-traditional introduction to the mathematics of scientific computation describes the principles behind the major methods, from statistics, applied mathematics, scientific visualization, and elsewhere, in a way that is accessible to a large part of the scientific community. Introductory material includes computational basics, a review of coordinate systems, an introduction to facets (planes and triangle meshes) and an introduction to computer graphics. The scientific computing part of the book covers topics in numerical linear algebra (basics, solving linear system, eigen-problems, SVD, and PCA) and numerical calculus (basics, data fitting, dynamic processes, root finding, and multivariate functions). The visualization component of the book is separated into three parts: empirical data, scalar values over 2D data, and volumes.

Anamorphic 3D geometry

An anamorphic image appears distorted from all but a few viewpoints. They have been studied by artists and architects since the early fifteenth century. Computer graphics opens the door to anamorphic 3D geometry. We are not bound by physical reality nor a static canvas. Here we describe a simple method for achieving anamorphoses of 3D objects by utilizing a variation of a simple projective map that is well-known in the computer graphics literature. The novelty of this work is the creation of anamorphic 3D digital models, resulting in a tool for artists and architects.