Ursula H. Augsdörfer

NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes

We present a subdivision framework that adds extraordinary vertices to NURBS of arbitrarily high degree. The surfaces can represent any odd degree NURBS patch exactly. Our rules handle non-uniform knot vectors, and are not restricted to midpoint knot insertion. In the absence of multiple knots at extraordinary points, the limit surfaces have bounded curvature.

Tuning Subdivision by Minimising Gaussian Curvature Variation Near Extraordinary Vertices

We present a method for tuning primal stationary subdivision schemes to give the best possible behaviour near extraordinary vertices with respect to curvature variation.

Isogeometric shell analysis with NURBS compatible subdivision surfaces

We present a discretisation of Kirchhoff-Love thin shells based on a subdivision algorithm that generalises NURBS to arbitrary topology. The isogeometric framework combines the advantages of both subdivision and NURBS, enabling higher degree analysis on watertight meshes of arbitrary geometry, including conic sections. Because multiple knots are supported, it is possible to benefit from symmetries in the geometry for a more efficient subdivision based analysis. The use of the new subdivision algorithm is an improvement to the flexibility of current isogeometric analysis approaches and allows new use cases.

Variations on the four-point subdivision scheme

A step of subdivision can be considered to be a sequence of simple, highly local stages. By manipulating the stages of a subdivision step we can create families of schemes, each designed to meet different requirements. We postulate that such modification can lead to improved behaviour. We demonstrate this using the four-point scheme as an example. We explain how it can be broken into stages and how these stages can be manipulated in various ways. Six variants that all improve on the quality of the limit curve are presented and analysed. We present schemes which perfectly preserve circles, schemes which improve the Holder continuity, and schemes which relax the interpolating property to achieve higher smoothness.

Deriving Box-Spline Subdivision Schemes

We describe and demonstrate an arrow notation for deriving box-spline subdivision schemes. We compare it with the z -transform, matrix, and mask convolution methods of deriving the same. We show how the arrow method provides a useful graphical alternative to the three numerical methods. We demonstrate the properties that can be derived easily using the arrow method: mask, stencils, continuity in regular regions, safe extrusion directions. We derive all of the symmetric quadrilateral binary box-spline subdivision schemes with up to eight arrows and all of the symmetric triangular binary box-spline subdivision schemes with up to six arrows. We explain how the arrow notation can be extended to handle ternary schemes. We introduce two new binary dual quadrilateral box-spline schemes and one new

Artifact analysis on B-splines, box-splines and other surfaces defined by quadrilateral polyhedra

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Artifacts in box-spline surfaces

box-spline scheme. With appropriate extensions to handle extraordinary cases, these could each form the basis for a new subdivision scheme.

Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra

When using NURBS or subdivision surfaces as a design tool in engineering applications, designers face certain challenges. One of these is the presence of artifacts. An artifact is a feature of the surface that cannot be avoided by movement of control points by the designer. This implies that the surface contains spatial frequencies greater than one cycle per two control points. These are seen as ripples in the surface and are found in NURBS and subdivision surfaces and potentially in all surfaces specified in terms of polyhedrons of control points. Ideally, this difference between designer intent and what emerges as a surface should be eliminated. The first step to achieving this is by understanding and quantifying the artifact observed in the surface. We present methods for analysing the magnitude of artifacts in a surface defined by a quadrilateral control mesh. We use the subdivision process as a tool for analysis. Our results provide a measure of surface artifacts with respect to initial control point sampling for all B-Splines, quadrilateral box-spline surfaces and regular regions of subdivision surfaces. We use four subdivision schemes as working examples: the three box-spline subdivision schemes, Catmull-Clark (cubic B-spline), 4-3, 4-8; and Kobbelt@?s interpolating scheme.

Isogeometric Analysis for Modelling and Design

Certain problems in subdivision surfaces have provided the incentive to look at artifacts. Some of these effects are common to all box-spline surfaces, including the tensor product B-splines widely used in the form of NURBS, and these are worthy of study. Although we use the subdivision form of box- and B-splines as the mechanism for this study, and also apply the same mechanism to the subdivision schemes which are not box-splines, we are looking at problems which are not specific to subdivision surfaces, but which afflict all Box- and B-splines.

Bounded curvature subdivision without eigenanalysis

Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra (Sabin et al., 2005; Augsdorfer et al., 2011). In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.