Numerical Geometry Ltd.

Ely, United Kingdom

Numerical Geometry Ltd.

Ely, United Kingdom
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Augsdorfer U.H.,Graz University of Technology | Dodgson N.A.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd.
Computer Aided Geometric Design | Year: 2011

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. © 2010 Elsevier B.V.


Augsdorfer U.H.,Graz University of Technology | Dodgson N.A.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd.
Computer Aided Geometric Design | Year: 2011

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; Augsdörfer 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. © 2011 Elsevier B.V.


Niu Z.,University of Cardiff | Martin R.R.,University of Cardiff | Langbein F.C.,University of Cardiff | Sabin M.A.,Numerical Geometry Ltd.
CAD Computer Aided Design | Year: 2015

Automatic feature recognition aids downstream processes such as engineering analysis and manufacturing planning. Not all features can be defined in advance; a declarative approach allows engineers to specify new features without having to design algorithms to find them. Naive translation of declarations leads to executable algorithms with high time complexity. Database queries are also expressed declaratively; there is a large literature on optimizing query plans for efficient execution of database queries. Our earlier work investigated applying such technology to feature recognition, using a testbed interfacing a database system (SQLite) to a CAD modeler (CADfix). Feature declarations were translated into SQL queries which are then executed. The current paper extends this approach, using the PostgreSQL database, and provides several new insights: (i) query optimization works quite differently in these two databases, (ii) with care, an approach to query translation can be devised that works well for both databases, and (iii) when finding various simple common features, linear time performance can be achieved with respect to model size, with acceptable times for real industrial models. Further results also show how (i) lazy evaluation can be used to reduce the work performed by the CAD modeler, and (ii) estimating the time taken to compute various geometric operations can further improve the query plan. Experimental results are presented to validate our main conclusions. © 2015 The Authors. Published by Elsevier Ltd.


Kosinka J.,University of Cambridge | Sabin M.,Numerical Geometry Ltd. | Dodgson N.,University of Cambridge
Computer Aided Geometric Design | Year: 2013

We investigate univariate and bivariate binary subdivision schemes based on cubic B-splines with double knots. It turns out that double knots change the behaviour of a uniform cubic scheme from primal to dual. We focus on the analysis of new bivariate cubic schemes with double knots at extraordinary points. These cubic schemes produce C1 surfaces with the original Doo-Sabin weights. © 2012 Elsevier B.V.


Shen J.,University of Cambridge | Kosinka J.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd. | Dodgson N.A.,University of Cambridge
Computer Aided Geometric Design | Year: 2014

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 Elsevier Ltd. All rights reserved.


Kosinka J.,University of Cambridge | Sabin M.,Numerical Geometry Ltd. | Dodgson N.,University of Cambridge
Graphical Models | Year: 2014

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points. © 2014 The Authors. Published by Elsevier Inc.


Kosinka J.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd | Dodgson N.A.,University of Cambridge
Computer Graphics Forum | Year: 2014

We deal with subdivision schemes based on arbitrary degree B-splines. We focus on extraordinary knots which exhibit various levels of complexity in terms of both valency and multiplicity of knot lines emanating from such knots. The purpose of truncated multiple knot lines is to model creases which fair out. Our construction supports any degree and any knot line multiplicity and provides a modelling framework familiar to users used to B-splines and NURBS systems. © 2013 The Authors Computer Graphics Forum published by John Wiley & Sons Ltd.


Kosinka J.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd. | Dodgson N.A.,University of Cambridge
Computer Graphics Forum | Year: 2014

We explore a method for generalising Pixar semi-sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high-degree univariate examples and explore analogous methods for subdivision surfaces. © 2014 The Eurographics Association and John Wiley & Sons Ltd.


Augsdorfer U.H.,University of Cambridge | Dodgson N.A.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd.
Computer Aided Geometric Design | Year: 2010

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 Hölder continuity, and schemes which relax the interpolating property to achieve higher smoothness. © 2009 Elsevier B.V. All rights reserved.


Kosinka J.,University of Cambridge | Sabin M.A.,Numerical Geometry Ltd. | Dodgson N.A.,University of Cambridge
CAD Computer Aided Design | Year: 2015

Traditionally, modelling using spline curves and surfaces is facilitated by control points. We propose to enhance the modelling process by the use of control vectors. This improves upon existing spline representations by providing such facilities as modelling with local (semi-sharp) creases, vanishing and diagonal features, and hierarchical editing. While our prime interest is in surfaces, most of the ideas are more simply described in the curve context. We demonstrate the advantages provided by control vectors on several curve and surface examples and explore avenues for future research on control vectors in the contexts of geometric modelling and finite element analysis based on splines, and B-splines and subdivision in particular. © 2014 The Authors. Published by Elsevier Ltd.

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