Triangulation of unordered data using trimmed NURBS computer aided design models M Ristic * , D Brujic and S Handayani Mechanical Engineering Department, Imperial College of Science, Technology and Medicine, London, UK Abstract: Computer aided design (CAD) models of many complex engineering parts, typically in the aerospace and the automotive industries, are de®ned using trimmed NURBS. The shape of the ®nished part can be captured by measuring a suciently large number of points. Triangulation of those points is a frequent requirement in the analysis of that shape as part of the quality assessment and of the product and manufacturing process development. The paper proposes a new method for ecient and robust triangulation of large sets of unordered three-dimensional points, based on the available CAD model. Unlike many other methods, it is not constrained by certain types of measurement distribution or object shapes. Examples involving real engineering parts are presented, with the conclusion that the method was well suited for the perceived applications in engineering manufacture. Keywords: triangulation, trimmed NURBS, CAD, reverse engineering, dimensional inspection, random data 1 INTRODUCTION Given a set of spatial points that lie on an object surface, triangulation is the process of establishing a correct connectivity structure between the points, such that the resulting triangular mesh is a correct representation of the object shape. The need to perform triangulation arises in many ®elds and it is generally performed for the purposes of: (a) shape modelling, (b) visualization of measured data, especially rendering (shading), (c) interpolation of the measured data involving re- sampling, usually to produce a regular grid from unordered data, or surface ®tting, (d) simulation, such as that using ®nite element methods. These needs also arise in applications involving engineer- ing manufacture, especially when dealing with free-form parts, such as aerofoils, turbomachinery components, car body panels and others. The nominal computer aided design (CAD) model of such parts is usually de®ned using NURBS (non-uniform rational B-spline), especially trimmed NURBS. The shape of the ®nished part may be captured using the conventional coordinate measuring machine (CMM), hand-held digitizers, laser scanners or other available metrology system. The measurement sets in such cases can be very large, sometimes containing more than a million points. Trian- gulation is often made dicult by the fact that the order- ing of the points is not necessarily known, because laser scanners and structured light systems tend to suer from the occlusion problem, while contact systems (such as the CMM) tend to produce unpredictable results in the vicinity of ®ne features [1]. These problems are further complicated by the presence of measurement noise. Under these circumstances, industrial applications require triangulation methods that are robust and computationally ecient. This paper proposes a new method for triangulation of unordered three-dimen- sional data aimed at applications in engineering manufacture, such as inspection and reverse engineering. It achieves its performance by utilizing the CAD model of the part as an approximation of the actual shape. 2 TRIANGULATION OF UNORDERED THREE- DIMENSIONAL DATA Among the many possible triangulations, Delaunay triangulation [2] is widely considered to provide the `best' triangles as it maximizes the smallest angle and minimizes the largest circumcircle of each triangle. This de®nition strictly applies only in two dimensions, leading 509 SC00200 # IMechE 2000 Proc Instn Mech Engrs Vol 214 Part B The MS was received on 1 December 1999 and was accepted after revi- sion for publication on 17 January 2000. * Corresponding author: Mechanical Engineering Department, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BX, UK.