: This paper proposes a method called meshless parameterization, for parameterizing and triangulating "single patch" unorganized point sets. The points are mapped into a planar parameter domain by solving a sparse linear system. By making a standard triangulation of the parameter points, we obtain a corresponding triangulation of the original data set.

By its dual representation, a developable surface can be viewed as a curve of dual projective 3-space. After introducing an appropriate metric in the dual space and restricting ourselves to special surface classes, we derive linear approximation algorithms for developable NURBS surfaces, including multiscale approximations. Special attention is paid to controlling the curve of regression. Keywords: computer aided geometric design, surface approximation, developable surface, dual representation, NURBS 1 Introduction A developable surface is a surface which can be unfolded (developed) into a plane without stretching or tearing. Mathematically speaking, there is a mapping of the surface into the Euclidean plane which is isometric, at least locally. Because of this property, developable surfaces possess a variety of applications in manufacturing with materials that are not amenable to stretching. These include the formation of aircraft skins, ship hulls, ducts and automobile parts such as...

Mesh segmentation has become an important component in many applications in computer graphics. In the last several years, many algorithms have been proposed in this growing area, offering a diversity of methods and various evaluation criteria. This paper provides a comparative study of some of the latest algorithms and results, along several axes. We evaluate only algorithms whose code is available to us, and thus it is not a comprehensive study. Yet, it sheds some light on the vital properties of the methods and on the challenges that future algorithms should face.

We propose a method for segmentation of 3D scanned shapes into simple geometric parts. Given an input point cloud, our method computes a set of components which possess one or more slippable motions: rigid motions which, when applied to a shape, slide the transformed version against the stationary version without forming any gaps. Slippable shapes include rotationally and translationally symmetrical shapes such as planes, spheres, and cylinders, which are often found as components of scanned mechanical parts. We show how to determine the slippable motions of a given shape by computing eigenvalues of a certain symmetric matrix derived from the points and normals of the shape. Our algorithm then discovers slippable components in the input data by computing local slippage signatures at a set of points of the input and iteratively aggregating regions with matching slippable motions. We demonstrate the performance of our algorithm for reverse engineering surfaces of mechanical parts.

We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a synthetic computer environment, including computer aided design and manufacturing. Our reconstruction approach is flexible enough to permit interpolation of both smooth surfaces and sharp features, while placing few restrictions on the geometry or topology of the object. Our algorithm is based on alpha-shapes to compute an initial triangle mesh approximating the object's surface. A mesh reduction technique is applied to the dense triangle mesh to build a simplified approximation, while retaining important topological and geometric characteristics of the model. The reduced mesh is interpolated with piecewise algebraic surface patches which approximate the original points. The process is fully automatic, and the reconstruction is guaranteed to be homeomorphic and error bounded with respec...

We introduce a method for approximating a given surface by a developable surface. It will be either a G 1 surface consisting of pieces of cones or cylinders of revolution or a G r NURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produce robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding. Keywords: computer aided design, computer aided manufacturing, surface approximation, reverse engineering, surface of revolution, developable surface, shipbuilding. 1 Introduction A developable surface is a surface which can be unfolded (developed) into a plane without stretching or tearing. Therefore, developable surfaces possess a wide range of applications, for example in sheet-metal and plate-metal based industries (see e.g. [1, 2]). It is well known in elementary differential geometry [3] that under the assumption of sufficient differentiabilit...

Aiming at robust surface structure recovery, we extend the powerful optimization technique of variational shape approximation by allowing for several different primitives to represent the geometric proxy of a surface region. While the original paper only considered planes, we also include spheres, cylinders, and more complex rollingball blend patches. The motivation for this choice is the fact that most technical CAD objects consist of patches from these four categories. The robust segmentation and global optimization properties which have been observed for the variational shape approximation carry over to our hybrid extension. Hence, we can use our algorithm to segment a given mesh model into characteristic patches and provide a corresponding geometric proxy for each patch. The expected result that we recover surface structures more robustly and thus obtain better approximations with a smaller number of primitives, is validated and demonstrated on a number of examples. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Curve, surface, solid and object representations

The authors are investigating reverse engineering for reconstructing the shape of simple mechanical parts. Currently, preliminary B-rep models can be created by fitting surfaces to point clouds obtained by scanning an actual part using a 3D laser scanner. The resulting model, although valid, is often not suitable for purposes such as redesign. This is because expected regularities and constraints are not present in the model. This report describes a number of aspects of the geometry of mechanical parts which should be exploited to adjust a B-rep model to improve its usefulness. Aspects considered are geometric constraints between surface parameters, regularly repeated substructures, symmetry, and the presence of features such as slots and holes. The results of a survey of a range of mechanical parts are presented and discussed, showing which of these aspects occur with a frequency that justifies their use in beautification algorithms intended to turn preliminary reverse engineered B-rep models into models engineers expect.

This paper deals with the constrained reconstruction of 3D geometric models of objects from range data. It describes a new technique of global shape improvement based upon feature positions and geometric constraints. It suggests a general incremental framework whereby constraints can be added and integrated in the model reconstruction process, resulting in an optimal trade-off between minimization of the shape fitting error and the constraint tolerances. After defining sets of constraints for planar and special case quadric surface classes based on feature coincidence, position and shape, the paper shows through application on synthetic model that our scheme is well behaved. The approach is then validated through experiments on different real parts. This work is the first to give such a large framework for the integration of geometric relationships in object modelling. The technique is expected to have a great impact in reverse engineering applications and manufactured object modelling where the majority of parts are designed with intended feature relationships. Keywords Reverse engineering, Geometric constraints, constrained shape reconstruction, shape optimization. 2

The authors are developing an automated reverse engineering system for reconstructing the shape of simple mechanical parts. B-rep models are created by fitting surfaces to point clouds obtained by scanning an object using a 3D laser scanner. The resulting models, although valid, are often not suitable for purposes such as redesign because expected regularities and constraints are not present. This information is lost because each face of the model is determined independently. A global approach is required, in particular one that is capable of finding symmetries originally present. This paper describes a practical algorithm for finding global symmetries in suitable B-rep models built from planes, spheres, cylinders, cones and tori. It has been implemented and used to determine approximate symmetries of models with up to about 200 vertices in reasonable time. The time performance of the algorithm in the worst case is bounded by O(n^3.5 log^4 n), and a justification is given that on common engineering objects it takes about O(n^2 log^4 n), making it a practical tool for use in a reverse engineering package. Details of the algorithm are given, along with some results from a number of illustrative test runs.