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.

This thesis presents methods for the automatic creation of boundary-representation models of polyhedral objects from single line drawings depicting the objects. This topic is important in that automated interpretation of freehand sketches would remove a bottleneck in current engineering design methods. The thesis does not consider conversion of freehand sketches to line drawings or methods which require manual intervention or multiple drawings. Thge thesis contains a number of...

We describe a two-stage approach for interpreting line drawings of curved objects. In the first stage, the user enters a natu-ral line drawing of a polyhedral template; this is automatically interpreted as the corresponding polyhedral object. In the second stage, the user enters freehand curves; by relating these to the template, a curved object can be constructed.

This paper considers the task of asymmetry rectification. We start by giving various reasons why the possession of symmetry may be beneficial for designed shapes, and mention how various construction methods may produce shapes which are less symmetric than desired.

Computer vision is concerned with the development of machines that can process visual information. Computational geometry is concerned with the design of algorithms for solving geometric problems. Most problems in computer vision can be couched in geometric terms. In this paper we outline how computational geometry may significantly contribute to almost every aspect of computer vision and we provide pointers to a selection of the computational geometry literature where some of the most relevant results can be found. 1. Introduction Computer vision has flourished now for some forty years as a sub-discipline of artificial intelligence and hundreds of books are readily available on the subject and will not be mentioned here. The best early book on computer vision, and still up to date from the point of view of discriminant function analysis, is the text by Duda & Hart [DH73]. Popular more recent books include Ballard & Brown [BB82] and Horn [Ho86]. Finally we mention the first two books ...

Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundary-representation (B-rep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the model to improve its shape, to simplify its analysis, or to constrain it during editing. We give an algorithm to detect local approximate symmetries in a discrete point set derived from a B-rep model: the output comprises the model’s potential local symmetries at various automatically detected tolerance levels. Non-trivial symmetries of subsets of the point set are found as unambiguous permutation cycles, i.e. vertices of an approximately regular polygon or an anti-prism, which are sufficiently separate from other points in the point set. The symmetries are detected using a rigorous, tolerance-controlled, incremental approach, which expands symmetry seed sets by one point at a time. Our symmetry cycle detection approach only depends on inter-point distances. The algorithm takes time O(n 4) where n is the number of input points. Results produced by our algorithm are demonstrated using a variety of examples.

. This paper considers the following problem: given two point sets A and B (jAj = jBj = n) in d-dimensional Euclidean space, determine whether or not A is congruent to B. First, this paper presents a randomized algorithm which works in O(n (d01)=2 (log n) 2 ) time. This improves the previous result (an O(n d02 log n) time deterministic algorithm) . The birthday paradox, which is a well-known property in combinatorics, is used effectively in our algorithm. Next, this paper shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too. 1 Introduction Recently, geometric pattern matching problems have been studied extensively in computational geometry [3, 4, 11]. Most of such studies have been done for approximate matchings in two or three dimensions. Few studies for exact matchings in higher dimensions have been done. This paper studies the problem of determining the exac...

This paper describes an O(log

Motivated by the need to detect design intent in approximate boundary representation models, we give an algorithm to detect incomplete symmetries of discrete points, giving the models ’ potential local symmetries at various automatically detected tolerances. Here, incomplete symmetry is defined as a set of incomplete cycles which are constructed by, e.g., a set of consecutive vertices of an approximately regular polygon, induced by a single isometry. All seven 3D elementary isometries are considered for symmetry detection. Incomplete cycles are first found using a tolerance-controlled point expansion approach. Subsequently, these cycles are clustered for incomplete symmetry detection. The resulting clusters have welldefined, unambiguous approximate symmetries suitable for design intent detection, as demonstrated experimentally.

This paper presents an algorithm that tests the congruence of two sets of n points in d-dimensional space in O(n d 1 3 de log n) time. This improves the previous best algorithm for dimensions d 6.