Thomas Schreiber

Surface interrogation algorithms

Various visualization techniques that identify unwanted curvature regions, such as inflection points and dents, are reviewed. Orthotomics for the convexity test, isophotes for the geometric continuity test for the boundaries of a patchwork, reflection lines for the aesthetic quality of a surface, and local surfaces for the detection of undesired curvature situations on a surface are discussed.<<ETX>>

The geometry of optimal degree reduction of Be´zier curves

Optimal degree reductions, i.e. best approximations of \(n\)-th degree Bezier curves by Bezier curves of degree \(n\) - 1, with respect to different norms are studied. It is shown that for any \(L_p\)-norm the euclidean degree reduction where the norm is applied to the euclidean distance function of two curves is identical to componentwise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree \(n\) - 1 of the original curve. This geometric situation is shown to hold also in the case of constrained degree reduction. The Bezier points of the degree reduction are explicitly given in the unconstrained case for \(p\) = 1 and \(p\) = 2 and in the constrained case for \(p\) = 2.

Methods for surface interrogation

The authors discuss various visualization techniques that have the goal of identifying unwanted curvature regions interactively on screen. The authors give a critical survey of surface interrogation methods. Several isoline and contouring techniques are presented, and the reflection line method, which simulates the so-called light cage by computer graphics, is presented. The isophote method analyzes surfaces by determining lines of equal light intensity. Silhouettes are special isophotes. A different approach to these problems is the mapping-technique. The mapping methods recognize unwanted curvature regions by detecting singularities of a special mapping of the curve or surface investigated. Curvature plots are a practical means of analyzing free-form surfaces. All these methods are effective, but generally need a lot of computational effort. The free-form surface visualization by ray tracing is discussed.<<ETX>>

Visualization and computation of curvature behaviour of freeform curves and surfaces

The paper consists of two parts. In the first, an interrogation method for a visualization of curvature behaviour is introduced. The method consists of a generalized focal analysis. The second part concerns the accurate computation of surface behaviour. Arithmetic operations on Bezier surfaces are used to achieve an exact representation of surface properties. With these property surfaces in Bezier form, qualitative and quantitative statements relating to the investigated surface are possible.

Visualisierung von Umweltdaten

Das Problem der Interpolation bzw. Approximation groser vollig unstrukturierter Datenmengen stellt sich in vielen technisch-wissenschaftlichen Bereichen. Ohne geeignete Aufarbeitung bleibt der Informationsgehalt der Daten weitgehend verborgen. Im Rahmen dieser Publikation wird ein Verfahren vorgestellt, das aus zwei Teilen besteht. Zunachst wird ein Algorithmus entwickelt, der durch iterative, verallgemeinerte Dirichlet Zerlegungen zu einer erheblichen Datenreduktion und einer guten Reprasentantenmenge fuhrt. Aufbauend auf dieser Reprasentantenmenge werden dann Scattered Data Algorithmen an realistischen Datensatzen aus dem Umweltbereich getestet.

Curve and Surface Interrogation

Free-form curves and surfaces are very important for sophisticated CAD/CAM Systems. Apart from the geometric modelling aspect of these curves and surfaces, the analysis of their quality is a necessary tool in the design and construction process. The purpose of this paper is to give a critical survey on curve and surface interrogation methods and to present generalized focal surfaces as a new surface interrogation tool.

SIMULATION BASED MODELLING

The generation of a 3D CAD model from a set of unstructured points of an unknown 3D object is an important problem of a reverse engineering process that can occur several times within a complete design process chain. Efficient 3D scanning technologies, such as laser range scanning systems become more and more sophisticated. Therefore the representation of 3D objects as a large set of digitized, unstructured point data became a common technique that builds the first step of a reverse engineering process. The main interest of this paper is to explain the further steps of the reverse engineering process, the filtering, clustering and segmentation of the point data and the final surface reconstruction out of point segmentations. This paper is concerned with structuring of point data as a pre-processing before the actual surface reconstruction and with the final surface reconstruction based on a variational design approach considering boundary conditions. First, the focus is set to an algorithm to reduce the data of large point sets. Subsequent a new method for curvature approximation, based on a Delaunay triangulation of the points, is described. By means of curvature discontinuities the points are grouped to represent separate surfaces of the model. After that variational design methods considering boundary conditions to generate the final surfaces from the group of points tangent to earlier created surfaces will be explained.

Scattered Data Algorithmen zur Umweltdatenvisualisierung

Interpolation and approximation of a large amount of unstructured data is a central issue in many applications. The purpose of this paper is to present an algorithm for this problem, consisting of two parts: in a first step we reduce data by creating a representation set based upon a generalized Voronoi diagram. Then we apply a scattered data algorithm on this reduced set. As a practical application of this method we visualize a SO2 distribution.

Polyhedral approximation and first order segmentation of unstructured point sets

The paper is concerned with the first two steps in a surface reconstruction process. Given a set of 3D points sampled from a physical model the first problem is that of creating a polyhedral approximation of the model. For that the authors introduce an algorithm which extends Boissonnats (1984) work. It allows the reconstruction of objects with arbitrary genus and proposes an automatic termination procedure. The next step in the process concerns the segmentation of the data points into regions for which each may be fitted by a single surface. They summarize some experiences with a region growing technique based on angle between normals criteria. Using just first order derivative estimations it is shown that the method is able to classify segments into predefined second order surface classes.

SEGMENTATION OF UNSTRUCTURED POINT SETS BASED ON TRIANGULAR SURFACE APPROXIMATIONS

This paper addresses the problem of segmenting an unstructured set of 3D points sampled from a physical model. Given such a set of points the first problem is that of creating a polyhedral approximation of the model. For that we introduce an algorithm which extends Boissonnats work and generates a triangular surface approximation. The proposed method allows the reconstruction of objects with arbitrary genus and proposes an automatic termination procedure. The next step in the process concerns the segmentation of the data points into regions for which each may be fitted by a single surface. Here we summarize our experiences with a region growing technique based on angle between normals criteria. Using just first order derivative estimations it is shown that the method is able to classify segments into predefined second order surface classes.