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June 1998
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36 Reads
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24 Citations
Reverse engineering is the reconstruction process of surfaces of physical objects into geometric surface models. The objects have to be digitized and these discrete data have to be converted into smooth surface models. We improve overall approximations of digitized set of points with help of feature lines. These feature lines are determined by the angular variation of estimated normal vectors.
January 1997
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20 Reads
In CAD-systems technical objects are given by set of points or analytic surface representations. Often the faces of these objects are unfair. In this paper energy oriented algorithms are developed to approximate a set of points by fair surfaces or to smooth automatically given unfair surfaces. To get linear (and fast) algorithms in both cases quadratic models of the bending energy of surfaces are used. Minimal values of these energy models lead to smooth and fairshaped surfaces.
February 1996
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9 Reads
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11 Citations
Computers & Graphics
The paper introduces two algorithms for the approximation of a digitized set of points from surfaces of revolution. In the first algorithm it is assumed that the rotation axis is known, in the second one the rotational axis is determined additionally.
January 1996
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436 Reads
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1,189 Citations
Mathematics of Computation
A classic reference and text, this book introduces the foundations used to create an accurate computer screen image using mathematical tools. This comprehensive guide is a handbook for students and practitioners and includes an extensive bibliography for further study.
January 1996
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22 Reads
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8 Citations
An algorithm for approximation of arbitrary clouds of points with integral tensor product B-spline surfaces is presented. The clouds may be scattered may have holes and may have arbitrary boundaries. The usual methods in Reverse Engineering subdivide the given cloud into rectangular parts and approximate these parts individually. In the presented paper an overall algorithm for tensor product B-spline approximation with free boundary curves is introduced.
January 1995
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10 Reads
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47 Citations
Computer-Aided Design
The paper discusses rational curve segments and surface patches on quadric surfaces. Detailed constructions of rational Bézier patches from given boundaries on a unit sphere and on a hyperbolic paraboloid are presented based on a generalization of the stereographic projection. The method is applied to interpolation with rational curves on quadrics. The results are extended to rational B-spline representations by discussion of products of B-spline functions. Finally, the generalization of the constructions to arbitrary nondegenerated quadric surfaces is outlined.
January 1995
Die Arthrose, d.h. die degenerative Gelenkerkrankung, ist eine der häufigen Erkrankungen des mittleren und höheren Lebensalters. Sie kann insbesondere im Bereich von Hüft- und Kniegelenk zu erheblichen Beeinträchtigungen des menschlichen Bewegungsvorganges und des allgemeinen Wohlbefindens führen.
January 1995
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37 Reads
Die erste Stufe der Entwicklung einer Automobilaußenhaut ist der Designprozeß. Über Skizzen und Zeichnungen (oft mit CAD-Hilfe) wird in der Design-Abteilung ein Plastikmodell gestaltet, das dann nach positiver Entscheidung des Vorstandes der Konstruktionsabteilung übergeben wird. Um nun von einem solchen Modell zu einer perfekten Auto-Oberfläche zu gelangen, sind zahlreiche Arbeitsgänge notwendig, die über geeignete mathematische Modelle mit Rechnerhilfe gesteuert werden.
November 1994
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29 Reads
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7 Citations
Conversion methods are required for data exchange. We introduce approximation and conversion algorithms for integral or rational B- spline surfaces which use least squares with discrete L 2 -norm and parameter correction. The method can be extended to trimmed surfaces.
January 1994
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16 Reads
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8 Citations
Computer Aided Geometric Design (CAGD) is concerned with the design, computation, and representation of curved objects on a computer. Therefore, not surprisingly, CAGD has traditionally had strong ties to some classical mathematical disciplines such as approximation theory (approximation by polynomial and piecewise polynomial functions), differential geometry (parametric surfaces), algebraic geometry (algebraic surfaces), functional analysis and differential equations (surface design by minimizing functionals), and numerical analysis. In addition, work in CAGD also requires a solid background in computer science.
... Kirbach et al. (2015), Pierron et al. (2007)) or B-splines can be used for this purpose. Basics of B-spline approximation can be found in (Hoschek & Lasser, 1992) and a selection of application examples are given in (Lehmann et al., 2018;Peretzki et al., 2022;Kanzenbach et al., 2022). Using an approximation-based method, strains are easily determined by derivatives and smoothing is achieved by the continuous mathematical description due to the functions (in most cases the displacement). ...
January 1992
... In der Tat halte ich für den notwendigen nächsten Schritt die Klärung des Bezugs von Kinematik und Oberflächengeometrie (unter Berücksichtigung der bekannten Tatsachen bis zurück zu O. Fischer [7], R. Fick [6] und H. Strasser [35]) am einzelnen Gelenk. Ansätze für den dazu nötigen theoretischen Hintergrund sehe ich bei H. R. Müller [25], J. Hoschek [15,17,18] und P. Meyer [22]. Indem sie das Problem als eben betrachteten und die Formen der Kondylen der Gelenke zu Kreisen annäherten, konnten H. Nägerl und Mitarbeiter das Problem auf die Getriebelehre bzw. ...
January 1987
... For surface approximation problem, least square method using B-splines is a simple approximation method, but it is sensitive to noisy data which may cause large errors. In order to reduce local error, Hoschek et al. [14] proposed an algorithm to construct a surface to fit an arbitrary point cloud and free boundary curves. It is found that in some cases they will not get the ideal fitting surface and might even fail, such as when the local density of sampling data points is high, or use high order polynomial to approximate, or the number of sampled data points is less than the number of control points of the B-spline. ...
January 1996
... The algorithm can be extended for C 0-continuous curves and possibly for open curves. Another potential application for using the theories that have been developed in this paper could be on blending of curves (see Chapter 14 in [29]). The application areas are being explored at present. ...
... To find an optimal spherical arc on S 2 that best approximates all sample points, we can also use the spherical spline interpolation/approximation method (e.g., in [4]) directly on S 2 . We have implemented both a spherical spline approximation method [14] and the above digital-image method. In our implementation, the spherical arc approximation directly on the spherical domain utilizes two spherical angles (i.e., many computations using the values of trigonometric functions like sine and cosine), which makes our implementation less stable. ...
January 1992
ESAIM Mathematical Modelling and Numerical Analysis
... space and to parametric surfaces. Using variational subdivision, instantaneous kinematics, and the relation to line geometry, we show how to construct such motions. Gliding motion design has applications e.g. in the NC machining of complex shapes (cf. Marciniak (1991) ), and in medical applications where the gliding motion of joints is studied (cf. Hoschek and Weber (1987)). ...
... The field of CAGD "is concerned with the design, computation, and representation of curved objects on a computer" [BHS94,p. 106]. This primarily involves (univariate) curves, which have a one-dimensional parameter space, as well as surfaces, whose parameter space is two-dimensional and which, therefore, require multivariate concepts. ...
Reference:
Delaunay Configuration B-Splines
January 1994
... i.e., a variation diminishing property (see [1,22,27,28,36,38]). ...
January 1985
... The method [1] produces splines of an arbitrarily high polynomial degree, which can be very critical for implementing an algorithm in a CAD system. Indeed, the geometric/solid modelers of CAD systems restrict the polynomial degrees to a fixed, relatively low, value (e.g., ACIS has a limit of 25 in some operations [20], AutoCAD fixes it to 10 [21], NLib to 28 [22]), an issue well known since the early days of CAD (see [23] and [24]). Based on the resulting curve of [1], in this work, we focus on techniques to reduce the degree of the VDS segments down to cubic in two ways: approximating the curve of [1] by (i) a parametric cubic polynomial spline (CS) or by (ii) a cubic (non-parametric) rational polynomial spline (CRS), both satisfying the interpolation and the shape-preservation conditions. ...
January 1990
... In this way, once a procedure for computing optimal weight values has been designed, control points turn out to be automatically defined and hence the best-fitting curve results completely determined. Therefore, differently from standard NURBS fitting procedures, which require a complicated and expensive iterative algorithm to minimize with respect to knots, control points and weights, a sum of squared Euclidean norms measuring the distance between the point set and the curve to be generated [3,4,9,11,12,13,14,15], the least-squares fitting method we are going to propose will be performed exclusively to identify the choice of weights that guarantees the best reconstruction of the original data. Moreover, while the output of existing algorithms cannot always guarantee a fitting curve with a fair shape (namely with a curvature plot consisting of only a small number of monotone pieces), due to the definition of the novel fitter this follows straightforward and, whenever the degree of the curve primitive is bigger than three, a curvature-continuous approximation of the original data is also ensured. ...
November 1994
