Georg Glaeser

Collision-free 3-axis milling and selection of cutting tools

Abstract This article deals with local and global conditions for collision-free 3-axis milling of sculptured surfaces and the selection of cutting tools for a given surface. We describe local and global millability results, the proof of which were published in a previous article. The theoretical background involves general offset surfaces. We present an algorithm here which, after evaluation of surface curvature, yields a differential inequality for the meridian curve of the cutting tool. This inequality is fulfilled if and only if the cutting tool is able to mill the entire surface. Besides this inequality, the optimal selection (or even design) of cutting tools involves a consideration of further characteristics, such as shape and size.

Geometric Criteria for Gouge-Free Three-Axis Milling of Sculptured Surfaces

The paper presents a geometric investigation of collision-free 3-axis milling of surfaces. We consider surfaces with a global shape condition: they shall be interpretable as the graphs of bivariate functions or shall be star-shaped with respect to a point. If those surfaces satisfy a local millability criterion involving curvature information, it is proved that this implies globally gouge-free milling. The proofs are based on general offset surfaces. The results can be applied to tool-motion planning and the computation of optimal cutter shapes.

Efficient Volume-Generation During the Simulation of NC-Milling

This paper presents an efficient and robust algorithm for the geometric determination of swept volumes during the simulation of numerical controlled (NC-) milling. The boundary Ψ of the volume swept by a cutter Φ is represented polygonally by using instantaneous helical motions to exactly determine the line of contact between Φ and Ψ.

Developable surfaces in contemporary architecture

Developable surfaces (tangential developables, in special cases cylinders and cones) are ruled surfaces with vanishing Gaussian curvature and can therefore be unfolded to the plane without distortions. In this article we will survey and discuss examples of the use of developable surfaces in contemporary architecture. We also discuss software for aiding architects, designers, engineers and artists to explore their ideas for the use of developable surfaces for such purposes.

Fast generation of curved perspectives for ultra-wide-angle lenses in VR applications

Classic ultra-wide-angle perspectives are not realistic and often misleading, nevertheless, they have to be used in many applications where the viewer needs a survey of the scene. Current hardware, however, only supports classic perspectives. We present a fast polygon-oriented algorithm that allows the use of curved perspectives in order to overcome several drawbacks, but at the cost of non-linearity. Space is projected onto a sphere rather than onto an image plane. The spherical coordinates are then interpreted two-dimensionally. We discuss the advantages and drawbacks of several approaches to curved perspectives. Time measurements show that real-time animation of reasonably complicated scenes can still be done; the overhead (additional cost of CPU-time) is less than 20%. Thus, curved perspectives are a good choice in virtual reality applications where ultra-wide-angle lenses have to be used.

Geometrie und ihre Anwendungen in Kunst, Natur und Technik

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Geometric Contribution to 3-axis Milling of Sculptured Surfaces

When we are trying to shape a surface X by 3-axis milling, we encounter a list of problems: First we have to decide if locally the milling tool Σ is able to move along the surface such that its envelope during the motion is the given surface. This is a question involving the curvatures of X and Σ. Second, we want to avoid that while milling in one part of X, Σ intersects another, already finished, part of the surface. This is a problem which involves global shape properties of the surface and can be successfully attacked by considering the general offset surface of X with respect to Σ. Third, in practice a cutting-tool is not able to perform a 2-dimensional motion along a surface. It has to trace out a finite number of piecewise smooth paths such that the resulting surface does not differ from X too much. This question again involves, in the limit case of very small error tolerance, only the curvatures of X and Σ. If we allow larger scallop heights, the path finding also requires the study of local and global properties.

Magnetism and minimal surfaces: a different tool for surface design

The design of free form surfaces is usually based on NURBS and it works well to quickly get shapes that a designer intends to create. Such surfaces then have desired properties like given border lines and C1 or C2 continuity along lines where several surfaces touch. Our approach is to create surfaces with certain physical properties that designers often need. Given a closed or not closed border line, can we then find an elastic surface (comparable with a rubber surface) with the property requiring that in each point the tension is equally distributed? This is – simplified spoken – the condition for a minimal surface. Our solution does not use any differential equations but rather the following idea: We start from a patch that may be planar or part of a cylinder or any easy to define surface. This patch is tesselated in such a way that the vertices have roughly equal distances. Each point is considered to be magnetic. Now we start a converging real-time-iteration that allows the points to move according to the rules of magnetism. Border lines or parts of them may be fixed and manipulated. The corresponding algorithm is adapted from earlier algorithms by Fruchterman et al. The result is an approximation to a minimal surface that is defined by the fixed border lines. The advantage of such a surface design is twofold: First, the problem is hard to solve exactly by means of differential equations, and second the algorithm works interactively in real time. This means that the designer can change shapes almost as quickly as with conventional free form surfaces. Finally, the surface is already suitably triangulated.

Statistik und Wahrscheinlichkeitsrechnung

Das Wort Statistik wurde im beginnenden 19. Jhdt. in England und Frankreich in der Folge als Synonym zur „numerischen Beschreibung der Gesellschaft“ gebraucht. Dabei wurden noch keine Schlusse von Daten auf Einzelpersonen gemacht. Erst im spaten 19. Jahrhundert erlebte die Statistik mit der zunehmenden Anwendung mathematischer Methoden in den Naturwissenschaften einen Aufschwung.

A Spatial Version of the Theorem of the Angle of Circumference

We try a generalization of the theorem of the angle of circumference to a version in three-dimensional Euclidean space and ask for pairs \(({\varepsilon },{\varphi })\) of planes passing through two (different skew) straight lines \(e\ni {\varepsilon }\) and \(f\ni {\varphi }\) such that the angle \(\alpha \) enclosed by \({\varepsilon }\) and \({\varphi }\) is constant. It turns out that the set of all such intersection lines is a quartic ruled surface \(\varPhi \) with \(e\cup f\) being its double curve. We shall study the surface \(\varPhi \) and its properties together with certain special appearances showing up for special values of some shape parameters such as the slope of e and f (with respect to a fixed plane) or the angle \(\alpha \).