Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Intersection of a ruled surface with a free-form surface

This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics. An erratum to this article can be found at     

Analysis of the core of multisided Böhm-Bawerk assignment markets

We introduce the class of multisided Böhm-Bawerk assignment games which generalizes the well-known two-sided Böhm-Bawerk assignment games to markets with an arbitrary number of sectors. We reach the core and the corresponding extreme allocations of any multisided Böhm-Bawerk assignment game by means of an associated convex game defined on the set of sectors instead of the set of sellers and buyers. We also study when the core of a multisided Böhm-Bawerk assignment game is stable in the sense of von Neumann–Morgenstern.     

On the Gauss Map of Minimal Graphs

We consider graphs of solutions to the minimal surface equation which are unbounded over subarcs of the domain boundary. An extensive study of such surfaces was made by Jenkins and Serrin. In this note, properties of the Gauss map are studied.     

BOUNDS ON COINCIDENCE INDICES ON NON-ORIENTABLE SURFACES

§ 1 . IntroductionQuestions about bounds for indices ?rst appeared in the ?xed point context. The ?rstresults appeared in studies of surface homeomorphisms (see [13, 18, 19]). In [12, 14] and[15] some results about bounds for Nielsen ?xed point class ind…     

Noether's Theorem on Surfaces

In this article we prove a version of Noether's Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noether's Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.     

Toric Surface Patches

We define a toric surface patch associated with a convex polygon, which has vertices with integer coordinates. This rational surface patch naturally generalizes classical Bézier surfaces. Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points. The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.     

Kinematic Generation of Ruled Surfaces

This paper presents a method for kinematic generation of free-form ruled surfaces. The method is based on the kinematic displacement of lines. The ruled surfaces are represented as curves on a dual unit sphere. The curves are created by using the Lie Group structure of the dual space to generate dual displacement matrices for the lines. Free-form surfaces are created by repeated geodesic interpolation using the displacement matrices. An application for these surfaces is presented in five-axis cylindrical milling.     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

Noninterpolatory Hermite subdivision schemes

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.

A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.

     

Approximating tensor product Bézier surfaces with tangent plane continuity

We present a simple method for degree reduction of tensor product Bézier surfaces with tangent plane continuity in L₂-norm. Continuity constraints at the four corners of surfaces are considered, so that the boundary curves preserve endpoints continuity of any order α. We obtain matrix representations for the control points of the degree reduced surfaces by the least-squares method. A simple optimization scheme that minimizes the perturbations of some related control points is proposed, and the surface patches after adjustment are C^∞ continuous in the interior and G¹ continuous at the common boundaries. We show that this scheme is applicable to surface patches defined on chessboard-like domains.     

On Surfaces of General Type with p g = q = 1 Isogenous to a Product of Curves

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with p_g = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces 𝔐_I, 𝔐_II, 𝔐_IV are irreducible, whereas 𝔐_III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples.     

A generic mathematical model of single point cutting tools in terms of grinding parameters

The paper presents the analytical geometric details of the mathematical modeling of a single point cutting tool with a generic profile. The grinding angles and the ground depths on the tool are allowed to vary along the tool flanks and face, altering the cutting angles from point to point. The surface modeling begins with the creation of a tool blank model. Then unbounded surfaces are considered and transformed to get the cutting tool surfaces. The intersection of these surfaces gives the complete model of the tool. Starting from the basic model where the tool face and flank are planar, the generalization of the geometric design has been done in two steps to give free-form shapes to the tool surfaces, termed as the two generations of the generic profile. Then a forward and inverse mapping has been presented for the basic model and the two generations of the generic tool to relate the grinding angles with the prevalent nomenclatures (ASA, ORS and NRS). The model has been validated and the variation of tool angles with the grinding parameters has been illustrated with an example.     

Relation between algebraic and geometric view on nurbs tensor product surfaces

NURBS (Non-Uniform Rational B-Splines) belong to special approximation curves and surfaces which are described by control points with weights and B-spline basis functions. They are often used in modern areas of computer graphics as free-form modelling, modelling of processes. In literature, NURBS surfaces are often called tensor product surfaces. In this article we try to explain the relationship between the classic algebraic point of view and the practical geometrical application on NURBS.     

两种Bézier曲面形式的转换及几何连续拼接

本文讨论两种Bézier曲面形式的转换关系和转换条件,并给出矩阵域Bézier曲面片和三角域Bézier曲面片拼接的几何连续条件。     

Mathematical modelling of influence functions in computer-controlled polishing: Part I

Computer-controlled polishing (CCP) is commonly used to finish high-quality surfaces, such as optical lenses. Based on magnetorheological finishing (MRF), a mathematical model to calculate the polishing tool characteristic (influence function) was developed and verified experimentally. The first part of this paper introduces the model to predict the size and shape of an influence function. The second part of this paper describes the calculation of the distribution of material removal within the size of an influence function. The model supersedes the current cumbersome procedure for determining an influence function and thus results in considerably improved and more economical manufacture. Furthermore, the model enables the quality of the final surface to be enhanced when polishing complex, for example aspherical or free-form, workpiece geometries and provides the first step in the application of time-variant influence functions.     

Surfaces of revolution associated with the kink-type solutions of the SIdV equation

In this paper, we study the evolution scenarios of surfaces of revolution associated with the kink-type solutions of an integrable equation, which is called the SIdV equation because of its scale-invariant property and relationship with the Korteweg-de Vries equation, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.     

Reconstruction of quasi developable surfaces from ribbon curves

This paper deals with the acquisition and reconstruction of physical surfaces by mean of a ribbon device equipped with micro-sensors, providing geodesic curves running on the surface. The whole process involves the reconstruction of these 3D ribbon curves together with their global treatment so as to produce a consistent network for the geodesic surface interpolation by filling methods based on triangular Coons-like approaches. However, the ribbon curves follow their own way, subdividing thus the surface into arbitrary n-sided patches. We present here a method for the reconstruction of quasi developable surfaces from such n-sided curvilinear boundary curves acquired with the ribbon device.     

Eine neue Variante einer Schachtelaufgabe

The following geometrical optimization problem is studied by methods of optimal control: How can we cut out from an assembly line congruent surface patches that are bent in pairs to cylindric surfaces and fixed to boxes having maximum volume relatively to the needed material?     

Power Series Solution for Isoscallop Tool Path Generation on Free-form Surface with Ball-end Cutter

In this paper we propose a method to compute symbolically a curve of constant scallop height on the surface swept by a ball-end cutter when it traces a free-form cutter-contact curve on a free-form surface. We give explicit formulas for power series solutions to the curve of constant scallop height and the next cutter-contact curve, when the initial cutter-contact curve is given. To make the symbolic results useful in practice, we propose a method to approximate the solutions unilaterally in the parametric plane by piecewise Chebyshev polynomials and exponential functions.