A short abstract will appear on the permanent conference web page (see below) as soon as accepted.
An extended abstract will appear on the permanent conference web page (see below) as soon as accepted.
It will also appear on the proceedings that will be distributed during the meeting.
If you would like to give a talk at ICMS, you need to submit first a short abstract and then later an extended abstract. See the guideline for the details.
Abstract: Geometric modeling aims at providing shape descriptions and at developing computational tools for processing these models. Strong interactions exist with other application domains such as graphical rendering and visualisation, Computer Aided Design and Computer Aided Manufacturing, numerical simulation, etc. Meshes, which are piecewise linear models, are classically used for shape representations. In many domains such as CAD-CAM or Isogeometric Analysis, this is not sufficient and higher order models are needed to describe accurately and efficiently the geometry. Axl is a platform, which provides tools for the manipulation, computation and visualisation of semi-algebraic models of higher order. This includes meshes, basic geometric objects such as spheres, cylinders, cones, ellipsoids, torus, piecewise polynomial parameterisations of curves, surfaces or volumes such as b-spline parameterisations, algebraic curves and surfaces defined by polynomial equations. Axl provides also algorithms to process these geometric representations such as computing intersection points or curves of parametric models, singularities of algebraic curves or surfaces, certified topology of curves and surfaces, etc. To cope with the versatility of shape representations, Axl integrates a generic extension mechanism, which allows to define new data types and new processes on this data, as well as, automatic visualisation and interaction facilities. Via the production of dedicated plugins, external tools can be easily embedded, tested and demonstrated in this framework. The design of the software, its main features and development techniques will be presented. The application capacities of the software will be illustrated by short descriptions of plugins on algebraic curves and surfaces, on splines for Isogeometric Analysis and on fitting methods for high quality modeling. It is a joint work with E. Christoforou, A. Mantzaflaris and J. Wintz.
Abstract: The problem of detecting when two moving ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc., where ellipsoids are often used for modelling (and/or enclosing) the shape of the objects under consideration. By analysing symbolically the sign of the real roots of the characteristic polynomial of the pencil defined by two ellipsoids A and B we derive a new closed formulae characterising when A and B overlap, are separate and touch each other externally. These conditions are defined by a minimal set of polynomial inequalities depending only on the entries of A and B, need only to compute the characteristic polynomial of the pencil defined by A and B and do not require the computation of the intersection points between them. Compared with the best available approach dealing with this problem the new formulae involve a smaller set of polynomials and less sign conditions. As an application, this characterisation provides a new approach for exact collision detection of two moving ellipsoids since the analysis of the univariate polynomials (depending on the time) in the previously mentioned formulae provides the collision events between them.
Abstract: A fundamental problem in Computer Graphics and Computer Aided Design is to convert between an implicit representation of a surface and a parameterization of it. Almost as fundamental is to derive a parameterization for the intersection of two surfaces. In these problems, it seems that resultants have been under-appreciated. Indeed, several well known papers from ten to twenty years ago reported failure and unsuitability of resultant techniques. To the contrary, we show by example that using the Dixon resultant to eliminate parameters is an extremely effective and efficient method to compute an implicit representation. To use resultants to compute a parameterization of an intersection, we introduce the concept of an "implicit parameterization." Unlike the conventional parameterization of a curve where x, y, and z are each explicitly given as functions of t, we have three implicit functions, one each for (x,t), (y,t), and (z,t). This concept has rarely been mentioned before. We show that given a (conventional) parameterization for one surface and either an implicit equation for the second, or a parameterization for it, it is straight-forward to compute an implicit parameterization for the intersection. Doing so is very easy for the Dixon resultant, but can be daunting even for well respected Groebner bases programs. Further, we demonstrate that such implicit parameterizations are useful. We use builtin 3D plotting utilities of a well known computer algebra system to graph the intersection from our implicit parameterization. We do this for examples that are more complex that the quadric examples usually discussed in intersection problems.
Abstract: Adaptive, subdivision-based methods recursively subdivide an input region until an algorithm-specific test is passed on each subregion. Such methods are common in practice because they are easy to implement via recursion, provide adaptive output, and are guaranteed to terminate (at least for smooth input). These useful properties, however, make the behavior of these algorithms dependent on the geometry of the embedding of the underlying variety. In this talk, I will describe various subdivision-based algorithms whose input is a curve or surface. In addition, I will discuss problem-specific features which affect the practical efficiency of these algorithms. In particular, I will illustrate how to use the technique of continuous amortization to describe which input instances subdivision-based algorithms will be efficient or inefficient.
Abstract: In recent years, various methods for geometric modeling and isogeometric analysis have been developed that allow local refinement of a control mesh. These methods include hierarchical B-splines, polynomial splines over T-meshes, locally refined B-splines, and specifically for this talk, T-splines. In our previous work, we investigated necessary and sufficient conditions under which the de Boor algorithm, originally for B-spline curve and surfaces, can be applied to evaluate points on a T-spline surface. We showed that the standard analysis-suitable conditions for T-spline surfaces are not strong enough to ensure that the de Boor algorithm can be applied to evaluate points on a T-spline surface. We introduced de Boor-suitable (DS) T-splines, which are a subset of the analysis-suitable T-splines and for which the de Boor algorithm is everywhere applicable. In this previous work, however, we discussed only bicubic T-splines. In this talk, we will generalize our previous work to de Boor-suitable T-splines of arbitrary bidegree. We will present necessary and sufficient conditions, which guarantee that the de Boor algorithm can be applied to compute points on a T-spline surface of this more general class.
Abstract: We present a new method to plot planar implicit curve in a given box in real plane. Based on analyzing the topology and geometry of the level sets of the given function, following the points with local maximum or minimum curvatures on the level sets, we plot all the components of the given function in the given box. We also promoted this method to find real zeros of bivariate function systems in a given box. It also works for non-polynomial systems. It is a joint work with Junyi Wen and Wenjian Zhang.
Abstract: We describe a new subdivision method to trace an $\epsilon$-isotopic approximation of a non-singular part of a planar curve $f(x,y)=0$ in a box $B_0\ib \RR^2$. It is based on two well-known predicates called $C_0$ and $C_1$. Unlike previous use of these predicates to compute isotopic approximations of curves, we do not search the entire box $B_0$ for these curves. It is assumed that we have an appropriate start box that identifies the curve to be traced. We indicate applications where this formulation is sufficient.