Christoph Fünfzig

Adaptive Tesselation of Subdivision Surfaces

Abstract For a variety of reasons subdivision surfaces have developed into a prominent member of the family of free-form shapes. Based on a standard polygonal mesh a modeller can build various kinds of shapes using an arbitrary topology and special geometrical features like creases. However, the interactive display of subdivision surfaces in current scenegraph systems based on static levels of detail is unpractical, because of the exponentially increasing number of polygons during the subdivision steps. Therefore, an adaptive algorithm choosing only the necessary quads and triangles is required to obtain high-quality images at high frame rates. In this paper we present a rendering algorithm which dynamically adapts to static surface properties like curvature as well as to view-dependent properties like silhouette location and projection size. Without modifying the base mesh, the method works patchwise and tesselates each patch recursively using a new data structure, called slate. Besides these geometric properties the algorithm can also adapt to the graphics load in order to achieve a desired frame rate in the scenegraph system OpenSG.

Dinus: Double insertion, nonuniform, stationary subdivision surfaces

The Double Insertion, Nonuniform, Stationary subdivision surface (DINUS) generalizes both the nonuniform, bicubic spline surface and the Catmull-Clark subdivision surface. DINUS allows arbitrary knot intervals on the edges, allows incorporation of special features, and provides limit point as well as limit normal rules. It is the first subdivision scheme that gives the user all this flexibility and at the same time all essential limit information, which is important for applications in modeling and adaptive rendering. DINUS is also amenable to analysis techniques for stationary schemes. We implemented DINUS as an Autodesk Maya plugin to show several modeling and rendering examples.

Nonlinear systems solver in floating-point arithmetic using LP reduction

This paper presents a new solver for systems of nonlinear equations. Such systems occur in Geometric Constraint Solving, e.g., when dimensioning parts in CAD-CAM, or when computing the topology of sets defined by nonlinear inequalities. The paper does not consider the problem of decomposing the system and assembling solutions of subsystems. It focuses on the numerical resolution of well-constrained systems. Instead of computing an exponential number of coefficients in the tensorial Bernstein basis, we resort to linear programming for computing range bounds of system equations or domain reductions of system variables. Linear programming is performed on a so called Bernstein polytope: though, it has an exponential number of vertices (each vertex corresponds to a Bernstein polynomial in the tensorial Bernstein basis), its number of hyperplanes is polynomial: O(n2) for a system in n unknowns and equations, and total degree at most two. An advantage of our solver is that it can be extended to non-algebraic equations. In this paper, we present the Bernstein and LP polytope construction, and how to cope with floating point inaccuracy so that a standard LP code can be used. The solver has been implemented with a primal-dual simplex LP code, and some implementation variants have been analyzed. Furthermore, we show geometric-constraint-solving applications, as well as numerical intersection and distance computation examples.

Technical Section: A comparison of local parametric C0 Bézier interpolants for triangular meshes

Parametric curved shape surface schemes interpolating vertices and normals of a given triangular mesh with arbitrary topology are widely used in computer graphics for gaming and real-time rendering due to their ability to effectively represent any surface of arbitrary genus. In this context, continuous curved shape surface schemes using only the information related to the triangle corresponding to the patch under construction, emerged as attractive solutions responding to the requirements of resource-limited hardware environments. In this paper we provide a unifying comparison of the local parametric C^0 curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bezier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.

Polytope-based computation of polynomial ranges

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n=1,2,3,4, and we compare the computed range widths for random n-variate polynomials for n=<10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

Using the witness method to detect rigid subsystems of geometric constraints in CAD

This paper deals with the resolution of geometric constraint systems encountered in CAD-CAM. The main results are that the witness method can be used to detect that a constraint system is over-constrained and that the computation of the maximal rigid subsystems of a system leads to a powerful decomposition method.n In a first step, we recall the theoretical framework of the witness method in geometric constraint solving and extend this method to generate a witness. We show then that it can be used to incrementally detect over-constrainedness. We give an algorithm to efficiently identify all maximal rigid parts of a geometric constraint system. We introduce the algorithm of W-decomposition to identify all rigid subsystems: it manages to decompose systems which were not decomposable by classical combinatorial methods.

G1 rational blend interpolatory schemes: A comparative study

Interpolation of triangular meshes is a subject of great interest in many computer graphics related applications, as, for example, gaming and realtime rendering. One of the main approaches to interpolate the positions and normals of the mesh vertices is the use of parametric triangular Bezier patches. As it is well known, any method aiming at constructing a parametric, tangent plane (G^1) continuous surface has to deal with the vertex consistency problem. In this article, we propose a comparison of three methods appeared in the nineties that use a particular technique called rational blend to avoid this problem. Together with these three methods we present a new scheme, a cubic Gregory patch, that has been inspired by one of them. Our comparison includes an analysis of their computational costs on CPU and GPU, a study of their capabilities of approximating analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.

Two different views on collision detection

In this article, we present two algorithms for precise collision detection between two potentially colliding objects. The first one uses axis-aligned bounding boxes (AABB) and is a typical representative of a computational geometry algorithm. The second one uses spherical distance fields originating in image processing. Both approaches addresses the following challenges of collision detection algorithms: just in time, little resources, inclusive etc. Thus both approaches are scalable in the information they give in collision determination and the analysis up to a fixed refinement level, the collision time depends on the granularity of the bounding volumes and it is also possible to estimate the time bounds for the collision test tightly

OPTIMIZATIONS FOR TENSORIAL BERNSTEIN–BASED SOLVERS BY USING POLYHEDRAL BOUNDS

The tensorial Bernstein basis for multivariate polynomials in n variables has a number 3n of functions for degree 2. Consequently, computing the representation of a multivariate polynomial in the tensorial Bernstein basis is an exponential time algorithm, which makes tensorial Bernstein-based solvers impractical for systems with more than n = 6 or 7 variables. This article describes a polytope (Bernstein polytope) with a number of faces, which allows to bound a sparse, multivariate polynomial expressed in the canonical basis by solving several linear programming problems. We compare the performance of a subdivision solver using domain reductions by linear programming with a solver using a change to the tensorial Bernstein basis for domain reduction. The performance is similar for n = 2 variables but only the solver using linear programming on the Bernstein polytope can cope with a large number of variables. We demonstrate this difference with two formulations of the forward kinematics problem of a Gough-Stewart parallel robot: a direct Cartesian formulation and a coordinate-free formulation using Cayley-Menger determinants, followed by a computation of Cartesian coordinates. Furthermore, we present an optimization of the Bernstein polytope-based solver for systems containing only the monomials xi and . For these, it is possible to obtain even better domain bounds at no cost using the quadratic curve (xi, ) directly.

Haptic manipulation of rational parametric planar cubics using shape constraints

In this paper, we show how to deform a planar rational cubic based on a local interpolation constraint while retaining the qualitative shape of the curve. An impedance-type, parallel haptic device is used to signal changes of the number of inflection points, cusps and loops during the deformation. In this way, the user is provided with an intuitive and natural guidance throughout the curves shape generation process in CAD.