Creating motions of objects or characters that are physically plausible and follow an animator's intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a framework for controlling the motion of deformable objects that offers interactive response times. This is achieved by a model reduction of the underlying variational problem, which combines dimension reduction, multipoint linearization, and decoupling of ODEs. After a preprocess, the cost for creating or editing a motion is reduced to solving a number of one-dimensional spacetime problems, whose solutions are the wiggly splines introduced by Kass and Anderson [2008]. We achieve interactive response times through a new fast and robust numerical scheme for solving the one-dimensional problems that is based on a closed-form representation of the wiggly splines.

We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT (centroidal Voronoi tessellation) optimization framework. We devise a CVT-CMC energy function defined as a combination of an extended CVT energy and a volume functional. We show that minimizing the CVT-CMC energy is asymptotically equivalent to minimizing mesh surface area with a fixed volume, thus defining a discrete CMC surface. The CVT term in the energy function ensures high mesh quality throughout the evolution of a CMC surface in an interactive design process for form finding. Our method is capable of modeling CMC surfaces with fixed or free boundaries and is robust with respect to input mesh quality and topology changes. Experiments show that the new method generates discrete CMC surfaces of improved mesh quality over existing methods

We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].

In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies, which operate on a function space on the surface, or of deformation energies, which operate on a shape space. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.

We present a discretization of Koiter's model of elastic thin shells based on a finite element that employs limit surfaces of Catmull-Clark's subdivision scheme. The discretization can directly be applied to control grids of Catmull-Clark subdivision surfaces, and, therefore, integrates modeling of Catmull-Clark subdivision surfaces with analysis and optimization of elastic thin shells. To test the discretization, we apply it to standard examples for physical simulation of thin shells and compute free vibration modes of thin shells. Furthermore, we use the discrete shell model to set up a deformation-based modeling system for Catmull-Clark subdivision surfaces. This system integrates modeling of subdivision surfaces with deformation-based modeling and allows to switch back and forth between the two different approaches to modeling.

This work concerns the approximation of the shape operator of smooth surfaces in R³ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions. The estimates are confirmed by numerical experiments.

Discrete Laplace--Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace--Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace--Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian.Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace--Beltrami operators.

Despite the success of quad-based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. CubeCover is a first approach for generating a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field.

The input of CubeCover is a tetrahedral volume mesh. First, a frame field is designed with manual input from the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and the hexahedral mesh are computed so as to align with the given frame field.

CubeCover has similarities to the QuadCover algorithm and extends it from 2D surfaces to 3D volumes. The paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties of the appropriate function space.

Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 7. For efficient compression of vertex positions we adapt popular wavelet-based coding schemes to the adaptive triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state-of-the-art coders, we use a zerotree to encode the resulting coefficients. Using improved context modeling we enhanced the zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression factors that are 4 times greater than those of coders which can handle irregular meshes.

In this paper we introduce hexagonal global parameterizations, a new type of parameterization in which parameter lines respect six-fold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with regular hexagonal texture and geometry patterns and can be used to generate high-quality triangular remeshing.

To construct a hexagonal parameterization given a surface, we provide an automatic technique to generate a 6-RoSy field that respects directional and singularity features in the surface. We also introduce a technique for automatically merging and cancelling singularities. This field will then be used to generate a hexagonal global parameterization by adapting the framework of QuadCover parameterization.

We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with regular textures and geometry patterns and triangular remeshing.

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in R^3 from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical projection onto R^2 and can therefore be considered as multivalued graphs over the plane – hence they provide a comprehensible visualization of the topological structure. Complex functions are elegantly visualized using domain coloring on a subset of C. By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.

In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of this energy are sensitive to the extrinsic curvature of the surface.

Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of the surface. In addition, we discuss a spectral quadrangulation scheme for surfaces.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

Structuring of surface meshes is a labor intensive task in reverse engineering. For example in CAD, scanned triangle meshes must be divided into characteristic/uniform patches to enable conversion into high-level spline surfaces. Typical industrial techniques, like rolling ball blends, are very labor intensive. We provide a novel, robust and quick algorithm for the automatic generation of a patch layout based on a topology consistent feature graph. The graph separates the surface along feature lines into functional and geometric building blocks. Our algorithm then thickens thickens the edges of the feature graph and forms new regions with low varying curvature. Further these new regions - so called llets and node patches - will have highly smooth boundary curves making it an ideal preprocessor for a subsequent spline tting algorithm.

We present a novel coder for lossless compression of adaptive multiresolution meshes that exploits their special hierarchical structure. The heart of our method is a new progressive connectivity coder that can be combined with leading geometry encoding techniques. The compressor uses the parent/child relationships inherent to the hierarchical mesh. We use the rules that accord to the refinement scheme and store bits only where it leaves freedom of choice, leading to compact codes that are free of redundancy. To illustrate our scheme we chose the widespread red-green refinement, but the underlying concepts can be directly transferred to other adaptive refinement schemes as well. The compression ratio of our method exceeds that of state-of-the-art coders by a factor of 2 to 3 on most of our benchmark models.

Classical surface parameterization algorithms often place singularities in order to enhance the quality of the resulting parameter map. Unfortunately, singularities of positive integral index (as the north pole of a sphere) were not handled since they cannot be described with piecewise linear parameter functions on a triangle mesh. Preprocessing is needed to adapt the mesh connectivity. We present an extension to the QuadCover parameterization algorithm, which allows to handle those singularities.

A singularity of positive integral index can be resolved using bilinear parameter functions on quadrilateral elements. This generalization of piecewise linear functions for quadrilaterals enriches the space of parameterizations. The resulting parameter map can be visualized by textures using a rendering system which supports quadrilateral elements, or it can be used for remeshing into a pure quad mesh.

We present a novel algorithm for automatic parameterization of tube-like surfaces of arbitrary genus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.

Die euklidische Geometrie von Eiflächen und Eikörpern bietet eine gute Gelegenheit, mathematischen Laien geometrische Begriffsbildungen und Sachverhalte aus der Flächentheorie plausibel zu machen. In dieser Arbeit erörtern wir die erstaunliche Stabilität von Eierschalen und dazu verwandte Probleme unter heuristisch-mathematischen Gesichtspunkten.

Complex-valued functions are fundamental objects in complex analysis, algebra, differential geometry and in many other areas such as numerical mathematics and physics. Visualizing complex functions is a non-trivial task since maps between two-dimensional spaces are involved whose graph would be an unhandy submanifold in four-dimensional space. The present paper improves the technique of “domain coloring” in several aspects: First, we lift domain coloring from the complex plane to branched Riemann surfaces, which are essentially the correct domain for most complex functions. Second, we extend domain coloring to the visualization of general 2-valued maps on surfaces. As an application of such general maps we visualize the Gauss map of surfaces as domain colored plots and establish a link to current surface parametrization techniques and texture maps. Third, we adjust the color pattern in domain and in image space to produce higher quality domain colorings. The new color schemes specifically enhance the display of singularities, symmetries and path integrals, and give better qualitative measures of the complex map.

We propose a method to identify planar regions in volume data using a specialized version of the discrete Radon transform operating on a structured or unstructured grid. The algorithm uses an efficient discretization scheme for the parameter space to obtain a running time of O(N(T log T)), where T is the number of cells and N is the number of plane normals in the discretized parameter space. We apply our algorithm in an industrial setting and perform experiments with real-world data generated by topology optimization algorithms, where the planar regions represent portions of a mechanical part that can be built using steel plate.

We present a method for parametric reconstruction of a piecewise defined pipe surface, consisting of cylinder and torus segments, from an unorganized point set. Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G1 continuous circular arcs and line segments. Our algorithm accurately outputs the parametric data required for bending machines to create the reconstructed tube.

We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2-manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2-manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.

We propose a constraint-based method for the fairing of surface meshes. The main feature of our approach is that the resulting smoothed surface remains within a prescribed distance to the input mesh. For example, specifying the maximum distance in the order of the measuring precision of a laser scanner allows noise to be removed while preserving the accuracy of the scan. The approach is modeled as an optimization problem where a fairness measure is minimized subject to constraints that control the spatial deviation of the surface. The problem is efficiently solved by an active set Newton method.

We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in R³. The notion of totally normal convergence is shown to be equivalent to the convergence of either one of the following: surface area, intrinsic metric, and Laplace-Beltrami operators. We further show that totally normal convergence implies convergence results for shortest geodesics, mean curvature, and solutions to the Dirichlet problem.

This work provides the justification for a discrete theory of differential geometric operators defined on polyhedral surfaces based on a variational formulation.

We introduce FreeLence, a novel and simple single-rate compression coder for triangle manifold meshes.  Our method uses free valences and exploits geometric information for connectivity encoding. Furthermore, we introduce a novel linear prediction scheme for geometry compression of 3D meshes. Together, these approaches yield a significant entropy reduction for mesh encoding with an average of 30% over leading single-rate region-growing coders, both for connectivity and geometry.

The use of point sets instead of meshes became more popular during the last years. We present a new method for anisotropic fairing of a point sampled surface using an anisotropic geometric mean curvature flow. The main advantage of our approach is that the evolution removes noise from a point set while it detects and enhances geometric features of the surface such as edges and corners. We derive a shape operator, principal curvature properties of a point set, and an anisotropic Laplacian of the surface. This anisotropic Laplacian reflects curvature properties which can be understood as the point set analogue of Taubin's curvature-tensor for polyhedral surfaces. We combine these discrete tools with techniques from geometric diffusion and image processing. Several applications demonstrate the efficiency and accuracy of our method.

Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance.

We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2d. This paper describes a curve smoothing flow that satisfies strict spatial constraints and allows simultaneous control of both curvature functions.

A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved edges and surface regions. Our method uses a non-linear anisotropic geometric diffusion flow for polyhedral surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on polyhedral surfaces that is fully consistent with the known discrete mean curvature representation, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behavior along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow converging to surfaces with prescribed mean curvature which preserves non-linear features. Additionally our discrete flow is very well suited to prevent boundary shrinkage at constrained and free boundary segments.

In differential geometry the study of smooth submanifolds with distinguished curvature properties has a long history and belongs to the central themes of this field. Modern work on smooth submanifolds, and on surfaces in particular, relies heavily on geometric and analytic machinery which has evolved over hundreds of years. However, non-smooth surfaces are also natural mathematical objects, even though there is less machinery available for studying them. Consider, for example, the pioneering work on polyhedral surfaces by the Russian school around Alexandrov [Aleksandrov/Zalgaller67Intrinsic], or Gromov's approach of doing geometry using only a set with a measure and a measurable distance function [Gromov99Metric]. Also in other fields, for example in computer graphics and scientific computing, we nowadays encounter a strong need for a discrete differential geometry of arbitrary meshes.

These tutorial notes introduce the theory and computation of discrete minimal surfaces which are characterized by variational properties, and are based on a part of the authors Habilitationsschrift [Polthier02Habilitationsschrift]. In Section we introduce simplicial surfaces and their function spaces. Laplace-Beltrami harmonic maps and the solution of the discrete Cauchy-Riemann equations are introduced on simplicial surfaces in Section . These maps are the basis for an iterative algorithm to compute discrete minimal and constant mean curvature surfaces which is discussed in Section . There we define the discrete mean curvature operator, derive the associate family of discrete minimal surfaces in terms of conforming and non-conforming triangles meshes, and present some recently discovered complete discrete surfaces, the family of discrete catenoids and helicoids.

Mathematics education strongly benefits from the interactivity and advanced features of the Internet. The presentation of mathematical concepts on the Internet may go far beyond what we could demonstrate in traditional mathematics textbooks. In this paper we demonstrate, in a number of examples, the additional insight into complex mathematical concepts that can be gained from 3D interactive visualization embedded into web pages. Mathematical visualization is improving our teaching environments and the communication between teachers and students.

We combine two mathematical software systems—MuPAD as a development platform and JavaView for computation and online visualization of interactive mathematics experiments. We discuss the practical aspects of online publications and show some technical details about how to develop mathematical experiments on your own. The conference presentation will demonstrate MuPAD and JavaView components in live-action.

JavaViewLib is a new Maple package combined with the JavaView visualization toolkit that adds new interactivity to Maple plots in both web pages and worksheets. It provides a superior viewing environment to enhance plots in Maple by adding several features to plots' interactivity, such as mouse-controlled scaling, translation, rotation in 2d, 3d, and 4d, auto-view modes, animation, picking, material colors, texture and transparency. The arc-ball rotation makes geometry viewing smoother and less directionally constrained than in Maple. Furthermore, it offers geometric modeling features that allow plots to be manipulated and imported into a worksheet. Several commands are available to export Maple plots to interactive web pages while keeping interactivity. JavaViewLib is available as an official Maple Powertool.

We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient are rotation-free respectively divergence-free components of the vector field. The sinks and sources respectively vortices are then automatically identified as the critical points of the corresponding scalar-valued potentials. The global nature of the decomposition avoids the approximation problem of the Jacobian and higher order tensors used in local methods, while the two potentials plus a harmonic flow component are an exact decomposition of the vector field containing all information.

In this paper we define the new alignment energy for non-conforming triangle meshes, and describes its use to compute unstable conforming discrete minimal surfaces. Our algorithm makes use of the duality between conforming and non-conforming discrete minimal surfaces which was observed earlier. In first experiments the new algorithm allows us the computation of unstable periodic discrete minimal surfaces of high numerical precision. The extraordinary precision of the discrete mesh enables us to compute the index of several triply periodic minimal surfaces.

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as discrete minimal catenoids and helicoids.

We use these discrete surfaces to study the index of unstable minimal surfaces, by numerically evaluating the spectra of their Jacobi operators. Our numerical estimates confirm known results on the index of some smooth minimal surfaces, and provide additional information regarding their area-reducing variations. The approach here deviates from other numerical investigations in that we add geometric interpretation to the discrete surfaces.

In this paper we present the implementation of a partial knot recognition algorithm as a mathematical web service on the internet. Knots may interactively be loaded and edited, and then checked for being unknotted. We use the approach by Dynnikov, which is based on a combinatorial representation of knots in a three-page book, and we use JavaView as visualization environment.

Modern mathematical visualization has always been related with special graphics workstation although visualization was always part of mathematics. Here we start from historical roots, bring interactive visualization into the classrooms and create online mathematical publications. The topics include Java applets and online videos, a new electronic journal for geometry models, an interactive mathematical dissertation and online experiments.

The archive Electronic Geometry Models is a new electronic journal for the publication of digital geometry models from a broad range of mathematical topics. The geometry models are distinguished constructions, counter examples, or results from elaborate computer experiments. Each submitted model has a self-contained textual description and is peer reviewed, and later reviewed by the Zentralblatt für Mathematik.

This paper gives in depth information about the principle ideas behind this service and discusses various technical issues. In particular, we show how XML related techniques are applied.

JavaView is a 3D geometry viewer and a numerical software library written in Java which allows one to publish interactive geometries and mathematical experiments in online web pages. Its numerical software library provides solutions and tools for problems in differential geometry and mathematical visualization. This allows the creation of one's own geometric experiments, while always profiting from the advanced visualization capabilities and the web integration. JavaView easily integrates with third-party software like Mathematica and Maple, and enables direct publication of experimental results online.

For the feature analysis of vector fields we decompose a given vector field into three components: a divergence-free, a rotation-free, and a harmonic vector field. This Hodge-type decomposition splits a vector field using a variational approach, and allows to locate sources, sinks, and vortices as extremal points of the potentials of the components. Our method applies to discrete tangential vector fields on surfaces, and is of global nature. Results are presented of applying the method to test cases and a CFD flow.

The first collection of reviewed electronic geometry models is available online at the new Internet server
http://www.eg-models.de. This archive is open for any geometer to publish new geometric models, or to browse this site for material to be used in education and research. Access to the server is free of charge.

The geometry models in this archive cover a broad range of mathematical topics from geometry, topology, and, to some extent, from numerics. Examples are geometric surfaces, algebraic surfaces, topological knots, simplicial complexes, vector fields, curves on surfaces, convex polytopes, and, in some cases, experimental data from finite element simulations.

All models of this archive are reviewed by an international team of editors. The criteria for acceptance follow the basic rules of mathematical journals and are based on the formal correctness of the data set, the technical quality, and the mathematical relevance. This strict reviewing process ensures that users of the EG-Models archive obtain reliable and enduring geometry models. For example, the availability of certified geometry models allows for the validation of numerical experiments by third parties. All models are accompanied by a suitable mathematical description. The most important models will be reviewed by the Zentralblatt für Mathematik.

EG-Models ( http://www.eg-models.de) ist ein neuartiger Internet-Server für digitale geometrische Modelle. Die grundsätzliche Funktionsweise des Archivs orientiert sich am Vorbild einer referierten mathematischen Fachzeitschrift, aber wesentliche Eigenschaften ergeben sich unmittelbar aus der Wahl des Mediums. Der Zugriff auf die Modelle des Servers ist frei.

The future of mathematical communication is strongly related with the internet. On a number of examples, the present paper gives a futuristic outlook how mathematical visualization imbedded in the internet will provide new insight into complex phenomena, influence the international cooperation of researchers, and allow to create online hyperbooks combining interactive experiments and mathematical texts.

Using the software JavaView we discuss practical aspects of online publications and give technical details on the ease of implementations. The online version of this paper is a sample interactive document with visualization examples and numerical experiments.

On a curved surface the front of a point wave evolves in concentric circles which start to overlap and branch after a certain time. This evolution is described by the geodesic flow and helps us to understand the geometry of surfaces.

In this paper we compute the evolution of distance circles on polyhedral surfaces and develop a method to visualize the set of circles, their overlapping, branching, and their temporal evolution simultaneously. We consider the evolution as an interfering wave on the surface, and extend isometric texture maps to efficiently handle the branching and overlapping of the wave.

View-dependent rendering allows interactive visualization of larger scenes. A well-known artifact is the popping problem in animations resulting from temporal differences in the level of detail between the view-dependent representations used in subsequent frames.

We solve the popping problem by computing view-dependent representations of a scene only at every n-th frame, and smoothly interpolate adjacent keyframe sections to obtain the representations for all in-between frames. Additionally, we accelerate rendering at only minor accuracy costs since interpolation is much faster than computing a view-dependent for each displayed frame.

We use scenes represented as triangle hierarchies and fulfilling a special constraint allowing for fast interpolation without remeshing. In contrast to other approaches our method naturally extends to animated scenes whose geometry and mesh may adaptively change in time.

We consider interpolation between keyframe hierarchies. We impose a set of weak constraints that allows smooth interpolation between two keyframe hierarchies in an animation or, more generally, allows the interpolation in an n-parameter family of hierarchies. We use hierarchical triangulations obtained by the Rivara element bisection algorithm and impose a weak compatibility constraint on the set of root elements of all keyframe hierarchies. We show that the introduced constraints are rather weak.

The strength of our approach is that the interpolation works in the class of conforming triangulations and simplifies the task of finding the intermediate hierarchy, which is the union of the two (, or more,) keyframe hierarchies involved in the interpolation process. This allows for an efficient generation of the intermediate connectivity and additionally ensures that the intermediate hierarchy is again a conforming hierarchy satisfying the same constraints.

Geodesic curves are the fundamental concept in geometry to generalize the idea of straight lines to curved surfaces and arbitrary manifolds. On polyhedral surfaces we introduce the notion of discrete geodesic curvature of curves and define straightest geodesics. This allows a unique solution of the initial value problem for geodesics, and therefore a unique movement in a given tangential direction, a property not available in the well-known concept of locally shortest geodesics.

An immediate application is the definition of parallel translation of vectors and a discrete Runge-Kutta method for the integration of vector fields on polyhedral surfaces. Our definitions only use intrinsic geometric properties of the polyhedral surface without reference to the underlying discrete triangulation of the surface or to an ambient space.

We present a new algorithm for computing discrete constant mean curvature surfaces in R³. It is based on the definition of a discrete version of the conjugate surface construction for CMC surfaces. Here we solve a Plateau problem for a discrete minimal surface in S³ by computing a sequence of discrete harmonic maps F : S³ to S³. The definition of a discrete conjugation allows to transform this sequence to a sequence of conjugate discrete maps which converges to a discrete CMC surface in R³. The algorithm is applicable to free boundary value problems for CMC surfaces and led to the recent discovery of new compact CMC surfaces.

We show how Wente tori and Delaunay surfaces can be used as building blocks to construct new surfaces of constant mean curvature. In a first part we give examples of periodic Wente tori and Wente tori with Delaunay ends. In a second part we study all embedded Delaunay-like surfaces with a fixed number of ends and some given reflectional symmetry.

Oorange is a virtual laboratory for experimental mathematics. It consists of a set of infrastructure services supporting the creation, execution, and dissemination of mathematical experiments. For each component of a traditional physical experiment, there is a corresponding Oorange infrastructure feature:

Object of study: High level software classes
Laboratory equipment: Foundation software classes and function libraries
Configuration of specific experiment: Computational network composed of objects
Monitor and control: Object inspection; 2D and 3D viewers
Running the experiment: Animation objects
Recording the experiment: Archiving and scripting
Disseminating result: Documentation

A hybrid language scheme underlies the design: interpreted scripts in Tcl manage tasks requiring high flexibility, while a compiled object library in Objective C supports the underlying mathematical algorithms. The resulting system is intended to be accessible to wide range of expertise levels. Oorange is free software distributed according to a GNU-like license agreement.

We describe numerical experiments that suggest the existence of compact constant mean curvature surfaces. Our surfaces come in three families with the genus ranging from 3 to 5, 7 to 10, and 3 to 9, respectively; there are further surfaces with the symmetry of the Platonic polyhedra and genera 6, 12, and 30. We use the numerical algorithm of Oberknapp and the second author that is based on a discrete version of the conjugate surface method.

We discuss the construction of triply period minimal surfaces. This includes concepts for constructing new examples as well as a discussion of numerical computations based on the new concept of discrete minimal surfaces. As a result we present a wealth of old and new examples and suggest directions for further generalizations.

We propose a simple concept for distributed computing. In a first stage this allows stand-alone programs to communicate and exchange data across computer networks. Existing stand-alone programs need very little adaptation to participate in such a system of (temporarily) connected programs. In a second stage the concept is extended to allow remote-objects and remote-methods. All network related functionality is located in the network manager, therefore adapted stand-alone programs may still run stand-alone without the network manager.

We construct new examples of compact constant mean curvature surfaces numerically. A conjugate surface method allows to explicitly construct examples. We employ the numerical algorithm of Oberknapp and Polthier based on discrete techniques to find area minimizers in the sphere S³ and to conjugate them to surfaces of constant mean curvature in R³. We compute examples of genus 5 and 30 and discuss a further example of genus 3.

We develop a new concept to extend a static interactive visualization package to a time-dependent animation environment by reusing as many as possible of the existing static classes and methods. The discussion is based on an object-oriented mathematical programming environment and is applied to parameter-dependent structures, time-dependent adaptive geometries and flow computations but most of the ideas apply to other environments in scientific visualization too. We define new classes describing dynamic processes (including e.g. time-dependent adaptive geometries) and specify a protocol mechanism they must understand. This allows the definition of a class TimeNode supervising an arbitrary dynamic processes as a time-dependent node in a formerly static data hierarchy. We discuss mechanisms of such time-dependent hierarchies, and additionally the problem of algorithms on time-dependent geometries in a number of examples.

We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R³, S³ and H³.. The algorithm makes no restriction on the genus and can handle singular triangulations. For a discrete harmonic map a conjugation process is presented leading in case of minimal surfaces additionally to instable solutions of the free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.

We prove existence of new complete embedded minimal surfaces in H³ having the symmetry of a regular tessellation by Coxeter orthoschemes. Each tetrahedron bounds a fundamental piece along four convex symmetry arcs. Its existence is proved by a conjugate surface construction.

In this note we describe the use of geometric data for the construction of triply periodic minimal surfaces in R³, S³, and H³. With a conjugate surface construction we obtain the Plateau solution of a fundamental piece for the symmetry group of the minimal surface. For some examples in R³ a method of H. Karcher and M. Wohlgemuth has led to the Weierstraß formula.

We consider discrete harmonic maps that are conforming or non-conforming piecewise linear maps, and derive a bijection between the minimizers of the two corresponding Dirichlet problems. Pairs of harmonic maps with a conforming and a non-conforming component solve the discrete Cauchy-Riemann equations, and have vanishing discrete conformal energy.

As an application, the results of this work provide a thorough understanding of the conjugation algorithms of Pinkall/Polthier and Oberknapp/Polthier used in the computation of discrete minimal and constant mean curvature surfaces.

Many interesting effects in scientific computing, such as shocks, flames, or vortex cores, are local in space and move in time. Recent numerical methods resolve these fine scales by adaptive meshes. We present a visualization approach to the handling of time-dependent data on grids with varying zones of refinement. This includes interactively working on arbitrary interpolated time cuts and an appropriate framework for vector field integration and structures for sceneries of icons representing tensor information at moving points of interest. We especially emphasize the relations between the numerical algorithms and visualization tools.

This text picks up, as a starting point, the concept developed by the authors in a previous paper and discusses applicability and extensions in the context of adaptive simulations.