


Georg Umlauf's Publications  



2010  
Adaptive and freaturepreserving subdivision for highquality tetrahedral meshes. Burkhart, Hamann, Umlauf To appear: Computer Graphics Forum, 2010. We present an adaptive
subdivision scheme for unstructured
tetrahedral meshes inspired by the sqrt(3)subdivision scheme for
triangular meshes. Existing tetrahedral subdivision schemes do
not support adaptive refinement and have traditionally been driven by
the need to generate smooth threedimensional deformations of solids.
These schemes use edge bisections to subdivide tetrahedra, which
generates octahedra in addition to tetrahedra. To split octahedra into
tetrahedra one routinely chooses a direction for the diagonals for the
subdivision step. We propose a new topologybased refinement operator
that generates only tetrahedra and
supports adaptive refinement. Our tetrahedral subdivision algorithm is
motivated by the need to have one representation for the modeling,
the simulation and the visualization and so to bridge the gap
between CAD and CAE. Our subdivision algorithm design emphasizes on
geometric quality of the tetrahedral meshes, local and adaptive
refinement operations, and preservation of sharp geometric features on
the boundary and in the interior of the physical domain.
Available formats: pdf Citation: BibTeX 

Realtime triangulation of point streams. Denker, Lehner, Umlauf To appear: Engineering with Computers, 2010. Handheld
laser
scanners
are
commonly
used
in
industry for reverse engineering and quality measurements. In
this
process,
it
is
difficult
for
the
human
operator to scan
the target object completely and uniformly. Therefore, an interactive
triangulation
of
the
scanned points can assist the operator
in
this
task.
Available
formats:
pdfIn this paper we describe the technical and implementational details of our realtime triangulation approach for point streams, presented at the 17th International Meshing Roundtable. Our method computes a triangulation of the point stream generated by the laser scanner online, i.e., the data points are added to the triangulation as they are received from the scanner. Multiple scanned areas and areas with a higher point density result in a finer mesh and a higher accuracy. On the other hand, the vertex density adapts to the estimated surface curvature. To guide the operator the resulting triangulation is rendered with a visualization of its uncertainty and the display of an optimal scanning direction. Citation: BibTeX 

Geometric properties of the adaptive Delauney triangulation. Bobach, Constantiniu, Steinmann, Umlauf To appear: Mathematical Methods for Curves and Surfaces, 2010. Recently, the
Adaptive Delaunay Tessellation (ADT) was introduced in the context of
computational mechanics as a tool to support Voronoibased nodal
integration schemes in the finite element method. While focusing on
applications in mechanical engineering, the former presentation lacked
rigorous proofs for the claimed geometric properties of the Adt
necessary for the computation of the nodal integration scheme. This
paper gives the missing proofs for the three main claims which are
uniqueness of the ADT, connectedness of the ADT, and coverage of the
Voronoi tiles by adjacent ADT tiles. Furthermore, this paper provides a
critical assessment of the ADT for arbitrary point sets.
Available formats: pdf Citation: BibTeX 

Generalized swap operation for tetrahedrizations. Lehner, Hamann, Umlauf To appear in: H.Hagen, A.Knoll (eds.): Proceedings of NN, 2010. Mesh optimization of
2D and 3D triangulations is used in multiple applications extensively.
For example, mesh optimization is crucial in the context of adaptively
discretizing geometry, typically representing the geometrical boundary
conditions of a numerical simulation, or adaptively
discretizing the entire space over which various dependent variables of
a numerical simulation must be approximated. Together with operations
applied to the vertices the socalled edge or face swap operations are the building block
of all optimization approaches. To speed up the optimization or to
avoid local minima of the function measuring overall mesh quality these
swaps are combined to generalized swap operations with a less local
impact on the triangulation. Despite the fact that these swap
operations change only the connectivity of a triangulation, it depends
on the geometry of the triangulation whether the generalized swap will
generate inconsistently oriented or degenerate simplices. Because these
are undesirable for numerical reasons, this paper is concerned with
geometric criteria that guarantee the generalized swaps for a 3D
triangulation to yield only valid, nondegenerate triangulations.


2009  
Natural neighbor extrapolation using ghost points. Bobach, Farin, Hansford, Umlauf Computer AidedDesign, 41(5): 350365, 2009. Among locally
supported
scattered data schemes, natural neighbor interpolation has some unique
features that makes it interesting for a range of applications.
However, its restriction to the convex hull of the data sites is a
limitation that has not yet been satisfyingly overcome. We use this
setting to discuss some aspects of scattered data extrapolation in
general, compare existing methods, and propose a framework for the extrapolation
of natural neighbor interpolants on the basis of dynamic ghost points.
Available formats: pdf Citation: BibTeX 

2008  
Online triangulation of laserscan data. Denker, Lehner, Umlauf In: R. Garimella (ed.), Proceedings of the 17th International Meshing Roundtable, SpringerVerlag, 415432, 2008. Handheld
laser
scanners
are
used
massively
in
industry
for
reverse
engineering
and quality measurements. In this process,
it is difficult for the human operator to cover the scanned object
completely and uniformly. Therefore, an interactive triangulation of
the scanned surface points can assist the human operator in this task.
Our method computes a triangulation of the point stream generated by the laser scanner online, i.e., the data points are added to the triangulation as they are received from the scanner. Multiple scanned areas and areas with a higher point density result in a finer mesh and a higher accuracy. On the other hand, the vertex density adapts to the estimated surface curvature. To assist the human operator the resulting triangulation is rendered with a visualization of its faithfulness. Additionally, our triangulation method allows for a levelofdetail representation to reduce the mesh complexity for fast rendering on lowcost graphics hardware. Available formats: pdf Citation: BibTeX Videos: avi 

Video compression using datadependent triangulations. Lehner, Umlauf, Hamann In: Y. Xiao, E. ten Thij (eds.), Computer Graphics and Visualization '08, 244248, 2008. Best shortpaper award. We present a method for
compression of video
clips using datadependent triangulations. This technique utilizes the
time coherence of a video to transfer information from one frame to the
next, reducing the computation time for the compression. The results of
this method are compared to those obtained with MJPEG and MPEG2.
Available formats: pdf Citation: BibTeX Videos: avi 

The adaptive Delaunay tessellation: A neighborhood covering meshing technique. Constantiniu, Steinmann, Bobach, Farin, Umlauf Computational Mechanics, 42(5), 655669, 2008. In this paper we
propose an unstructured
hybrid tessellation of a scattered point set
that
minimally covers the proximal space around each point. The mesh is
automatically obtained in a bounded period of time by transforming an
initial Delaunay tessellation. Novel types of polygonal interpolants
are
used for interpolation applications and geometric quantities of the
elements make them also useful for discretization schemes. The approach
proves to be superior to classical Delaunay one in a finite element
context.
Available formats: pdf Citation: BibTeX 


Local energyoptimizing subdivision algorithms. Ginkel, Umlauf Computer Aided Geometric Design, 25(3): 137147, 2008. In this paper a
method is presented to fair the limit surface of a subdivision
algorithm
locally around an extraordinary point. The dominant six eigenvalues of
the
subdivision matrix have to satisfy linear and quadratic equality and
inequalityconstraints in order to guarantee normalcontinuity and
bounded curvature at the extraordinary point. All other eigenvalues
can be chosen arbitrarily within certain intervals and therefore can be
used
to optimize
the shape of the subdivision surface by minimizing quadratic energy
functionals.
Additionally, if the sub and subsubdominant eigenvalues vary within
predefined intervals, C^{1}regularity
of
the
surface
and
locality
of
the
stencils
can be guaranteed, although
eigenvectors are changed.
Available formats: pdf Citation: BibTeX 

Symmetry of shape charts. Ginkel, Umlauf Computer Aided Geometric Design, 24(3): 131136, 2008. For subdivision
surfaces, the socalled shape chart can be used to characterize the
curvature behavior at an extraordinary point a priori from the
initial control net. Of late, it has been used in different approaches
to tune subdivision algorithms to handle the socalled hybrid
shapes. For this the shape charts are computed numerically. In this
paper, symmetries of shape charts are analyzed that can be used to
simplify the computations and to reduce the computation costs
significantly.
Available formats: pdf Citation: BibTeX 

2007  
Discrete harmonic functions from local coordinates. Bobach, Farin, Hansford, Umlauf In: R. Martin, M. Sabin, J. Winkler (eds.), Mathematics of Surfaces XII, SpringerVerlag, 93103, 2007. In this work we focus
on approximations of continuous
harmonic functions by discrete
harmonic functions based on the discrete Laplacian in a triangulation
of a point set. We show how the choice
of edge weights
based on generalized barycentric coordinates influences the
approximation quality of discrete harmonic functions. Furthermore, we
consider a
varying point set to demonstrate that generalized barycentric
coordinates based on natural neighbors admit discrete harmonic
functions that
continuously depend on the point set.
Citation: BibTeX 

Natural neighbor concepts in scattered data interpolation and discrete function approximation. Bobach, Umlauf In: H. Hagen, M. HeringBertram, C. Garth (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 2335, 2007. The concept of natural neighbors employs the notion of
distance to define local neighborhoods in discrete data. Especially
when querying and accessing large scale data, it
is important to limit the amount of data that has to be
processed for an answer. Because of its implicit definition on
distances, the natural neighbor concept is extremely well suited to
provide meaningful neighborhoods in spatial data with a scattered,
inhomogeneous distribution.
This paper revisits some unique properties of natural neighbor based methods and summarizes important findings for their successful application to scattered data interpolation, and the computation of discrete harmonic functions. Citation: BibTeX 

Analyzing a generalized Loop subdivision scheme. Ginkel, Umlauf Computing, 79(24), 353363, 2007. In this paper a class
subdivision schemes generalizing the algorithm of Loop is presented.
The stencils have same support as those from the algorithm of Loop, but
allow a variety of weights. By varying the weights a class of C^{1}
regular subdivision schemes is obtained. This class includes the
algorithm of Loop and the midpoint schemes of order one and two for
triangular nets. The proof of C^{1}
regularity of the limit surface for arbitrary triangular nets is
provided for any choice of feasible weights.
The purpose of this generalization of the subdivision algorithm of Loop is to demonstrate the capabilities of the applied analysis technique. Since this class includes schemes that do not generalize box spline subdivision, the analysis of the characteristic map is done with a technique that does not need an explicit piecewise polynomial representation. This technique is computationally simple and can be used to analyze classes of subdivision schemes. It extends previously presented techniques based on geometric criteria. Available formats: pdf Citation: BibTeX 

Tuning subdivision algorithms using constrained energy minimization. Ginkel, Umlauf In: R. Martin, M. Sabin, J. Winkler (eds.), Mathematics of Surfaces XII, SpringerVerlag, 166176, 2007. In this paper a method
is presented to fair the limit surface of a subdivision algorithm
around an extraordinary point. The eigenvalues and
eigenvectors of the subdivision matrix determine the continuity and
shape of the limit surface. The dominant, subdominant and
subsubdominant eigenvalues should satisfy linear and quadratic
equality and inequalityconstraints to guarantee continuous normal and
bounded curvature globally. The remaining eigenvalues need only satisfy
linear inequalityconstraints. In general, except for the dominant
eigenvalue, all eigenvalues can be used to optimize the shape of the
limit surface with our method.
Available formats: pdf Citation: BibTeX 

Normals of subdivision surfaces and their control polyhedra. Ginkel, Peters, Umlauf Computer Aided Geometric Design, 24(2): 112116, 2007. For planar spline curves
and bivariate boxspline functions, the cone of normals of a polynomial
spline piece is enclosed by the cone of normals of its spline control
polyhedron. This note collects some concrete examples to show that this
is not true for subdivision surfaces, both at extraordinary points and
in the regular, boxspline setting. A larger set, the cross products of
families of control net edges, has to be considered.
Available formats: pdf Citation: BibTeX 

Image compression using datadependent triangulations. Lehner, Umlauf, Hamann In: G. Bebis et al. (eds.), Advances in Visual Computing, Part I, SpringerVerlag, 352362, 2007. We present a method to
speed up the computation of a highquality
datadependent
triangulation
approximating
an
image
using
simulated
annealing
by
probability distributions guided by local approximation
error and its variance. The triangulation encodes the image, yielding
compression rates comparable to or even superior to JPEG and JPEG2000
compression.
The specific contributions of our paper are a speedup of the simulated annealing optimization and a comparison of our approach to other image approximation and compression methods. Furthermore, we propose an adaptive vertex insertion/removal strategy and termination criteria for the simulated annealing to achieve specified approximation error bounds. Available formats: pdf Citation: BibTeX 

Survey of techniques for datadependent triangulations. Lehner, Umlauf, Hamann In: H. Hagen, M. HeringBertram, C. Garth (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 178187, 2007. We present a survey of different
techniques to approximate a color image using a piecewise linear
interpolation induced by a triangulation of the image domain. We also
include a detailed description of a method we designed. We give a short
overview of possible applications and extensions.
Available formats: pdf Citation: BibTeX 

SoPC for 3D point rendering. Middendorf, Mühlbauer, Bobda, Umlauf To appear: DASIP, 2007. Embedded vertex shader in FPGA. Middendorf, Mühlbauer, Umlauf, Bobda In: A. Rettberg et al. (eds.), Embedded System Design: Topics, Techniques and Trendes, Springer, 155164, 2007. Realtime
3D visualization of objects
or information becomes increasingly important in everyday life e.g. in
cellular phones or mobile systems. Care should be taken in the design
and implementation
of 3D rendering in such embedded devices like handhelds devices in
order to meet the performance requirement,
while maintaining power consumption low. In this work, the design and
implementation of a vertex shader on a reconfigurable
hardware is presented.
The main focus is placed on the efficient
hardware/software partitioning of the vertex
shader computation, in order
to maximize the performance while maintaining a high flexibility. The
resulting solution must be compatible to existing vertex shaders in
oder to allow the large
amount of existing program to be easily ported to our platform.
A prototype
consisting of a PowerPC, peripherals and some custom hardware modules
is
realized a on
an FPGAboard. The implementation of a point rendering
shows considerable speed up compared to a pure software solution.
Available formats: pdf 

Comparison of Voronoi based scattered data interpolation schemes. Bobach, Bertram, Umlauf In: J.J. Villanueva (ed.), Proceedings of International Conference on Visualization, Imaging and Image Processing, 342349, 2006. Voronoi based
interpolation employs the concept of natural neighbors to define an
interpolating function over discrete data known at scattered sample
points. In this work we review the two main concepts for improving
interpolation continuity inside the convex hull of the sample domain
and compare four natural neighbor interpolants of C^{1}
and C^{2}
continuity. We give a visual presentation of all interpolants to
provide insight into their overall behavior in addition to a comparison
of their analytical and practical properties.
Available formats: pdf Citation: BibTeX 

Issues and implementation of C^{1} and C^{2} natural neighbor interpolation. Bobach, Bertram, Umlauf In: G. Bebis et al. (eds.), Advances in Visual Computing, Part II, SpringerVerlag, 186195, 2006. Smooth local coordinates have been
proposed by Hiyoshi and Sugihara 2000 to improve the classical Sibson’s
and Laplace coordinates. These smooth local coordinates are computed by
integrating geometric quantities over weights in the power diagram. In
this paper we describe how to efficiently& implement the Voronoi
based C^{2} local coordinates. The globally C^{2} interpolant that Hiyoshi and Sugihara
presented in 2004 is then compared to Sibson’s and Farin’s C^{1} interpolants when applied toscattered
data interpolation.
Available formats: pdf Citation: BibTeX 

Natural neighbor interpolation and order of continuity. Bobach, Umlauf In: H. Hagen, A. Kerren, P. Dannenmann (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 6986, 2006. In this paper we give a
survey on natural neighbor based interpolation, a class of local
scattered data interpolation schemes that define their support on
natural neighbors in the Voronoi diagram of the input data sites. We
discuss the existing work with respect to common aspects of scattered
data interpolation and focus on smoothness of the interpolant.
Available formats: pdf Citation: BibTeX 

Controlling a subdivision tuning method. Ginkel, Umlauf In: A. Cohen, J.L. Merrien, L.L. Schumaker (eds.), Curve and Surface Fitting, 170179, 2006. In this paper the
problem of curvature behavior around extraordinary points of a Loop
subdivision surface is addressed. For subdivision surfaces,
configurations of the initial control net exist in which neither the
elliptic nor the hyperbolic component of the initial control net
becomes dominant. This leads to socalled hybrid shapes which often
cause visible artifacts. A solution to this problem is to split it into
two parts: first an eigenvalue tuning to allow for bounded Gauss
curvature with arbitrary sign and, second, an eigencoefficient tuning
to avoid hybrid shapes. The techniques for eigencoefficient tuning will
now be analyzed in detail. The analysis allows to quantify the
difference between the original and the modified surfaces.
Additionally, extensions to the eigencoefficient tuning techniques are
given to solve various problems that might be imposed by the topology
of the initial control net.
Available formats: pdf Citation: BibTeX 

Loop subdivision with curvature control. Ginkel, Umlauf In: K. Polthier, A. Sheffer (eds.), Eurographics Symposium on Geometry Processing, Eurographics Association, 163171, 2006. In this paper the
problem of curvature behavior around extraordinary points of a Loop
subdivision surface is addressed. A variant of Loop’s algorithm with
small stencils is used that generates surfaces with bounded curvature
and prescribed elliptic or hyperbolic behavior. We present two
different techniques that avoid the occurrence of hybrid
configurations, so that an elliptic or hyperbolic shape can be
guaranteed. The first technique uses a symmetric modification of the
initial controlnet to avoid hybrid shapes in the vicinity of an
extraordinary point. To keep the difference between the original and
the modified mesh as small as possible the changes are formulated as
correction stencils and spread to a finite number of subdivision steps.
The second technique is based on local optimization in the frequency
domain. It provides more degrees of freedom and so more control over
the global shape.
Available formats: pdf Citation: BibTeX 

Topographic distance functions for interpolation of meteorological data. Lehner, Umlauf, Hamann, Ustin In: H. Hagen, A. Kerren, P. Dannenmann (eds.), GI Lecture Notes in Informatics, Visualization of Large and Unstructured Data Sets, 119131, 2006. The reference
evapotranspiration ET0 is an important meteorological quantity in
agriculture and water resource management. It is usually estimated from
other meteorological quantities measured at weather stations. To
estimate ET0 at an arbitrary geographical position these quantities
must be interpolated. The Center for Spatial Technologies And Remote
Sensing (CSTARS) at UC Davis uses the DayMet approach for this task.
We discuss some inconsistencies within the DayMet approach and suggest improvements. One significant problem of DayMet is the lack of consideration of terrain topography. We define new distance functions that are elevationdependent and show preliminary results of the comparison of the classic and the improved DayMet approach. Available formats: pdf Citation: BibTeX 

Parametrizations for triangular G^{k} spline surfaces of low degree. Prautzsch, Umlauf ACM Transactions on Graphics, 25(4), 12811293, 2006. In this paper, we
present regularly parametrized G^{k}
free form spline surfaces that extend box and halfbox splines over
regular triangular grids. The polynomial degree of these splines is
max{4k+1; floor(3k/2 +1)}, where r \in N can be chosen arbitrarily
and determines the flexibility at extraordinary points.
The G^{k}
splines presented in this paper depend crucially on lowdegree
(re)parametrizations of piecewise polynomial hole fillings. The
explicit construction of such parametrizations forms the core of this
paper and we present two classes of singular and regular
parametrizations. Also we show how to build box and halfbox spline
surfaces of arbitrarily high smoothness with holes bounded by only n
patches in principle.
Available formats: pdf Citation: BibTeX 

2005 

On normals and control nets. Ginkel, Peters, Umlauf In: R. Martin, H. Bez, M. Sabin (eds.): The Mathematics of Surfaces XI, SpringerVerlag, 2005, 233239. This paper characterizes when the normals of a spline curve or spline surface lie in the more easily computed cone of the normals of the segments of the spline control net. Available formats: pdf Citation: BibTeX 

Analysis and tuning of subdivision schemes. Umlauf In: B. Jüttler (ed.), Proceedings of Spring Conference on Computer Graphics SCCG 2005, ACM Press, 3340, 2005. This paper surveys
the current state in analyzing and tuning of subdivision algorithms.
These two aspects of subdivision algorithms are very much
intertwined with the differential geometry of the subdivision surface.
This paper deals with the interconnection of these different aspects of
subdivision algorithms and surfaces. The principal idea for the
analysis of a subdivision algorithm dates back to the late 70s although
the overall technique is only well understood since the early 90s. Most
subdivision algorithms are analyzed today but the proofs involve time
consuming computations. Only recently simple proofs for a certain class
of subdivision algorithms were developed that are based on geometric
reasoning. This allows for easier smoothness proofs for new developed
or tuned subdivision algorithms.
The analysis of the classical algorithms, such CatmullClark, Loop, etc., shows that the subdivision surfaces at the extraordinary points are not as smooth as the rest of the surface. It was also shown that the subdivision surfaces of these classical algorithms cannot model certain basic shapes. One way to tune a stationary subdivision algorithms to overcome this problem is to drop the stationarity while at the same time using the smoothness proof of the stationary algorithms. Available formats: pdf Citation: BibTeX 

2004 

Postprocessing polygonal voxel data from numerical simulation. Bobach, Umlauf Interner Bericht 332/04, Fachbereich Informatik, TU Kaiserslautern, 2004. Many applications
dealing with geometry acquisition and processing produce polygonal
meshes that carry artifacts like discretization noise. While there are
many approaches to remove the artifacts by smoothing or filtering the
mesh, they are not tailored to any specific application demanding
certain restrictive constraints. We show how to incorporate smoothing
schemes based on the general Laplacian approximation to process the
results of flow simulation in car manufacturing.
In the presented application setting the major restrictions come from the bounding volume of the flow simulation, the socalled installation space. In particular, clean mesh regions (without noise) should not be smoothed while at the same time the installation space must not be violated by the smoothing of the noisy mesh regions. Additionally, aliasing effects at the boundary between clean and noisy mesh regions must be prevented. To address the fact that the meshes come from flow simulation, the presented method is versatile enough to preserve their exact volume and to apply anisotropic filters using the flow information. Available formats: pdf Citation: BibTeX 

A technique for verifying the smoothness of subdivision schemes. Umlauf In: M.L. Lucian, M. Neamtu (eds.), Geometric Modeling and Computing: Seattle 2003, Nashboro Press, 513521, 2004. In recent years,
subdivision schemes for surfaces of arbitrary topology were developed
that do not generalize boxsplines subdivision schemes. Examples for
this kind of subdivision schemes are the sqrt(3)scheme and certain
averaging schemes for triangular and hexahedral nets. In order to
analyze the smoothness of the limit surface, it is necessary to know if
its characteristic map is regular and injective. For boxspline based
schemes this can be done based on the explicit piecewise polynomial
representation. In this paper, a general approach is introduced
that allows analyzing the characteristic map even if no explicit
representation is available. The proposed technique requires that the
first divided differences schemes are scalar and use convex
combinations. Then simple geometric properties of the subdominant
eigenvectors of the subdivision matrix can be used to prove regularity
and injectivity of the characteristic map for any valence. This is
demonstrated for a midpoint scheme for triangular nets.
Available formats: pdf Citation: BibTeX 

before 2003 

Computing curvature bounds for bounded curvature subdivision. Peters, Umlauf Computer Aided Geometric Design, 18 (5): 455462, 2001. For a stationary,
affine invariant, symmetric, linear and local subdivision scheme that
is C^{2}
apart from the
extraordinary point the curvature is bounded if the square of the
subdominant eigenvalue equals the subsubdominant eigenvalue. This paper
gives a technique for quickly establishing an interval to which the
curvature is confined at the extraordinary point. The approach
factors the work into precomputed intervals that depend only on the
scheme and a meshspecific component. In many cases the intervals are
tight enough to be used as a test of shapefaithfulness of the given
subdivision scheme; i.e. if the limit surface in the neighborhood of
the extraordinary point of the subdivision scheme is
consistent with the geometry implied by the input mesh.
Available formats: pdf Citation: BibTeX 

Triangular G^{k} splines. Prautzsch, Umlauf Interner Bericht Nr. 20018, Fakultät für Informatik, Universität Karlsruhe, 2001. In this paper a new
approach is presented to construct piecewise polynomial G^{k}surfaces
of arbitrary topology and smoothness order k>0 of degree O(k).
This
approach
generalizes
some
results
presented
in
1997
in CAGD and in
1999 at the SaintMalo conference, respectively. In our construction
only 4n polynomial patches are
needed to fill an nsided
hole in a generalized C^{k}(half)box
spline
surface.
This
is
achieved
by
coalescing
certain
control points
while at the same time maintaining a regular parametrization.


Gaussian and mean curvature of subdivision surfaces. Peters, Umlauf In: R. Cipolla, R. Martin (eds.), The Mathematics of Surfaces IX, SpringerVerlag, 5969, 2000. By explicitly deriving
the curvature of subdivision surfaces in the extraordinary points, we
give an alternative, more direct account of the criteria
necessary and sufficient for achieving curvature continuity than
earlier approaches that locally parametrize the surface by
eigenfunctions. The approach allows us to rederive and thus survey the
important lower bound results on piecewise polynomial subdivision
surfaces by Prautzsch, Reif, Sabin and Zorin, as well as explain the
beauty of curvature continuous constructions like Prautzsch’s. The
parametrization neutral perspective gives also additional insights into
the inherent constraints and stiffness of subdivision surfaces.
Available formats: pdf Citation: BibTeX 

A G^{1} and a G^{2} subdivision scheme for triangular nets. Prautzsch, Umlauf International Journal on Shape Modelling, 6(1): 2135, 2000. In this article we
improve the butterfly and Loop's algorithm. As a result we obtain
subdivision algorithms for triangular nets which can be used to
generate G^{1}
and
G^{2}surfaces,
respectively.
Citation: BibTeX 

Analyzing the characteristic map of triangular subdivision schemes. Umlauf Constructive Approximation, 16(1): 145155, 2000. Tools for the analysis
of generalized triangular box spline subdivision schemes are developed.
For the first time the full analysis of Loop's algorithm can be carried
out with these tools.
Available formats: pdf Citation: BibTeX 

Glatte Freiformflächen und optimierte Unterteilungsalgorithmen. Umlauf April 1999. Verlag Shaker, Aachen, 1999. In German. 

Triangular G^{2} splines. Prautzsch ,Umlauf In: P.L. Laurent, P. Sablonniere, L.L. Schumaker (eds.), Curve and Surface Design, Vanderbilt University Press, 335342, 1999. We introduce curvature
continuous regular freeform surfaces with triangular control nets.
These surfaces are composed of quartic box spline surfaces and are
piecewise polynomial multisided patches of total degree 8 which
minimize some energy integral. The Bézier nets can be computed
efficiently form the spline control net by some fixed masks, i.e.
matrix multiplications.
Available formats: pdf Citation: BibTeX 

Improved triangular subdivision schemes. Prautzsch, Umlauf In: F.E. Wolter and N.M. Patrikalakis (eds.), Proceedings of the CGI '98, Hannover, 626632, 1998. In this article we improve the butterfly and Loop’s algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G^{1} and G^{2}surfaces, respectively. Available formats: pdf Citation: BibTeX 

Konstruktion von G^{k}Flächen. Umlauf In: T. Ertl (ed.), "Trends und Höhepunkte der Graphischen Datenverarbeitung", Gesellschaft für Informatik, FA 4.1, Bonn, 1998. In German. 

A G^{2}subdivision algorithm. Prautzsch, Umlauf Computing, 13: 217224, 1998. In this paper we present
a method to optimize the smoothness order of subdivision algorithms
generating surfaces of arbitrary topology. In particular
we
construct
a
subdivision
algorithm
with
some
negative
weights
producing G^{2}surfaces.
These
surfaces
are
piecewise
bicubic and are flat at their
extraordinary points. The underlying ideas can also be used to improve
the smoothness order of subdivision algorithms for surfaces of higher
degree or triangular nets.
Available formats: pdf Citation: BibTeX 

