This research area has been explored within the
framework of the
the National Research Network
Industrial Geometry,
funded by FWF in 20082011. Currently we are exploring
Semidiscrete Surfaces (FWFProject I705, 20122016) which
is part of the
SFBTransregio Discretization
in Geometry and Dynamics, which is funded by DFG and in part by
FWF.
Computational Differential Geometry means methods of both numerical and
discrete mathematics with the purpose of investigating and modeling curves
and surfaces. The main theme of this research project is the robust
analysis of differential properties of surfaces, the creation of discrete
and semidiscrete models of freeform surfaces, and the study of geometric
properties of such models.
It is only recently that the wealth of interesting geometry
connected to applications in, say, architecture, has come to the attention of
mathematicians, and presumably only a small part of it has been
investigated.
We are investigating topics of Discrete Differential Geometry:
discrete curvatures based on parallel meshes, quadbased and hexbased
discrete surfaces, Christoffel duality, and others. New lines
of research of semidiscrete surfaces and inverse problems
in connection with integral invariants.
The list of publications below also includes work performed before
the runtime of the abovementioned grants.


D. Pellis, M. Kilian, J. Wallner, and H. Pottmann.
Materialminimizing forms and structures.
ACM Trans. Graphics 36/6 (2017), article 173, Proc. SIGGRAPH
Asia.
[doi].

H. Pottmann and J. Wallner.
Freeform architecture and discrete differential geometry.
In W. Kropatsch, N. Artner, and I. Janusch, editors, Discrete Geometry for
Computer Imagery  DGCI 2017, volume 10502 of Lecture Notes in
Computer Science, pages 38. Springer, 2017, ISBN 9783319662718.
[doi].

H. Pottmann and J. Wallner.
Geometry and
freeform architecture.
In W. König, editor, Mathematics and Society, pages 131151.
European Math. Soc., 2016.
[MR], [doi].

C. Jiang, F. Günther, J. Wallner, and H. Pottmann.
Measuring and
controlling fairness of triangulations.
In Advances in Architectural Geometry 2016, pages 2439. vdf
Hochschulverlag, Zürich, 2016.
[doi].

C. Tang, M. Kilian, P. Bo, J. Wallner, and H. Pottmann.
Analysis and
design of curved support structures.
In Advances in Architectural Geometry 2016, pages 823. vdf
Hochschulverlag, Zürich, 2016.
[doi].

W. Carl.
On semidiscrete constant
mean curvature surfaces and their associated families.
Monatsh. Math. 182 (2017), 537563.
[doi].

W. Carl and J. Wallner.
Variational
Laplacians for semidiscrete surfaces.
Adv. Comp. Math 42 (2016), 14911509.
[MR], [doi].

C. Tang, P. Bo, J. Wallner, and H. Pottmann.
Interactive design of
developable surfaces.
ACM Trans. Graphics 35/2 (2016), #12, 112.
[doi].

O. Karpenkov.
Finite and infinitesimal flexibility of semidiscrete surfaces.
Arnold Math. J. 1/4 (2015), 403444.
[doi].

W. Carl.
A Laplace
operator on semidiscrete surfaces.
Foundations Comp. Math. 16 (2016), 11151150.
[MR], [doi].

X. Sun, C. Jiang, J. Wallner, and H. Pottmann.
Vertex normals and
face curvatures of triangle meshes.
In A. I. Bobenko, editor, Advances in Discrete Differential Geometry,
pages 267286. Springer, 2016.
[MR], [doi].

C. Jiang, J. Wang, J. Wallner, and H. Pottmann.
Freeform
honeycomb structures.
Computer Graphics Forum 33/5 (2014), 185194, Proc. Symp.
Geom. Processing.
[doi].

C. Jiang, J. Wang, P. Bompas, H. Pottmann, and J. Wallner.
Freeform shading
and lighting systems from planar quads.
In C. Gengnagel, A. Kilian, J. Nembrini, and F. Scheurer, editors,
Rethinking Prototyping  Proc. Design Modelling Symposium Berlin 2013,
pages 335346. epubli GmbH, Berlin, 2013, ISBN 9783844268454.

J. Wang, C. Jiang, P. Bompas, J. Wallner, and H. Pottmann.
Discrete line
congruences for shading and lighting.
Computer Graphics Forum 32/5 (2013), 5362, Proc. Symposium
Geometry Processing.
[doi].

C. Müller and J. Wallner.
Semidiscrete
isothermic surfaces.
Results Math. 63 (2013), 13951407.
[Zbl], [MR], [doi].

O. Karpenkov and J. Wallner.
On offsets and
curvatures for discrete and semidiscrete surfaces.
Beitr. Algebra Geom. 55 (2014), 207228.
[Zbl], [MR], [doi].

E. Vouga, M. Höbinger, J. Wallner, and H. Pottmann.
Design of
selfsupporting surfaces.
ACM Trans. Graphics 31/4 (2012), #87, 111, Proc. SIGGRAPH.
[doi].

S. Flöry, Y. Nagai, F. Isvoranu, H. Pottmann, and J. Wallner.
Ruled free
forms.
In L. Hesselgren et al., editors, Advances in Architectural Geometry
2012, pages 5766. Springer, 2013.
[doi].

J. Wallner.
On the semidiscrete
differential geometry of Asurfaces and Ksurfaces.
J. Geometry 103 (2012), 161176.
[Zbl], [MR], [doi].

M. Skopenkov, H. Pottmann, and P. Grohs.
Ruled
Laguerre minimal surfaces.
Math. Zeitschrift 272 (2012), 645674.
[doi], [arxiv:1011.0272].

B. Deng, H. Pottmann, and J. Wallner.
Functional webs for
freeform architecture.
Computer Graphics Forum 30/5 (2011), 13691378, Proc. Symp.
Geometry Processing.
[doi].

J. Wallner and H. Pottmann.
Geometric
computing for freeform architecture.
J. Math. Industry 1 (2011), #4,119.
[MR], [doi].

P. Bo, H. Pottmann, M. Kilian, W. Wang, and J. Wallner.
Circular arc
structures.
ACM Trans. Graphics 30 (2011), #101,111, Proc. SIGGRAPH.
[doi].

O. Ebner and S. Haller.
Harmonic cohomology of
symplectic fiber bundles.
Proc. Amer. Math. Soc. 139 (2011), 28272931.
[MR], [doi], [arxiv:1004.2218].

H. Pottmann and J. Wallner.
Freiformarchitektur
und Mathematik.
Mitt. DMV 18/2 (2010), 8895.
[Zbl], [MR].

H. Pottmann, Q. Huang, B. Deng, A. Schiftner, M. Kilian, L. Guibas, and J.
Wallner.
Geodesic
patterns.
ACM Trans. Graphics 29/4 (2010), #43,110, Proc. SIGGRAPH.
[doi].

C. Müller.
Conformal
hexagonal meshes.
Geom. Dedicata 154 (2011), 2746.
[doi].

A. Bobenko, H. Pottmann, and J. Wallner.
A curvature theory
for discrete surfaces based on mesh parallelity.
Math. Annalen 348 (2010), 124.
[Zbl], [MR], [doi], [arxiv:0901.4620].

A. Schiftner, M. Höbinger, J. Wallner, and H. Pottmann.
Packing circles
and spheres on surfaces.
ACM Trans. Graphics 28/5 (2009), #139,18, Proc. SIGGRAPH
Asia.
[doi].

H. Pottmann, P. Grohs, and N. J. Mitra.
Laguerre minimal surfaces, isotropic geometry and linear elasticity.
Adv. Comp. Math. 31 (2009), 391419.
[MR].

O. Karpenkov.
On the flexibility of Kokotsakis
meshes.
Geom. Dedicata 147 (2010), 1528.
[doi], [arxiv:0812.3050].

C. Müller and J. Wallner.
Oriented mixed
area and discrete minimal surfaces.
Discrete Comput. Geom. 43 (2010), 303320.
[Zbl], [MR], [doi].

J. Wallner.
Semidiscrete surface representations.
In A. Bobenko et al., editors, Discrete Differential Geometry,
Oberwolfach Reports. 2009.
Abstracts from the workshop held January 1217, 2009.
[doi].

H. Pottmann, P. Grohs, and B. Blaschitz.
Edge offset
meshes in Laguerre geometry.
Adv. Comp. Math. 33 (2010), 4573.
[MR], [doi].

H. Pottmann, A. Schiftner, and J. Wallner.
Geometry of
architectural freeform structures.
Int. Math. Nachr. 209 (2008), 1528.

H. Pottmann, J. Wallner, Q. Huang, and Y.L. Yang.
Integral invariants
for robust geometry processing.
Comput. Aided Geom. Design 26 (2009), 3760.
[Zbl], [MR], [doi].

H. Pottmann, A. Schiftner, P. Bo, H. Schmiedhofer, W. Wang, N. Baldassini, and
J. Wallner.
Freeform surfaces
from single curved panels.
ACM Trans. Graphics 27/3 (2008), #76,110, Proc. SIGGRAPH.
[doi].

H. Pottmann, Y. Liu, J. Wallner, A. Bobenko, and W. Wang.
Geometry of
multilayer freeform structures for architecture.
ACM Trans. Graphics 26/3 (2007), #65,111, Proc. SIGGRAPH.
[doi].

J. Wallner and H. Pottmann.
Infinitesimally
flexible meshes and discrete minimal surfaces.
Monatshefte Math. 153 (2008), 347365.
[Zbl], [MR], [doi].

H. Pottmann, S. BrellCokcan, and J. Wallner.
Discrete surfaces
for architectural design.
In P. Chenin, T. Lyche, and L. L. Schumaker, editors, Curves and Surface
Design: Avignon 2006, pages 213234. Nashboro Press, 2007, ISBN
9780972848275.
[Zbl], [MR].

H. Pottmann and J. Wallner.
The focal geometry
of circular and conical meshes.
Adv. Comp. Math 29 (2008), 249268.
[Zbl], [MR], [doi].

Y. Liu, H. Pottmann, J. Wallner, Y.L. Yang, and W. Wang.
Geometric modeling
with conical meshes and developable surfaces.
ACM Trans. Graphics 25/3 (2006), 681689, Proc. SIGGRAPH.
[doi].

W. Wang, J. Wallner, and Y. Liu.
An angle criterion
for conical mesh vertices.
J. Geometry Graphics 11 (2007), 199208.
[Zbl], [MR].

J. Wallner.
Computing
quadrilateral and conical meshes.
In A. Bobenko et al., editors, Discrete Differential Geometry, volume 3
of Oberwolfach Reports. 2006.
Abstracts from the workshop held March 610, 2006.
[Zbl].

H. Pottmann, J. Wallner, Y.L. Yang, Y.K. Lai, and S.M. Hu.
Principal
curvatures from the integral invariant viewpoint.
Comput. Aided Geom. Design 24 (2007), 428442.
[MR], [doi].
