To appear in the ACM SIGGRAPH conference proceedings
Packing circles and spheres on surfaces
Alexander Schiftner∗
Evolute, TU Wien
Mathias Ho¨binger†
TU Wien
Johannes Wallner‡
TU Graz
Helmut Pottmann§
KAUST, TU Wien
Figure 1: Resolving a freeform shape into circle-bearing hexagons. This is one of the many possible structures derived from a newly
introduced type of triangle mesh, where the incircles of triangular faces form a packing. Such derived structures enjoy interesting properties
relevant to both geometry and building construction.
Abstract
Inspired by freeform designs in architecture which involve circles
and spheres, we introduce a new kind of triangle mesh whose faces’
incircles form a packing. As it turns out, such meshes have a rich
geometry and allow us to cover surfaces with circle patterns, sphere
packings, approximate circle packings, hexagonal meshes which
carry a torsion-free support structure, hybrid tri-hex meshes, and
others. We show how triangle meshes can be optimized so as to
have the incircle packing property. We explain their relation to con-
formal geometry and implications on solvability of optimization.
The examples we give confirm that this kind of meshes is a rich
source of geometric structures relevant to architectural geometry.
CR Categories: I.3.5 [Computer Graphics]: Computational Ge-
ometry and Object Modeling—Geometric algorithms, languages,
and systems; I.3.5 [Computer Graphics]: Computational Geometry
and Object Modeling—Curve, surface, solid, and object represen-
tations
Keywords: computational differential geometry, architectural ge-
ometry, computational conformal geometry, freeform surface, cir-
cle packing, sphere packing, supporting structures.
∗email: schiftner@evolute.at
†email: mathias.hoebinger@gmail.com
‡email: j.wallner@tugraz.at
§email: helmut.pottmann@kaust.edu.sa
1 Introduction
The motivation for the present paper comes from architectural
projects which contain packing-like arrangements of circles on sur-
faces (see Fig. 2). In view of the beautiful geometry of circle pack-
ings in the plane and on the sphere (see [Stephenson 2005]), the
question arises whether one can extend these results to surfaces.
This is the main topic of our paper.
We present a new type of triangle meshes, where the incircles of
the triangles form a packing, i.e., the incircles of two triangles with
a common edge have the same contact point on that edge (Figures
3 and 4). These circle packing (CP) meshes exhibit an aesthetic
balance of shape and size of their faces and thus are great can-
didates for applications in architecture. They are closely tied to
sphere packings on surfaces and to various remarkable structures
and patterns which are also of interest in art, architecture, and de-
sign. The focus of our work lies on computation and applications.
Some of the underlying mathematics seems to be very difficult, so
in some places we cannot offer proofs but only numerical evidence.
Figure 2: Arrangement of circles in freeform architecture. This
image shows two differnt views of the Selfridges building, Birm-
ingham, by Future Systems (right hand image courtesy P. Terjan).
1

To appear in the ACM SIGGRAPH conference proceedings
Figure 3: A CP mesh is a triangle mesh whose incircles (orange)
form a packing. Then spheres (blue), which are centered at mesh
vertices and are orthogonal to the incircles of neighboring triangles
form a packing, too. Centers and axes of incircles define a hexago-
nal support structure with torsion-free nodes (red, see Section 3).
Related work. The present paper revolves around meshes which
are endowed with additional circles or spheres. Relevant prior work
therefore is research on circle packings, conformal mappings, and
their geometry processing applications.
Complex analysis received a major impulse when W. Thurston pro-
posed to study conformal mappings via circle packings. For in-
stance, He and Schramm [1993] used this device to show the con-
formal equivalence of a quite general class of domains to ‘circle do-
mains’ whose boundary is a possibly infinite collection of circles,
thus proving an old conjecture of Koebe’s. Circle packings in gen-
eral and especially their interpretation as discrete analytic functions
/ discrete conformal mappings are beautifully described in textbook
form by K. Stephenson [2005]. For other circle patterns as discrete
conformal mappings, see [Bobenko and Suris 2008].
Polyhedral surfaces endowed with circles are studied extensively
[Gu and Yau 2008; Jin et al. 2008; Luo et al. 2008; Jin et al. 2009;
Yang et al. 2009], in the context of topics like circle packing metrics
and applications to discrete Ricci flow and optimized conformal
parametrization. The CP meshes of the present paper are related
in so far as they are meshes where the induced metric is a circle
packing metric.
The Koebe meshes of Bobenko and Springborn [2004] have the CP
property, and so do the isothermic meshes and discrete minimal
surfaces of Bobenko et al. [2006]. In contrast to our work these
are quad meshes with planar faces, describing isothermic surfaces
(which means shape restrictions). All these constructions implicitly
deal with conformal mappings, at least approximately so. An exact
notion of conformal equivalence for triangle meshes appeared first
in [Luo 2004]; it has recently been studied by [Springborn et al.
2008] for conformal flattening of surfaces whose topological type
is a disk or sphere.
It recently turned out that both geometry and computational math-
ematics bear a great potential to advance the field of freeform ar-
chitecture. The large number of unsolved problems and the in-
creasing level of complexity in design are the source of the new
research area Architectural Geometry, currently emerging at the
border of differential geometry, computational mathematics and
architectural design/engineering (see for example the monograph
[Pottmann et al. 2007a] and the proceedings volume [Pottmann
et al. 2008]). On a technical level, this paper deals with meshes
with planar faces which approximate freeform shapes, as well as
with meshes which allow for a torsion-free support structure. This
means the layout of beams following the edges of a mesh, such that
symmetry planes of beams intersect at node axes passing through
the vertices (for these node axes, see [Pottmann et al. 2007c] and
Figures 10, 18).
Contributions. The notion of CP mesh (i.e., a triangle mesh
whose incircles constitute a packing) generalizes – to the extent
possible – the notion of planar circle packing to the surface case.
Our main focus is optimization algorithms for the creation of CP
meshes and to derive from them geometric structures relevant to ar-
chitectural geometry. We do not attempt the difficult task of recre-
ating the strong theory available in the planar case but nevertheless
employ the conformal geometry of surfaces when explaining solv-
ability of optimization. We believe that this research on meshes
combined with circle packings, sphere packings, and torsion-free
support structures is a substantial contribution to the processing of
freeform geometry which goes beyond meshes and which is impor-
tant for applications. In particular, our main results are:
• The geometry and optimization of CP meshes and their relation
to conformal geometry.
• Hybrid meshes (tri-hex and others) derived from CP meshes
which exhibit geometric properties relevant to architecture and
building construction (such as planar faces, design flexibility, struc-
tural properties). Other kinds of derived meshes allow for a support
structure with torsion-free nodes.
• Approximate circle packings and patterns on arbitrary freeform
shapes.
2 Triangle meshes with an incircle packing
We start our study of CP meshes with some basic relations and
elementary geometric properties. Consider a triangle with ver-
tices v1,v2,v3, whose edges are tangent to the incircle at points
t12, t23, t13 (see Fig. 4). The distances of these contact points
to the vertices vi are denoted by ri, e.g., r1 = ‖v1 − t12‖ =
‖v1 − t13‖. With the edge lengths lij := ‖vi − vj‖, we have
lij = ri + rj , 2ri = lij + lik − ljk. (1)
Associated packing of spheres. Consider the star of triangles ad-
jacent to the vertex vi. For a CP mesh, all contact points of in-
circles on edges through vi must lie at the same distance ri from
vi. Hence, ri is a value associated with a vertex in the triangula-
tion. Viewing ri as radius of a sphere Si with center vi, we arrive
at a sphere packing (Fig. 3): Adjacent spheres Si, Sj touch at tij .
Moreover, the sphere Si is orthogonal to all incircles of triangles
which share the vertex vi. Hence, we can say that the sphere pack-
ing is orthogonal to the packing of incircles.
Remark: This process of associating incircles and orthogonal
spheres to a CP mesh can be reversed: it is clear that any collec-
tion of circles and orthogonal spheres yields a CP mesh, with one
face per circle, and one vertex per sphere. When representing a CP
mesh by this circle-sphere collection instead of its vertices, it be-
comes an object of Mo¨bius geometry (see [Cecil 1992]): A Mo¨bius
Figure 4: We introduce
notation for a pair of
adjacent triangles in a
CP mesh: Vertices vj ,
points of tangency with
incircles tij , distances
ri = ‖vi − tij‖. Note
that t14, t12, t34, and
t23 are co-planar.
v1
t14
r1r1
t12
r1r1
t13
r1r1
v3
v2
v4
t23
t34
2