DOI: 10.1007/s00454-003-2948-z
Discrete Comput Geom 31:37–59 (2004)
Discrete & Computational
Geometry
©
2003 Springer-Verlag New York Inc.
Optimally Cutting a Surface into a Disk
∗
Jeff Erickson and Sariel Har-Peled
Department of Computer Scienc, University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA
{jeffe,sariel}@cs.uiuc.edu
http://www.cs.uiuc.edu/˜{jeffe,sariel}
Abstract. We consider the problem of cutting a subset of the edges of a polyhedral
manifold surface, possibly with boundary, to obtain a single topological disk, minimizing
either the total number of cut edges or their total length. We show that this problem is NP-
hard in general, even for manifolds without boundary and for punctured spheres. We also
describe an algorithm with running time n
O(g+k)
, where n is the combinatorial complexity,
g is the genus, and k is the number of boundary components of the input surface. Finally,
we describe a greedy algorithm that outputs a O(log
2
g)-approximation of the minimum
cut graph in O(g
2
n log n) time.
1. Introduction
Several applications of three-dimensional surfaces, such as parameterization [21], [50]
and texture mapping [3], [47], require information about the underlying topological
structure in addition to the geometry. In some cases we wish to simplify the surface
topology, to facilitate algorithms that can be performed only if the surface is a topological
disk.
In the texture mapping problem, we wish to find a continuous and invertible mapping
from the texture, usually a two-dimensional rectangular image, to the surface. Unfortu-
nately, if the surface is not a topological disk, no such map exists. In such a case, the
only feasible solution is to cut the surface so that it becomes a topological disk. (Haker
et al. [28] present an algorithm for directly texture mapping models with the topology
∗
A preliminary version of this paperwas presented at the 18th Annual ACM Symposium on Computational
Geometry [20]. See http://www.cs.uiuc.edu/ jeffe/pubs/schema.html for the most recent version of this paper.
Jeff Erickson was partially supported by a Sloan Fellowship, NSF CAREER Award CCR-0093348, and NSF
ITR Grant DMR-0121695. Sariel Har-Peled was partially supported by NSF CAREER Award CCR-0132901.
38 J. Erickson and S. Har-Peled
of a sphere, where the texture is also embedded on a sphere.) Of course, when cutting
the surface, one would like to find the best possible cut under various considerations.
For example, one might want to cut the surface so that the resulting surface can be tex-
ture mapped with minimum distortion [21], [25], [50]. To our knowledge, all previous
approaches for the texture-mapping problem either rely on cutting heuristics with no
quality guarantees [25], [49], [52] or require the user to cut the surface beforehand [21],
[47].
One method for reducing a manifold to a topological disk is to cut along the boundary
of a so-called canonical polygonal schema. This boundary is a set of 2g cycles, all passing
through a common base point in M, such that cutting along those cycles breaks M into
a topological disk. (We give more formal definitions in Section 2.) Lazarus et al. [40]
presented and implemented two algorithms for computing a canonical polygonal schema
of an orientable surface of complexity n and with genus g, in time O(gn), simplifying an
earlier algorithm of Vegter and Yap [60]. Since those cycles must share a common point,
it is easy to find examples where the overall size of those cycles is (gn). Furthermore,
those cycles share several edges and are visually unsatisfying.
Formost applications, however, canonical schemata are overkill.It is usually sufficient
to find an arbitrary collection of edges whose removal transforms the surface into a
topological disk. We call such a set of edges a cut graph; see Fig. 1 for an example. Cut
graphs have several advantages. First, they are compact. Trivially, any cut graph contains
at most n edges of the surface mesh, much less than any canonical schema in the worst
case, and we expect it to be even smaller in practice. Second, it is quite easy to construct
a cut graph for an arbitrary polyhedral surface in O(n) time, using a breadth-first search
of the dual graph [14], or simply taking a maximal set of edges whose complement is
connected [40]. Finally, the cut graph has an extremely simple structure: a tree with
O(g) additional edges. As such, it should be easier to manipulate algorithmically than
other representations. For example, Dey and Schipper [14] describe fast algorithms to
determine whether a curve is contractible, or two curves are homotopic, using an arbitrary
cut graph instead of a canonical schema.
In this paper we investigate the question of how find the “best” such cutting of a
surface, restricting ourselves to cuts along the edges of the given mesh. Specifically, we
c
b
d
e
a
e
d
c
c
d
b
a
e
a
b
c
a
bc
d
e
Fig. 1. A cut graph for a two-holed torus and its induced (non-canonical) polygonal schema.
Optimally Cutting a Surface into a Disk 39
want to find the smallest subset of edges of a polyhedral manifold surface M, possibly
with boundary, such that cutting along those edges transforms Minto a topological disk.
We also consider the weighted version of this problem, where each edge has an arbitrary
non-negative weight and we want to minimize the total weight of the cut graph. The most
natural weight of an edge is its Euclidean length, but we could also assign weights to
take problem-specific considerations into account. For example, if we want to compute a
texture mapping for a specific viewpoint, we could make visible edges more expensive,
so that the minimum cut graph would minimize the number of visible edges used in the
cuts. Our algorithms do not require the edge weights to satisfy the triangle inequality.
We show that the minimum cut graph of any polyhedral manifold M with genus g
and k boundary components can be computed in n
O(g+k)
time. This is exponential in
the worst case, since there are manifolds with both g = (n) and k = (n). We also
show that computing a minimum cut graph is NP-hard, even if g or k is fixed (but not
both). Finally, we present a simple and efficient greedy approximation algorithm for this
problem. Our algorithm outputs a cut graph whose weight is a factor O(log
2
g) larger
than optimal, in O(g
2
n log n) time.
1
If g = 0, the approximation factor is exactly 2. As
a tool in our approximation algorithm, we also describe efficient algorithms to compute
shortest and nearly shortest nontrivial cycles in a manifold; we believe these algorithms
are of independent interest.
2. Background
Before presenting our new results, we review several useful notions from topology
and describe related results in more detail. We refer the interested reader to [31], [46],
or [53] for further topological background and more formal definitions. For related
computational results, see the recent surveys by Dey et al. [12] and Vegter [59].
2.1. Topology
A2-manifold with boundary is a set M such that every point x ∈ M lies in a neigh-
borhood homeomorphic to either the plane R
2
or a closed halfplane. The points with
only halfplane neighborhoods constitute the boundary of M; the boundary consists of
zero or more disjoint circles. This paper considers only compact manifolds, where every
infinite sequence of points has a convergent subsequence.
The genus of a 2-manifold M is the maximum number of disjoint non-separating
cycles γ
1
,γ
2
,...,γ
g
in M; that is, γ
i
∩ γ
j
= ∅ for all i and j, and M\(γ
1
∪···∪γ
g
)
is connected. For example, a sphere and a disk each have genus 0, a torus and a M¨obius
strip each have genus 1, and a Klein bottle has genus 2.
A manifold is orientable if it has two distinct sides, and non-orientable if it has only
one side. Although many geometric applications use only orientable 2-manifolds (pri-
marily because non-orientable manifolds without boundary cannot be embedded in R
3
1
To simplify notation, we define log x = max{1, log
2
x}.
40 J. Erickson and S. Har-Peled
without self-intersections) our results apply to non-orientable manifolds as well. Every
(compact, connected) 2-manifold with boundary is characterized by its orientability, its
genus g, and the number k of boundary components [22]. For example, the disk is the
only oriented 2-manifold with genus 0 and one boundary component.
A polyhedral 2-manifold is constructed by gluing closed simple polygons edge-to-
edge into a cell complex: the intersection of any two polygons is either empty, a vertex of
both, or an edge of both. We refer to the component polygons as facets. (Since the facets
are closed, every polyhedral manifold is compact.) For any polyhedral manifold M, the
number of vertices and facets, minus the number of edges, is the Euler characteristic χ
of M. Euler’s formula [19] implies that χ is an invariant of the underlying manifold,
independent of any particular polyhedral representation: χ = 2 −2g −k if the manifold
is orientable, and χ = 2 − g − k if the manifold is non-orientable. Euler’s formula
implies that if M has v vertices, then M has at most 3v − 6 + 6g edges and at most
2v − 4 + 4g − k facets, with equality for orientable manifolds where every facet and
boundary circle is a triangle. We let n ≤ 6v −10 + 10g −k denote the total number of
facets, edges, and vertices in M.
The 1-skeleton M
1
of a polyhedral manifold Mis the graph consisting of its vertices
and edges. We define a cut graph G of M as a subgraph of M
1
such that M\G is
homeomorphic to a disk. (Cut graphs are generalizations of the cut locus of a manifold
M, which is essentially the geodesic medial axis of a single point.) The disk M\G is
known as a polygonal schema of M. Each edge of G appears twice on the boundary of
polygonal schema M\G, and we can obtain M by gluing together these corresponding
boundary edges.Findinga cut graph of Mwith minimum total lengthisclearly equivalent
to finding a polygonal schema of M with minimum perimeter.
Any 2-manifold has a so-called canonical polygonal schema, whose combinatorial
structure depends only on the genus g, the number of boundary components k, and
whether the manifold is orientable.
2
The canonical schema of an orientable manifold is
a (4g + 3k)-gon with successive edges labeled
x
1
, y
1
, ¯x
1
, ¯y
1
, ...,x
g
, y
g
, ¯x
g
, ¯y
g
, z
1
, e
1
, ¯z
1
, ...,z
k
, e
k
, ¯z
k
;
for a non-orientable manifold, the canonical schema is a (2g +3k)-gon with edge labels
x
1
, x
1
, ...,x
g
, x
g
, z
1
, e
1
, ¯z
1
,...,z
k
, e
k
, ¯z
k
.
Every pair of corresponding edges x and ¯x is oriented in opposite directions. Gluing
together corresponding pairs in the indicated directions recovers the original manifold,
with the unmatched edges e
i
forming the boundary circles. For a manifold M without
boundary, a reduced polygonal schema is one where all the vertices are glued into a
single point in M; canonical schemata of manifolds without boundary are reduced.
We emphasize that the polygonal schemata constructed by our algorithms are neither
necessarily canonical nor necessarily reduced.
2
There are actually several different ways to define canonical schemata; the one described here is merely
the most common. For example, the canonical schema for an oriented surface without boundary could also be
labeled x
1
, x
2
,...,x
2g
, ¯x
1
, ¯x
2
,..., ¯x
2g
.
Optimally Cutting a Surface into a Disk 41
2.2. Previous and Related Results
Dey and Schipper [14] describe a linear-time algorithm to construct a reduced, but
not necessarily canonical, polygonal schema for any triangulated orientable manifold
without boundary. Essentially, their algorithm constructs an arbitrary cut graph G by
depth-first search, and and then shrinks a spanning tree of G to a single point. (See also
[13].)
Vegterand Yap [60] developedan algorithm to construct a canonical schema in optimal
O(gn) time and space. Two simpler algorithms with the same running time were later
developed by Lazarus et al. [40]. The “edges” of the polygonal schemata produced
by all these algorithms are (possibly overlapping) paths in the 1-skeleton of the input
manifold. Since there are polyhedral manifolds where every canonical schema contains
(gn) edges, these algorithms are optimal. We will modify one of the algorithms of
Lazarus et al. to construct short nontrivial cycles and cut graphs.
Very recently, Colin de Verdi´ere and Lazarus considered the problem of optimizing
canonical polygonal schemata [11]. Given a canonical polygonal schema for a triangu-
lated oriented manifold M, their algorithm constructs the shortest canonical schema in
the same homotopy class. Surprisingly (in light of our Theorem 3.1) their algorithm runs
in polynomial time under some mild assumptions about the input. As a by-product, they
also obtain a polynomial-time algorithm to construct the minimum-length simple loop
homotopic to a given path.
Surface parameterization is an extremely active area of research, thanks to numerous
applications such as texture mapping, remeshing, compression, and morphing. For a
sample of recent results, see [1], [18], [21], [52], [25], [42], [43], [49], [51], [50], [63],
and references therein. In most of these works, surfaces of high genus are parameterized
by cutting them into several (possibly overlapping) patches, each homeomorphic to a
disk, each with a separate parameterization. A recent exception is the work of Gu et
al. [25], which computes an initial cut graph in O(n log n) time by running a shortest
path algorithm on the dual of the manifold mesh, starting from an arbitrary seed triangle.
Essentially the same algorithm was independently proposed by Steiner and Fischer [52].
Once a surface has been cut into a disk (or several disks), further (topologically trivial)
cuts are usually necessary to reduce distortion [25], [49], [51]. Many of these algorithms
include heuristics to minimize the lengths of the cuts in addition to the distortion of the
parameterization [25], [43], [49], but none with theoretical guarantees.
All of our algorithms are ultimately based on Dijkstra’s single-source shortest path
algorithm [15], [55]. Many previous results have used Dijkstra’s algorithm or one of its
continuous generalizations [35], [44], [58] to discover interesting topological structures
in 2-manifolds, such as cut graphs [25], [52], small handles (“topological noise”) [27],
texture atlases [43], contour trees [2], [41], and Reeb graphs [32], [52].
3. Computing Minimum Cut Graphs is NP-Hard
In this section we prove that finding a minimum cut graph of a triangulated manifold is
NP-hard. We consider two versions of the problem. In the weighted case the manifold is
assumed to be a polyhedral surface in R
3
and we want to compute the cut graph whose