South Bohemia Mathematical Letters
Volume 19, (2011), No. 1, 6-17.
ON THE RIGIDITY OF POLYGONAL MESHES
HELLMUTH STACHEL
Dedicated to Prof. Pavel Pech at the occasion of his 60th birthday
Abstract. A polygonal mesh is a connected subset of a polyhedral surface.
We address the problem whether the intrinsic metric of a mesh, i.e., its development, can determine the exterior metric. If this is the case then the mesh
is rigid. Among the non-rigid cases even flexible versions are possible. We
concentrate on quadrangular meshes and in particular on a mesh with a flat
pose in which the quadrangles belong to a tesselation. It is proved that this
mesh admits a self-motion and that all its flexions represent discrete models of cylinders of revolution. These flexions can be generated from a skew
line-symmetric hexagon by applying iterated coaxial helical motions.
Introduction
In the following we understand under a polygonal mesh a connected subset of
any polyhedral surface in the Euclidean 3-space E3 . This means, a polygonal mesh
is a surface consisting of planar polygonal faces, edges and vertices. The edges are
either internal when they are shared by two faces, or they belong to the boundary
of the mesh. The term combinatorial structure of the mesh stands for the list of
faces and the identification of those pairs (fi , fj ) of faces which share an internal
edge eij . In this sense a polyhedron is a polygonal mesh with internal edges only.
Suppose that this mesh undergoes a transformation which acts on the faces
as isometry and preserves their planarity. We call this mesh rigid when under
this transformation also all dihedral angles between adjacent faces are preserved.
The question whether the intrinsic metric of the mesh determines its spatial shape
uniquely or not is also important for many engineering applications, e.g., for mechanical or constructional engineers, for biologists in protein modelling or for the
analysis of isomers in chemistry.
Polygonal meshes, in particular quadrangular meshes, play an important role in
discrete differential geometry and in new architecture where they serve as a discrete
model of freeform surfaces.
In the following we present different kinds of rigidity and we characterize the
flexions of Kokotsakis’ flexible quadrangular tesselation of the plane. It should
be noted that we only focus on geometric aspects of flexibility. We do not treat
technical aspects like stiffness of faces and edges or clearances along hinges.
Received by the editors September 15, 2011.
1991 Mathematics Subject Classification. 51M20, 52B10, 53A17.
Key words and phrases. flexible polyhedra, snapping polyhedra, Kokotsakis meshes, origami
mechanisms.
6
ON THE RIGIDITY OF POLYGONAL MESHES
7
1. The definitions of rigidity
Definition 1. A polyhedron or a polygonal mesh is called “globally rigid” when its
development (unfolding) defines its spatial shape uniquely — apart from movements
in space.
The development of the polygonal mesh defines its intrinsic metric, i.e., the true
shape of the faces and the combinatorial structure. Global rigidity means that the
intrinsic metric defines the exterior metric in space including the dihedral angle
ϕij at each internal edge eij shared by fi and fj . Conversely, the development
together with all dihedral angles defines the spatial shape completely as we can
assemble the mesh face by face, provided angle ϕij is signed with respect to a
prescribed orientation of edge eij . Of course, the dihedral angles cannot be given
independently; they must be compatible with the intrinsic metric.
f
f
6
4
f
f7
4
6
f
f5
f1
f2
f1
f2
Figure 1. Two realizations of the same net.
From everybody’s experience with assembling cardboard model of cubes, prisms
or pyramids during school-days one tends to conjecture that each polyhedron is
globally rigid. In fact, it is true, e.g., for a three-sided pyramid (tetrahedron) or
for a cube or — more generally — for all polyhedra where at each vertex exactly
three faces are meeting. However, the example in Fig. 1 shows two incongruent
realizations with the same development. The polyhedron on the left-hand side is
convex; it is built from a cube where the top face is replaced by pyramid. The
polyhedron on the right-hand side contains edges along which the interior dihedral
angle is > π.
When the height of the attached pyramid is sufficiently small, we can transform
the convex version into the concave one by applying a slight force to the apex of the
pyramid. In this case we speak of “snapping” polyhedra; we can vary the spatial
shape between two possibilities when admitting small deformations in between, e.g.,
by slight bending of faces and edges. Theoretically, both realizations are locally
rigid according to the following definition.
Definition 2. A polygonal mesh is called “locally rigid”, if its intrinsic metric
admits no other realization with dihedral angles sufficiently close to that of the
given one.
Surprisingly, there are examples of polyhedra where the development admits
even infinitely many incongruent realizations.
Definition 3. A polygonal mesh is called flexible if there is a continuous family
of mutually incongruent meshes sharing the intrinsic metric. In this case the mesh
admits a self-motion; each pose obtained during this self-motion is called a flexion.
HELLMUTH STACHEL
f6
f8
f2
6
f8
f5
f7
8
f
f1
f1
f1
Figure 2. The regular octahedron and its re-assembled continuously flexible versions.
In Fig. 2 a trivial example of a flexible polyhedron is displayed. The intrinsic
metric originates from that of a regular octahedron. We can re-assemble this polyhedron by putting one four-sided pyramid into the other. This gives a twofold covered quadratic pyramid without basis, which of course admits a self-motion. From
a flat pose of this four-sided pyramid with congruent faces we can even switch to
realizations consisting of a fourfold covered mesh of two triangles.
05
02
06
03
08
05
01
08
02
05
01
02
13
09
10
12 09
16
06
14
12
15
16
07
08
05 01 0
2
03
15
11
12 09 1
0
16
09
16
04
13
Figure 3. This polyhedron called “Vierhorn” is locally rigid, but
snaps between its spatial shape and two flat realizations; Below:
Development of the “Vierhorn”; dashes indicate valley folds.
ON THE RIGIDITY OF POLYGONAL MESHES
9
It turns out that the computation of the spatial shape of any four-sided doublepyramid from given internal metric, i.e., from its 12 edge lengths, is an algebraic
problem of degree 8. Hence, there are either up to 8 different realizations, or the
edge lengths admit a flexible form.
One needs to be careful when any polygonal mesh looks like a flexible one. A
famous example is described in C. Schwabe’s and W. Wunderlich’s article [12] on
a polyhedron exposed at the science exposition “Phänomena” 1984 in Zürich (see
Fig. 3). At that time it was falsely stated that this polyhedron is flexible, but
it is only snapping between two different flat realizations and one spatial shape.
The development reveals that all faces of this polyhedron are congruent isoceless
triangles.
There are four mile-stones in the theory of flexible polyhedra:
• The first important result in the theory of rigidity claims that every convex
polyhedron is rigid [2]. This is due to A. L. Cauchy 1813. The example
presented in Fig. 2 is no contradiction since the convex form, the regular
octahedron, is locally rigid.
• 1897 R. Bricard [1] classified all flexible octahedra, i.e., all flexible foursided double-pyramids with a not necessarily coplanar equator. However,
all these polyhedra have self-intersections. A real-world model can only be
built either as a wireframe or as a cardboard polyhedron where two faces
are omitted.
• R. Connelly detected 1977 the first flexible sphere-homeomorphic polyhedron without self-intersections and without twofold covered faces [3]. A
simplified flexing sphere with 9 vertices was presented 1980 by K. Steffen
[10] (compare [5, p. 347]) as a compound of two Bricard’s polyhedra.
• 1996 I. Sabitov [6] proved the famous Bellows Conjecture stating that for
every flexible polyhedron in E3 the oriented volume keeps constant during
the self-motion [6]. This was a consequence of his generalization of Heron’s
formula: For any orientable polyhedron with triangular faces in E3 there
exists a polynomial whose coefficients are polynomials in the squared edgelengths over Q and which has the square V 2 of the volume as a root; this
polynomial depends only on the combinatorial structure of the polyhedron.
There is only an algorithm available for determining this polynomial.
If Sabitov’s result had been known at the exposition of “Vierhorn” (Fig. 3),
then it would have been evident that this is not really flexible because the volume
changes drastically during the transition from the spatial form to the flat pose.
There is still another kind of flexibility placed between riditity and continuous
flexibility: According to W. Whiteley’s principle of averaging (see [11]) it can be
seen as a limit of snapping polyhedra. In the following definition we make use of a
standard notation of kinematics (see, e.g., [8]): Each infinitesimal rotation1 about
an oriented axis with normalized dual coordinate vector b
a ∈ R6 and signed angular
b. The twist of the composition
velocity ω can be represented by the twist vector ω a
of two infinitesimal rotations is the sum of the two twist vectors.
Definition 4. Suppose that to each internal edge eij of a polygonal mesh we can
assign an angular velocity ωij for the relative motion of the adjacent fi against fj
1This is an affine map which appoints to each point in E3 a velocity vector.
10
HELLMUTH STACHEL
in such a way that for each loop f0 , . . . , fs = f0 of faces, where any two consecutive
faces share an edge ei−1 i , i = 1, . . . , s , the sum of corresponding twists vanishes,
Ps
b . Then the mesh is called “infinitesimally flexible”.
i.e., i=1 ωi−1 i b
ei−1 i = 0
When the only compatible appointment of angular velocities to all internal edges
is the trivial one with ωij = 0, then the mesh is called infinitesimally rigid.
A real-word model of an infinitesimally flexible mesh shows an apparent, but
somehow confined flexibility. At the Vierhorn both flat positions are infinitesimally
flexible. A classical extension of Cauchy’s result states that each convex polytope
is even infinitesimally rigid.
Remark 1. a) Infinitesimal flexibility is usually defined for frameworks, but this
works only for triangular meshes. Before applying it, e.g., to quadrangular meshes,
we have to split all quadrangles by a diagonal into two triangles and to build
pyramids over them in order to guarantee the planarity of the quadrangle.
b) There are even different orders of infinitesimal rigidity to distinguish (e.g., [7]).
But for the sake of brevity we focus here only on first-order infinitesimal flexibility.
2. Flexible polygonal meshes
We are now concentrating on quadrangular meshes and start with a Kokotsakis
mesh (German: Vierflach), the compound of 3 × 3 planar quadrangles, which is
named after A. Kokotsakis [4]. In Fig. 4, left, the scheme of a quadrangular Kokotsakis mesh is shown with a central face f0 and a belt of 8 quadrangles around it.
On the right hand side a flexion is displayed.
f6
f7
f3
V2
f4
f8
f0
V3
V1
f1
f0
f
5
f7
f2
V4
f2
f3
f6
f1
f4
f8
f5
Figure 4. Left: Scheme of a quadrangular Kokotsakis mesh.
Right: Flexion of a flexible version; dashes indicate valley folds.
A complete classification of all continuously flexible Kokotsakis meshes is still
open (compare, e.g., [9]). However, the geometric characterization of infinitesimally
flexible meshes has already been given in [4] (see Fig. 5). We follow Kokotsakis’
ideas and use in our proof of this characterization standard results from Kinematics
(e.g., [8]): For any two faces fi , fj sharing an edge, this edge — here denoted by
ij — is the axis of the relative motion. Due to the Three-Pole-Theorem for any
three faces fi , fj , fk with rotations as pairwise relative motions the three relative
axes ij, ik and jk must be coplanar and share a point, which is denoted by ijk.
ON THE RIGIDITY OF POLYGONAL MESHES
11
The angles αj , αk which jk enclosed with ij and ik, respectively, define the ratio
of angular velocities of fj and fk against fi by
ωji : ωki = sin αk : sin αj .
This implies, e.g., that at the Kokotsakis mesh (see Fig. 5) the axis 12 of the relative
motion between f1 and f2 is the line of intersection between the planes spanned
by f5 and f0 . We call the line of intersection between the planes of fi and f0 the
trace of fi , i = 1, . . . , 8 .
)
024
f7
234
f3
134
123
03
34
f0
04
)
13
f4
24
02
01
013
f2
f5
f1
14
12
f8
f6
23













124
Figure 5. Infinitesimally flexible Kokotsakis mesh; the bounding
faces f1 , . . . , f8 are cut by a plane parallel to that of f0 .
Theorem 5 (Kokotsakis, 1932). A Kokotsakis mesh is infinitesimally flexible if
and only if the following three points are collinear: the points of intersection 013,
123 and 134 between the traces of the pairs of faces (f1 , f3 ), (f5 , f6 ) and (f7 , f8 ),
respectively. This is equivalent to the statement that the points of intersection 024,
234 and 124 between the traces of the pairs of faces (f2 , f4 ), (f6 , f7 ) and (f5 , f8 ),
resp., are aligned.
Proof. Due to the Three-Pole-Theorem the traces 12 of f5 and 23 of f6 meet at
point 123; the traces 14 of f8 and 34 of f7 meet at point 134; the traces 01 of f1 and
03 of f3 meet at point 013. In the infinitesimal case the three points 013, 123 and
134 must be located on the relative axis 13. Also by Desargues’ Theorem we can
conclude that the collinearity of these three points is equivalent to the collinearity
of points 024, 124, and 234.
Conversely, the position of 13 defines the ratio of angular velocities of the faces
f1 and f3 with respect to f0 . The other relative axes define the angular velocities
of the other faces. Hence, the stated collinearity implies a compatible assignment
of angular velocities to all internal edges.
An interesting continuously flexible quadrilateral mesh dates also back to Kokotsakis [4]. We start with a flat pose which consists of congruent quadrangles of a
planar tessellation (Fig. 6). Any two quadrangles sharing a side (e.g., f3 and f4 )
change place under a rotation through 180◦ (= half-turn) about the midpoint (C)
12
HELLMUTH STACHEL
f2
f1
f3
C
f4
Figure 6. Kokotsakis’ flexible tessellation.
of the common side. Any two adjacent quadrangles form a centrally symmetric
hexagon, and the complete tesselation can also be generated by translations of this
hexagon. The arrows in Fig. 6 indicate the directions of these translations.
When the quadrangles are convex, then this polygonal mesh is flexible (Kokotsakis [4, p. 647]. We call this mesh a tesselation mesh, for short.
In the following theorem we extend Kokotsakis’ result by characterizing the
flexions of this polygonal mesh. We are interested on constrained motions of the
mesh. Therefore we exclude additional degrees of freedom of single faces by the
request: Whenever the tesselation mesh includes three faces with a common vertex,
then also the fourth face of this pyramid must be included.
On the other hand, when the basic quadrangle is a trapezoid, then there are
aligned edges along which the mesh can be folded. We exclude these trivial flexes
by requiring a generic basic quadrangle.
Theorem 6. Let a polygonal mesh be extracted from the planar tesselation displayed in Fig. 6 in such a way, that with any three faces with a common vertex also
the fourth face through this vertex is included.
a) This quadrangular mesh is continuously flexible if and only if the initial quadrangle is convex.
b) In the generic flexible case, at each non-planar pose of a continuous self-motion
all vertices are located on a cylinder of revolution (Figs. 8 and 9).
c) The faces of the flexion can be obtained from a line-symmetric hexagon composed
from two adjacent quadrangles by applying iterated coaxial helical motions. In the
flat pose these helical motions convert into the translations applied to a centrally
symmetric hexagon in order to generate the planar tessellation.
Proof. First we pick out the four faces f1 , . . . , f4 with the common vertex V1
(Fig. 7). These congruent faces form a four-sided pyramid which is flexible, provided the fundamental quadrangle is convex. Otherwise, one interior angle of a face
at V1 would be greater than the sum of the other 3 interior angles so that the only
realization is the flat pose.
Let any non-planar flexion of this pyramid be given (Fig. 7, left). For any
pair (f1 , f2 ), . . . , (f4 , f1 ) of adjacent faces there is a respective half-turn ρ1 , . . . , ρ4
ON THE RIGIDITY OF POLYGONAL MESHES
13
f4
ρ3
ρ2
V1
ρ1
ρ4
f
3
f1
f2
f2
ρ1 (f4 )
ρ1 (f1 ) ρ2
f3
V2
ρ1
ρ3
f6 ρ1 ρ2 ρ1
f1
V
1
ρ1 (f3 )
ρ1 (f2 )
f4
ρ4 ρ (f
ρ4 (f4 )
4 1)
e56
V3 ρ 4 ρ 3 ρ 4
V4
f5
ρ4 (f3 )
e58
ρ4 (f2 )
f8 f9
f7
Figure 7. The complete flexion can be generated by applying
iterated half-turns ρi to an initial face f1 .
which swaps the two faces. So, e.g., f2 = ρ1 (f1 ) and f1 = ρ1 (f2 ). The axis of ρ1
(see Fig. 7, left) is perpendicular to the common edge V1 V2 , and it is located in a
plane which bisects the dihedral angle between f1 and f2 .
After applying all four half-turns ρ1 , . . . , ρ4 consecutively to the quadrangle f1 ,
this is mapped via f2 , f3 , and f4 onto itself; hence the product ρ4 . . . ρ1 equals the
identity. (We indicate the composition of mappings by left multiplication.) Because
of ρ−1
= ρi we obtain
i
(2.1)
ρ1 ρ2 = ρ4 ρ3 .
Lemma 7. The product of two half-turns about non-parallel axes a1 , a2 is a helical
motion. Its axis is the common perpendicular of a1 and a2 ; its angle of rotation is
twice the angle and the length of translation is twice the distance of the axes a1 , a2 .
When our pyramid with apex V1 is not flat, then the axes of the half-turns are
pairwise skew; the common perpendicular for any two of these axes is unique. Hence
(2.1) implies that the axes of the four half-turns have a common perpendicular s.
The motions ρ1 ρ2 = ρ4 ρ3 and ρ1 ρ4 = ρ2 ρ3 are helical motions with the common
axis s.
Now we extend the flexion of the pyramid with apex V1 stepwise to our polygonal
mesh by adding congruent copies of the initial pyramid without restricting the
flexibility:
The rotation ρ1 exchanges not only f1 with f2 but maps the pyramid with apex
V1 onto a congruent copy with apex V2 sharing two faces with its preimage. This
is the area which is hatched in Fig. 7, right. Analogously, ρ4 generates a pyramid
with apex V4 and sharing the faces f1 and f4 with the initial pyramid.
Finally there are two ways to generate a pyramid with apex V3 . Either, we
transform ρ2 by ρ1 and apply ρ1 ρ2 ρ1 , which exchanges ρ1 (f2 ) = f1 with ρ1 (f3 )
and swaps V2 and V3 . Or we proceed with ρ4 ρ3 ρ4 , which exchanges ρ4 (f4 ) = f1
with ρ4 (f3 ) and swaps V4 and V3 .
Thus we obtain mappings (ρ1 ρ2 ρ1 )ρ1 = ρ1 ρ2 and (ρ4 ρ3 ρ4 )ρ4 = ρ4 ρ3 with f1 7→
f5 and V1 7→ V3 . Both displacements are equal by (2.1), and we notice
(2.2)
ρ1 ρ2 = ρ4 ρ3 : f1 7→ f5 , f2 7→ ρ1 (f3 ), f3 7→ f1 , f4 7→ ρ4 (f3 ).
14
HELLMUTH STACHEL
Hence each flexion of the initial pyramid with apex V1 is compatible with a flexion
of the complete 3 × 3 tesselation mesh. Is this the only flexion of this mesh induced
by the given flex of the initial pyramid?
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are placed on a cylinder of rotation; the marked points are located
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Lemma 8. A generic flexion of a 3 × 3 tesselation mesh is uniquely defined by the
flexion of one included pyramid consisting of four faces with a common vertex V1 .
Proof. At our previously defined flexion the pyramids with vertices V2 , V3 and V4
are congruent to that with vertex V1 . However, there is another possibility at
V2 , which is compatible with that at V1 : We can reflect the two left faces f6 , f7
(Fig. 7, right) in the plane spanned by the edges e27 and e16 . In the same way it is
possible at V4 to replace the faces f8 , f9 by their mirrors with respect to the plane
e18 ∨ e49 . In total this gives 3 alternatives: Reflect either one pair of faces or both
simultaneously. Does any of these alternatives keep place for inserting the last face
f5 with given interior angle at V3 ?
(i) When only f6 and f7 are reflected, the edge e56 is replaced by its mirror in the
plane e27 ∨ e16 . This preserves the angle with e58 only if the plane e27 ∨ e16 passes
through e58 . After applying ρ1 , this is equivalent to the statement that ρ2 (e14 ) is
placed in e23 ∨ e14 , which means that the diagonal plane e23 ∨ e14 of our initial
pyramid is the exterior bisector of the faces f2 and f3 .
(ii) When reflecting f8 and f9 only, then the mirror plane e18 ∨ e49 must pass
through e56 in order to preserve the interior angle of f5 at V3 . After applying ρ3 ,
this is equivalent to the condition that e12 ∨ e43 is the exterior bisector of the faces
f3 and f4 .
(iii) When e56 is reflected in e27 ∨ e16 and e58 is reflected in e18 ∨ e49 , then the
angle between e56 and e58 is preserved if and only if there is a rotation mapping
e56 and e58 onto their respective mirror images. The axis a if this rotation is the
line of intersection between the two mirror planes. In the case of a rotation the
axis a must span planes with e56 and e58 which enclose with e27 ∨ e16 and e18 ∨ e49 ,
respectively, the same oriented angle.
ON THE RIGIDITY OF POLYGONAL MESHES
15
Only in particular poses condition (i), (ii) or (iii) can be fulfilled, but we excluded
this by the request for a generic pose. Hence the flexion of the 3 × 3 tesselation
mesh is uniquely defined.
By Lemma 8 we can uniquely extend the flexion of the initial pyramid to the
3 × 3 tesselation mesh and furtheron to the complete polygonal mesh, apart from
particular poses. But under a continuous self-motion it is not possible to switch into
one of the exceptional reflected positions listed above in (i), (ii) or (iii). Hence this
mesh is continuously flexible, and all included four-sided pyramids are congruent.
We detect at the flexion also the spatial analogues of the translations in the
plane: The product ρ1 ρ4 = ρ2 ρ3 maps the pyramid with apex V1 onto that with
apex V2 . On the other hand we have
(2.3)
ρ1 ρ4 : f1 7→ ρ1 (f4 ), f4 7→ f2 , ρ4 (f2 ) 7→ f1 , ρ4 (f3 ) 7→ ρ1 (f3 ).
When f3 and f4 are glued together, we obtain a skew hexagon, one half of our initial
pyramid with apex V1 . The half-turn ρ4 maps this hexagon onto itself; hence it
is line-symmetric. By (2.2) the helical motion ρ1 ρ2 maps this hexagon onto the
compound of f1 and ρ4 (f3 ) and furthermore f1 onto f5 . The inverse ρ2 ρ1 is the
spatial analogon of the translation indicated in Fig. 6 by the red arrow pointing
upwards to the right. On the other hand, ρ4 ρ1 maps the compound of f1 and
ρ4 (f3 ) onto ρ1 (f4 ) and ρ1 (f3 ). It these two helical motions act repeatedly on the
line-symmetric hexagon, the complete flexion is obtained.
Since all vertices of the flexion arise from V1 by motions which keep the common
perpendicular s of the half-turn axes fixed, e.g., V2 = ρ1 (V1 ), V3 = ρ1 ρ2 (V1 ),
V4 = ρ4 (V1 ), they all have the same distance to s, i.e., they are located on a
cylinder of revolution with axis s.
Remark 2. a) When starting from the flat initial pose of the pyramid with apex
V1 , there are two self-motions possible since there are two edges of the pyramid
where the adjacent interior angles at V1 have a sum smaller than 180◦ . These
edges become valley-folds in Fig. 8. Hence our polygonal mesh admits two kinds of
differentiable self-motions. Figure 9 shows snapshots of these two self-motions.
b) In the case of a trapezoid f1 one kind of generating motion is a rotation about
s, the other a translation along s. However, in this case we have a higher degree of
freedom since the mesh can be bended along each serie of aligned edges.
There is also a direct way to get a flexion of a tesselation mesh: We can start
with any point V1 and with three half-turns ρ1 , ρ2 , ρ3 such that the axes have a
common perpendicular. However, the quadrangle V1 . . . V4 will be planar only if V1
obeys an additional condition.
Conclusion
Not each polyhedron is uniquely defined by its development. We shed light on
the different types of rigidity of polygonal meshes und characterize the flexions of
Kokotsakis’ tesselation meshes.
HELLMUTH STACHEL
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ON THE RIGIDITY OF POLYGONAL MESHES
17
Acknowledgment. This research is partly supported by Grant No. I 408-N13
of the Austrian Science Fund FWF within the project “Flexible polyhedra and
frameworks in different spaces”, an international cooperation between FWF and
RFBR, the Russian Foundation for Basic Research.
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Institute of Discrete Mathematics and Geometry, Vienna University of Technology,
Vienna, Austria
E-mail address: [email protected]