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doi:10.1093/comjnl/bxh075
Analytical Honeycomb Geometry for
Raster and Volume Graphics
Valentin E. Brimkov1 and Reneta P. Barneva2
1 Fairmont State University, Fairmont, WV 26554, USA
2 State University of New York, Fredonia, NY 14063, USA
Email: [email protected], [email protected]
In this paper we investigate the advantages of using hexagonal grids in raster and volume graphics.
In 2D, we present a hexagonal graphical model based on a hexagonal grid. In 3D, we introduce two
honeycomb graphical models in which the voxels are hexagonal prisms, and we show that these are
the only possible models under certain reasonable conditions. In the framework of the proposed
models we design the 2D and 3D analytical honeycomb geometry of linear objects as well as of circles
and spheres. We demonstrate certain advantages of the honeycomb models and address algorithmic
and complexity issues.
Received 29 March 2004; revised 14 September 2004
1.
INTRODUCTION
Various advantages of volume graphics over surface graphics
have been widely discussed in the literature [1]. However,
certain disadvantages of the former have been recognized
quite early. One can list the following, among others [1]:
• Loss of geometric information after voxelization of a
surface object. The voxels composing a digital surface
do not retain the geometric information contained in the
definition of the surface.
• High aliasing artifacts due to the discrete nature of the
data from which the continuous object is reconstructed
(such as appearance of holes in the object after
performing some operations).
• Large amount of memory and computational resources
are required in order to store and manipulate digital
objects.
Over the last few decades numerous useful theoretical results
have been obtained in digital topology and digital geometry.
These are disciplines providing theoretical foundations for
computer graphics and digital image analysis [2, 3, 4, 5, 6]. A
promising approach, which may help resolve problems such
as those listed above, is based on developing an analytical
digital geometry (see, e.g. [7] for introduction). The main
objective here is to obtain a simple analytical description of
the basic Euclidean primitives and to develop algorithms for
modeling using such primitives.
Digital geometry (as well as analytical digital geometry)
deals with geometrical properties of ‘digital images’, which
are sets of points of the digital space Zn . The raster computer
graphics is modeled upon a regular square grid, where the
square tiles are called pixels. Square grids are physically
implemented in computer displays. In 3D, the graphical
models are based on regular cubic tilings, in which the cubes
are called voxels. Alternatives to the square/cubic grid have
also been considered. In 2D, such an alternative is provided
by the hexagonal grid whose cells are regular hexagons.
Certain properties of hexagonal tilings of the plane have
been first studied in relation to research on covering and
packing of the plane. Studies on these last topics seem to
originate from a work of Gauss in 1831 introducing the idea
of lattice. Over the years contributions to the subject have
been made by mathematicians like Minkowski, Davenport
and Rogers, among others. (See the classical monograph
of Rogers [8] for historical and other details related to this
early stage.) In particular, one can see that regular hexagons
provide economic (and under certain conditions optimal)
plane covering and packing. Hexagonal rasters are well
known in image processing [9, 10], distance transforms
[11, 12, 13, 14], image analysis architectures [15] and other
applications [16, 17]. Topological properties of hexagonal
grids (in particular, tunnel-freedom issues) have been studied
long ago by the mathematical morphology community [18].
Recent works elucidate various interesting points. For
instance, algorithmic comparison between hexagonal- and
square-based models is available in [19]. This last work
also provides a discussion on the possibility of implementing
hexagonal grids to graphical devices. Algorithms for drawing
lines and circles are presented in [20, 21].
Alternatives to the cubic grid in 3D have also been
considered. Some properties and advantages of grids based
on rhombic dodecahedral or truncated octahedral tilings have
been studied, e.g. in [22, 23].
In the present paper we consider the case when the tile
is a hexagonal prism. We call the corresponding models
the ‘3D honeycomb models’. To our knowledge of the
available literature, such kind of models have not been studied
seriously, although they indeed expose certain advantages
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Analytical Honeycomb Geometry
over the classical cubic model. In brief, we show that the
honeycomb models ensure tunnel-freedom of digital objects
in a more direct way than the cubic models. Honeycomb
models provide certain advantages in defining ‘good pairs’
of adjacency relations in view of the discrete version of the
well-known Jordan curve theorem (see the end of Section 2.1
for a related discussion). Moreover, the introduced models
ensure better approximation to a continuous object (in terms
of the Hausdorff distance) and are ‘more economic’ in some
respects. The proposed honeycomb models employ simple
mathematics and provide possibilities for a straightforward
transfer of notions and results from classical digital geometry.
Our main contribution is developing the analytical honeycomb geometry of some fundamental Euclidean primitives,
such as straight lines, segments, polygons, circles, planes,
spheres etc. Their compact analytical description provides
simple drawing algorithms in a straightforward manner. Our
analytical approach makes the theoretical considerations
either obvious or rather simpler than those of [20, 21].
The paper is organized as follows. In the next section we
recall some definitions and introduce notions to be used in the
sequel. In Section 3, we provide theoretical foundations for
developing the 2D digital analytical honeycomb geometry
of linear objects and circles. We also discuss some
advantages of the hexagonal grid. In Section 4, we
extend our considerations to the 3D space by introducing
two honeycomb graphical models in which the voxels are
hexagonal prisms. We show that these are the only possible
models under certain reasonable conditions. We also develop
the analytical digital honeycomb geometry within these
models. In Section 5, we address algorithmic and complexity
issues. We conclude with some final remarks in Section 6.
2.
2.1.
PRELIMINARIES
Basic notions of digital topology and geometry
A tiling P of Rn by convex polytopes is called normal [24]
if the intersection of any two tiles from P is either empty or
appears to be their common (n − d)-dimensional facet, for
some d, 1 ≤ d ≤ n.
Let T be a normal tiling of Rn , n = 2 or 3, where the tiles
P of T are convex polytopes. For a set of tiles A we denote
U (A) = ∪P ∈A P . A set of tiles will be regarded as a digital
object.
A j -dimensional facet of a tile P ∈ T is called j -facet,
for some j , 0 ≤ j ≤ n − 1. Thus the 0-facets of P are its
vertices, the 1-facets are its edges and the 2-facets of a 3D
polytope are its 2D faces.
Two tiles are called j -adjacent if they share a j -facet. A
k-path in a digital object A is a sequence of tiles from A such
that every two consecutive tiles are k-adjacent. Two tiles
are k-connected if there is a k-path between them. A digital
object A is k-connected if there is a k-path connecting any
two tiles of A. A digital object is connected if it is at least 0connected. Otherwise it is disconnected.1 A k- component of
1 Classically, 0-adjacent/connected (resp. 1-adjacent/connected) pixels
are called 8-adjacent/connected (resp. 4-adjacent/connected). In 3D,
A
A
D
D
(a)
(b)
FIGURE 1. (a) A set D that is tunnel-free and 0-separating in a
set A. (b) A set D that is 1-separating in a set A. D is 1-tunnel-free
and has 0-tunnels.
a digital object A is a maximal (non-extendable) k-connected
set of tiles of A.
Let D be a subset of a digital object A. If A − D is
not k-connected then the set D is called k-separating in A
(Figure 1). A supercover of an Euclidean object M is the set
Sup(M) of all tiles which are intersected by M. Let be a
curve in R2 or a surface in R3 . In a very general sense one
can consider a subset of Sup() as a digitization of or as a
discrete curve/surface corresponding to .
An important concept in discrete geometry for computer
imagery is that of tunnel. Intuitively, a k-tunnel is a location
in a discrete object through which a discrete k-path can
penetrate. Tunnels in a discrete object (in particular, digital
curve or surface) are usually defined on the basis of the notion
of separability. Let a set of tiles A be k-separating in a digital
object B but not j -separating in B. Then A is said to have
j -tunnels for any j < k. A digital object without k-tunnels is
called k-tunnel-free (Figures 1 and 2). An object that has no
tunnels for any k, 0 ≤ k ≤ 2, is called tunnel-free, in short.2
The notion of tunnel plays an important role in rendering
voxelized scenes by casting digital rays from the image to
the scene [25]. As Kaufman remarks, thinner rays are much
more attractive for ray traversal. Therefore, it is desirable to
construct k-tunnel-free digital objects where k is as small as
possible. The ideal situation is when the object is tunnel-free.
The 2D case is comparatively simple since the connectivity of
a digital object fully characterizes the topology of the tunnels.
More precisely, an object A contains 1-tunnels if and only if
it is disconnected; A contains 0-tunnels but no 1-tunnels if
and only if it is connected but not 1-connected (i.e. if and
only if it is strictly 0-connected); and A is tunnel-free if and
only if it is 1-connected. In 3D, however, the situation is more
complicated. Sometimes this complexity has certain negative
impacts on the design of discretization algorithms. The
variety of possibilities (presence of 0-, 1-, 2-tunnels or tunnelfreedom) may cause certain difficulties when constructing ktunnel-free discretizations of more complex objects. Thus, it
0-adjacent/connected (resp. 1 or 2-adjacent/connected) voxels are called
26-adjacent/connected (resp. 18 or 6-adjacent/connected).
2 Classically, in 2D, a 0-tunnel (resp. 1-tunnel) is called 8-tunnel (resp.
4-tunnel). In 3D, a 0-tunnel (resp. 1- or 2-tunnel) is called 26-tunnel (resp.
18- or 6-tunnel). A 2-tunnel is sometimes called a hole.
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V. E. Brimkov and R. P. Barneva
(i)
(ii)
(i)
(ii)
(iii)
(a)
(b)
FIGURE 2. Different types of tunnels in the case of square/cubic tiles. (a) Tunnels in a 2D digital object: (i) 1-tunnel, (ii) 0-tunnel. (b)
Tunnels in a 3D object: (i) 2-tunnel, (ii) 1-tunnel, (iii) 0-tunnel (in two different orientations).
may be difficult to control the object connectivity. Moreover,
it might even be a problem to secure that a digital object is
connected [7]. Ensuring tunnel-freedom can be a difficult
task as well [7, 26, 27, 28, 29, 30, 31, 32].
A source for such sort of difficulties is the topology of the
square grid. In Sections 4 and 5 we will see how they can be
easily overcome by using honeycomb models.
The notion of tunnels is closely related to the one known
in digital topology as ‘good pairs’ of adjacency relations
[33, 34]. The latter arises in particular when one works with
closed digital curves (in 2D) and closed digital surfaces (in
3D). It turns out that in order to make the considerations
meaningful, one should use different topologies when dealing
with the curve/surface or with its complement. For instance,
in 2D, the following discrete version of the well-known
Jordan curve theorem has been proved [35]:
Theorem 1. If C is the set of points of a simple closed
1-curve (0-curve) and |C| > 4(|C| > 3), then the complement
C of C has exactly two 0-components (1-components).
In this context, (1, 0) and (0, 1) are good pairs of adjacency
relations, while (1, 1) and (0, 0) are not. A similar situation
arises in 3D: (2, 0), (0, 2), (2, 1) and (1, 2) are good pairs,
while (2, 2), (1, 1) and (0, 0) are not. Thus, it is necessary
to use different adjacencies when processing a digital object
and its complement. However, if one uses a hexagonal grid
with an adjacency relation µ defined by the nearest neighbor,
such an undesirable asymmetry does not exist anymore as
(µ, µ) turns out to be a good pair [33].
2.2.
Analytical digital geometry
Analytical geometry is an important method for studying
geometric objects by algebraic means. A geometric object is
analytically represented with respect to a coordinate system
in the plane or space (Figure 3).
A novel approach in raster and volume graphics is the one
based on developing analytical digital geometry. The main
objective is to obtain simple analytical definitions of the
basic Euclidean primitives, such as lines and line segments,
polygons, circles, planes, spheres etc., and, on this basis,
to create tools for efficient modeling of more sophisticated
objects composed by such primitives.
In recent years analytical digital geometry has been
developed by several authors (see [36] and the bibliography
in [29, 37]). This approach is very successful especially
y
y
g
O (0,0)
O (0,0)
x
x
(a)
(b)
FIGURE 3. (a) A straight line through the points (5, 0) and (0, 4)
consists of all points (x, y) in the plane, whose coordinates satisfy
the equation 4x +5y −20 = 0 with respect to a Cartesian coordinate
system. (b) A circle with radius 5 and center (0, 0) in a Cartesian
coordinate system has equation x 2 + y 2 = 52 .
when dealing with linear objects. Analytical descriptions
and efficient digitization algorithms based on them have been
obtained for 2D and 3D digital straight lines or their segments
[32, 36, 38, 39], digital planes [36, 39, 40, 41], 2D and 3D
digital polygons as well as tunnel-free meshes of 3D polygons
[7, 30, 31, 32]. These results are of significant practical
importance since for various applications it suffices to work
with a reasonable polyhedral approximation to the given real
object. Circles and spheres have been analytically defined as
well [29, 42]. In the rest of this section we introduce some
basic notions and recall the analytical definitions of a digital
straight line, plane and circle.
A discrete coordinate plane consists of unit squares (pixels)
centered on the integer points of the Cartesian coordinate
system in the plane. A discrete coordinate space consists
of unit cubes (voxels) centered on the integer points of the
Cartesian coordinate system in the 3D space. The coordinates
of pixels/voxels are the coordinates of their centers. The
edges of a pixel/voxel are parallel to the coordinate axes.
A 2D arithmetic line is a set of pixels L(a1 , a2 , µ, ω) =
{(x, y) ∈ Z2 |0 ≤ a1 x + a2 y + µ + ω/2 < ω}, where
a1 , a2 , µ ∈ Z, ω ∈ N. ω is called the arithmetic thickness
or width of the line and µ is its internal translation constant.
L(a1 , a2 , µ, ω) can be considered as the discretization of a
straight line with equation ax1 + ax2 + µ = 0. The vector
(a1 , a2 ) is the normal vector to the line. An arithmetic
line L(a1 , a2 , µ, ω) is called naive if ω = max(|a1 |, |a2 |)
The Computer Journal Vol. 48 No. 2, 2005
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Analytical Honeycomb Geometry
y
y
g
y
g
x
O (0,0)
O(0,0)
x
O (0,0)
(b)
(a)
x
(c)
FIGURE 4. (a) A naive line L(4, 5, −18, 5). (b) A standard line L(4, 5, −16, 9). (c) A digital circle C(5) with center (0, 0) and radius 5.
(a)
(b)
(c)
FIGURE 5. (a) Tiling by a parallelogram. (b) Brick-built tiling. (c) Tiling by a quasi-regular hexagon.
(Figure 4a), and standard if ω = |a1 | + |a2 | (Figure 4b).
It is well-known [36] that an arithmetic line is (at least)
0-connected (classically, 8-connected) and 1-tunnel-free
if and only if ω ≥ max(|a1 |, |a2 |). Thus a naive line
L(a1 , a2 , µ, max(|a1 |, |a2 |)) is the ‘thinnest possible’ arithmetic line that is 0-connected and 1-tunnel-free (in a sense
that any arithmetic line of thickness less than max(|a1 |, |a2 |)
is disconnected and has 1-tunnels). Similarly, an arithmetic line is 1-connected (classically, 4-connected) and
0-tunnel-free iff ω ≥ |a1 | + |a2 |. Thus a standard line
L(a1 , a2 , µ, |a1 | + |a2 |) is the ‘thinnest possible’ arithmetic
line that is 1-connected and 0-tunnel-free (in a sense that
any arithmetic line of thickness less than |a1 | + |a2 | has
0-tunnels).3
An arithmetic digital plane is a set of voxels
P (a1 , a2 , a3 , µ, ω) = {x ∈ Z3 : 0 ≤ a1 x1 + a2 x2 + a3 x3 +
µ + ω/2 < ω}. The parameters a1 , a2 , a3 , µ ∈ Z, and
ω ∈ N have the same interpretation as for lines. A digital
plane with ω = max(|a1 |, |a2 |, |a3 |) is naive, while one with
ω = |a1 | + |a2 | + |a3 | is standard (see [41]). Properties of
naive/standard planes are similar to those of naive/standard
lines [7, 41].
parallelograms, such as in Figure 5a. Such a grid is
topologically equivalent to one with square-grid cells.
An arithmetic digital circle with center (0, 0) and
radius r ∈ Z+ is a set of pixels C(r) = {(x, y) ∈
Z2 : (r − 21 )2 ≤ x 2 + y 2 < (r + 21 )2 } (Figure 4c).
The analytical approach to raster and volume graphics
may lead to important advantages. The analytical digital
geometry is purely discrete and involves simple integer
arithmetics. A digital object is compactly defined by
formulas, which ensures a very economic object encoding.
Analytical definition may help achieve better accuracy of
certain operations, for which one can take advantage of
the corresponding analytical formulas used in the definition.
It should also be noted that after performing voxelization,
the distinct voxels still retain certain information about the
original object. In particular, it is usually straightforward to
determine whether a voxel belongs to an analytical digital
object, to its interior or boundary, or to the intersection or the
union of two or more analytically defined objects. Moreover,
negative effects and phenomena like object disconnectedness
or appearance of holes and tunnels can be understood more
deeply and as a result avoided.
Remark 1. It is easy to realize that the above definitions
extend to the case of a grid whose cells are equivalent
2.3.
3 Note that if one of the coefficients a , a is 0, then the corresponding
1 2
naive and standard lines coincide.
Consider a normal tiling P of Rn , n = 2 or 3, by copies of the
same tile P , so that the following conditions are met: (i) P is
a convex set, (ii) for any P1 , P2 ∈ P there is a translation τ
Hexagonal and honeycomb tilings
The Computer Journal Vol. 48 No. 2, 2005
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V. E. Brimkov and R. P. Barneva
(a)
(b)
(c)
FIGURE 6. (a) Square grid. (b) Regular brick-built grid. (c) Regular hexagonal grid.
(a)
(b)
(c)
(d)
FIGURE 7. (a) Parallelepiped. (b) Hexagonal prism. (c) Rombic dodecahedron. (d) Truncated octahedron.
such that τ (P1 ) = P2 and (iii) if p is a point of a tile P ∈ P,
then the set of the duplicas of p in all tiles from P is a lattice
in Rn . A tiling with these properties is called a uniform tiling.
First we consider uniform tilings of R2 . It is well known
(and also easy to see) that a uniform tiling is possible only if
the tile P is a parallelogram (Figure 5a and b) or a hexagon
whose opposite sides are equal and parallel (Figure 5c). Such
a hexagon is called quasi-regular. The vertices and sides of
the tiles form a grid. The center of a tile is the intersection
point of its diagonals. Sometimes we will identify a tile with
its center. A grid can be regarded as an infinite plane graph
G = (V , E) whose vertex set V and edge set E consist of the
polygons’ vertices and sides respectively. Note that the graph
of the hexagonal grid of Figure 5c is isomorphic to the one of
the parallelogram tiling of Figure 5b. Thus, these two grids
have the same topology, despite the different shape of their
tiles. A tiling/grid such as the one in Figure 5b will be called
a brick-built tiling/grid. If we require the tiles to be regular
polygons, then we obtain the square tiling/grid depicted in
Figure 6a, the regular brick-built tiling/grid in Figure 6b and
the regular hexagonal tiling/grid in Figure 6c.
Note that the brick-built tilings combine the advantages
of the square grid with those of the hexagonal grids. As
demonstrated later, the brick-built topology (as well as the
topology of the hexagonal tiling) supports obtaining tunnelfree objects and better approximations to the continuous
object. On the other hand, as distinct from the hexagonal grid,
the brick-built grid allows easier drawing by graphic devices.
It is well known that R3 can be uniformly tiled by
parallelepiped, hexagonal prism, rhombic dodecahedron or
truncated octahedron (Figure 7). In what follows we will
study and compare models on grids generated by a tile P that
is either a parallelepiped (in particular, cube) or a hexagonal
prism whose bases are quasi-regular hexagons. The notions
of a grid and a tile center are defined analogous to the
2D case.
3.
3.1.
2D ANALYTICAL HONEYCOMB GEOMETRY
Hexagonal coordinate system
Consider the tilings in Figure 6c and b. As mentioned, the
corresponding grids are isomorphic. An important property
of these tilings is that any two non-disjoint tiles are
1-adjacent. In contrast to the case of rectangular grids, here
0-adjacency is impossible. Thus, a connected digital object
is always 1-connected and tunnel-free. We now develop a
basis for analytical digital geometry on hexagonal grids.
We call the hexagonal tiles 2-hexels. The hexels form a
digital hexagonal space M. On it, we define a hexagonal
coordinate system, as follows. We choose a hexel and define
its center O to be the origin of the coordinate system. The
origin’s coordinates are both zeros, i.e. O = (0, 0). Next we
fix two coordinate axes Ob1 and Ob2 as shown in Figure 8a.
The basis vectors b1 and b2 of the coordinate system are
aligned with the coordinate axes.
The centers of the 2-hexels of the digital hexagonal space
form a sublattice L of Z2 , as b1 and b2 form a basis of L. The
cells of this lattice are unit rhombuses obtained by linking the
lattice centers along the Ob1 and Ob2 axes only (Figure 8a).
The so defined hexagonal coordinate system is denoted by
Ob1 b2 . The coordinate axes Ob1 and Ob2 divide the plane
into four quadrants, denoted Quad I, Quad II, Quad III and
Quad IV. The origin O is common for Quad II and Quad IV.
The positive part of Ob1 and the negative part of Ob2 belong
to Quad IV, while the positive part of Ob2 and the negative
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Analytical Honeycomb Geometry
Quad II
b2
b2
Quad I
g
n
O
b1
b1
C2
Quad III
(a)
Quad IV
C1
(b)
FIGURE 8. (a) Hexagonal coordinate system and Quad I, II, III, IV. (b) The cones C1 and C2 . The normal vector n = (a1 , a2 ) to the straight
line g belongs to C1 .
b2
r(x,y)
b1
h(x,y)
(a)
(b)
FIGURE 9. (a) 2-hexel h(x, y) and its adjunct rhombus r(x, y). (b) Hexagonal space M̄ and its adjunct space M̄.
r(−1, 1), r(−1, −1) and r(1, −1)) are 0-adjacent
to r(0, 0).
part of Ob1 belong to Quad II. A 2-hexel with center at a
point (x, y) will be denoted h(x, y).
Now with the help of Figure 10 we observe that:
3.2.
Adjunct rhomboidal space
With the digital hexagonal space M defined above, one can
associate a subsidiary rhomboidal space called the adjunct
rhomboidal space and denoted M̄. We will use it in the
proofs of some statements from the subsequent sections.
With any 2-hexel h(x, y) we associate a rhombus r(x, y),
called adjunct to h(x, y), with the same center and with sides
determined by the basis vectors of the space M, as shown in
Figure 9a. This correspondence defines a rhomboidal digital
space M̄ with the same center and basis vectors (Figure 9b).
We now consider in more detail the relation between the
spaces M and M̄ in view of the adjacency of their cells. We
label a 2-hexel from M (resp. a rhombus from M̄) by the
coordinates of its center. Let h(x, y) be a 2-hexel from M
and r(x, y) the corresponding rhombus from M̄. Without
loss of generality assume that x = y = 0. With a reference
to Figure 10a we notice that:
Fact 1.
• h(0, 0) has six neighbors: h(1, 0), h(0, 1), h(−1, 1),
h(−1, 0), h(0, −1) and h(1, −1). Each of them is
1-adjacent to h(0, 0).
• r(0, 0) has eight neighbors: r(1, 0), r(1, 1), r(0, 1),
r(−1, 1), r(−1, 0), r(−1, −1), r(0, −1) and r(1, −1).
Four of them (r(1, 0), r(0, 1), r(−1, 0) and r(0, −1))
are 1-adjacent to r(0, 0), while the others (r(1, 1),
Fact 2.
(a) Any pair of rhombuses of the form r(i, j ), r(i +
1, j ) or r(i, j ), r(i, j + 1) are 1-adjacent, and the
corresponding pairs of 2-hexels h(i, j ), h(i + 1, j ) or
h(i, j ), h(i, j + 1) are 1-adjacent as well (Figure 10b).
(b) Any pair of rhombuses of the form r(i, j ), r(i − 1, j +
1) are 0-adjacent, while the corresponding 2-hexels
h(i, j ), h(i − 1, j + 1) are 1-adjacent (Figure 10c).
(c) Any pair of rhombuses of the form r(i, j ), r(i + 1, j +
1) are 0-adjacent, while the corresponding 2-hexels
h(i, j ), h(i + 1, j + 1) are disjoint (Figure 10d).
3.3.
Lines, segments and polygons
With respect to the coordinate system Ob1 b2 , one can build
an analog of the analytical discrete geometry in a square grid.
Let a1 x1 + a2 x2 + b = 0 (a1 , a2 , b—rational numbers) be
the equation of a straight line g with respect to the hexagonal
coordinate system. To simplify some considerations we
assume that the coefficients a1 , a2 and b are integers and
that gcd(a1 , a2 ) divides b. This ensures that the line g
contains infinitely many equidistant lattice points. The vector
n = (a1 , a2 ) is the normal vector to g. The line g is collinear
with the vectors v = (a2 , −a1 ) and v = (−a2 , a1 ). The
normal vectors to straight lines g for which the vectors v and
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V. E. Brimkov and R. P. Barneva
b2
–1,1
–1,0
0,1
0,0
1,0
1,1
b1
–1,–1 0,–1 1,–1
(a)
(b)
(c)
(d)
FIGURE 10. Illustration of Fact 2.
to which the normal vector belongs, the width equals the
thickness ω of a standard or a naive line in a square grid (see
Section 2.2). Formally, we have the following theorem.
b2
b1
FIGURE 11. Two ‘orthogonal’ digital line segments with
endpoints (0, 0), (7, 4) and (0, 0), (−4, 5) respectively. The normal
vector to the former belongs to the cone C1 , while the normal vector
to the latter belongs to the cone C2 .
v belong to Quad I and Quad III, form a cone C1 (Figure 8b).
Analogously, the normal vectors to straight lines g for which
the vectors v and v belong to Quad II and Quad IV, form a
cone C2 (Figure 8b). From the definition of Quad I, Quad II,
Quad III and Quad IV it follows that the cone C1 is open
while the cone C2 is closed. Clearly, C1 ∩ C2 = ∅ and
C1 ∪ C2 = R2 .
We now define a digital line corresponding to g, as follows:
g D (a1 , a2 , b) = {(x1 , x2 ) ∈ L : 0 ≤ a1 x1
+ a2 x2 + b + t/2 < t},
where
|a1 | + |a2 |,
if n = (a1 , a2 ) ∈ C1 ,
t=
max(|a1 |, |a2 |), if n = (a1 , a2 ) ∈ C2 .
See Figure 11.
The parameter t is called the width of the line g D . It
equals the number of parallel equidistant straight lines, which
contain centers of hexels from g D .
The so defined digital line g D is the thinnest possible
connected line of this type. For smaller values of t the line
becomes disconnected, while for larger values it contains
extra hexels which, however, do not influence the line
connectivity. We remark that, depending on the quadrant
Theorem 2. A digital line g D (a1 , a2 , b) in a 2D digital
hexagonal space M is 1-connected and tunnel-free.
Moreover, it is the thinnest possible connected digital line
of this type, in a sense that decreasing the line width t
leads to removal of 2-hexels from the line, which makes it
disconnected.
Proof. The proof relies on the observations from Section 3.2
and well-known facts from the theory of arithmetic lines in a
square grid.
Given a line g D (a1 , a2 , b) with a normal vector n =
(a1 , a2 ), we distinguish the cases n ∈ C1 and n ∈ C2 .
Case 1. Let n ∈ C1 , i.e. g D (a1 , a2 , b) has width
t = |a1 | + |a2 |. In the adjunct space M̄ we consider the
set ḡ of rhombuses that correspond to the 2-hexels of g D .
Their centers (x1 , x2 ) satisfy the double linear Diophantine
inequality 0 ≤ a1 x1 + a2 x2 + b + t/2 < t, where
t = |a1 | + |a2 |. Hence, ḡ is a standard line in M̄. Since
a standard line is 1-connected, by Fact 2 we obtain that any
two consecutive 2-hexels of g D are 1-adjacent. Hence, the
whole line g D is 1-connected and tunnel-free as well.
It is easy to see that if an arbitrary pixel is removed from
a standard line, it becomes either disconnected or strictly 0connected (Figure 12a, top). In view of Remark 1 the same
happens in a grid with parallelogram cells (see Figure 12a,
middle). If ḡ is disconnected then the corresponding line
g D is disconnected. If ḡ is strictly 0-connected then g D is
disconnected, as implied by Fact 2. Thus removal of a 2hexel from g D always causes occurrence of a 1-tunnel and
disconnectedness (Figure 12a, bottom.)
It is well known [36] that any arithmetic line of thickness
ω < |a1 | + |a2 | has 1- or 0-tunnels. Consequently, if
we decrease the thickness t = |a1 | + |a2 | of g D , the
corresponding line ḡ in M̄ will have 0- or 1-tunnels and
therefore g D will be disconnected.
Case 2. Let n ∈ C2 , i.e. g D (a1 , a2 , b) has width
t = max(|a1 |, |a2 |). Consider in the adjunct space M̄ the set
ḡ of rhombuses corresponding to the 2-hexels of g D . Their
centers (x1 , x2 ) satisfy 0 ≤ a1 x1 + a2 x2 + b + t/2 < t,
The Computer Journal Vol. 48 No. 2, 2005
Analytical Honeycomb Geometry
(a)
187
(b)
FIGURE 12. Connectivity and tunnels in lines. (a) Standard lines: (top and middle) removal of a pixel makes the line either disconnected
or strictly 0-connected; (bottom) removal of an arbitrary 2-hexel makes the line disconnected. (b) Naive lines: removal of an arbitrary cell
makes the line disconnected.
where t = max(|a1 |, |a2 |). Hence, ḡ is a naive line in M̄.
A naive line is 0-connected and by Fact 2 it follows that any
two consecutive 2-hexels of g D are 1-connected. Hence, g D
is 1-connected and tunnel-free.
If we remove an arbitrary pixel of a naive line ḡ (in a square
or parallelogram grid), it becomes disconnected (Figure 12b,
left and middle). Then the corresponding line g D in M is
disconnected as well (Figure 12b, right).
It is well known [36] that any arithmetic line of thickness
ω < max(|a1 |, |a2 |) has 1-tunnels and, thus, is disconnected.
Hence, if we decrease the thickness t = max(|a1 |, |a2 |) of
g D , the corresponding line ḡ in M̄ will be disconnected.
Then g D will be disconnected as well.
Let us mention that a digital line in a 2D hexagonal space is
always 1-connected, even if its thickness matches the width of
a naive line in a square grid. In this last case a digital line g D
still possesses the connectivity of a standard line in a square
model. Note that this superior connectivity is achieved with
essentially fewer 2-hexels. This is an important advantage
for obtaining higher algorithmic efficiency of digital line
generation.
Given two hexels h1 and h2 one can obtain a tunnel-free
digital straight line segment with endpoints h1 and h2 . On
this basis one can construct tunnel-free digital polygons as a
sequence of digital line segments determined by the polygon
vertices.
3.4.
Digital circles
The circle is another basic Euclidean primitive admitting an
easy analytical definition. In a square grid a digital circle
has been analytically defined and studied in [29, 42, 43].
As already mentioned in Section 2.2 one can consider the
following digital circle with center at O(0, 0) and radius
r ∈ N.
2
2 2
2
2
1
1
C(r) = (x, y) ∈ Z : r − 2 ≤ x + y < r + 2
.
(1)
Note however that, in contrast to the case of lines and planes,
the above definition cannot be directly applied to a hexagonal
grid since it may define a set of 2-hexels which is too far from
the intuitive idea of a circle and, in fact, can be disconnected
(Figure 13a). Therefore, a new definition is needed. To
this end consider a hexagonal coordinate system Ob1 b2 with
hexagonal
unity α.4 We first observe that a point (x, y) of the √
coordinate system has coordinates (α(x + 21 y), α 23 y) with
respect to the ordinary square coordinate system with unity
1, origin (0, 0) and abscissa Ob1 . Then the equation of a
Euclidean circle in the hexagonal system becomes
2
√ 2
α 2 x + 21 y + α 2 23 y = r 2 ,
i.e. α 2 (x 2 + xy + y 2 ) = r 2 .
Assuming the unity α to be equal to one, we obtain the
following analytical definition of a digital circle with center
O(0, 0) and radius r.
2
D
C (r) = (x, y) ∈ L : r − 21 ≤ (x 2 + xy + y 2 )
2 1
< r+2
.
(2)
See Figure 13b. We have the following theorem.
Theorem 3. (a) A digital circle defined by Equation (2) is
always 1-connected and tunnel-free. (b) Concentric digital
circles with radii r = 1, 2, 3, . . . fill the whole plane.
Proof. In order to prove the statement, it suffices to show
that C D (r) has no 1-tunnels. This last fact follows by simple
argument which, however, requires some preparation.
C D (r) is symmetric with respect to the axes b1 , b2 and the
line with equation
√ (with respect to the orthogonal coordinate
system) y = 3x (Figure 14a). Therefore, it is enough to
restrict our considerations to the portion of C D (r) delimited
by the positive directions of b1 and b2 (Figure 14b).5 Let us
D (r).
denote this portion by C+
4 The unity α is the length of the basis vectors b and b . It is easy to
1
2
√
see that α = 3a, where a is the length of a hexel side.
5 Note that in [21] all three segments of a half circle are handled
separately.
The Computer Journal Vol. 48 No. 2, 2005
188
V. E. Brimkov and R. P. Barneva
b2
O
b1
(a)
(b)
FIGURE 13. (a) The set of hexels satisfying condition (1) with r = 3. (b) Concentric digital circles defined by formula (2), for
r = 1, 2, 3, 4, 5.
b2
6
5
4
3
2
B
3
2
1
(a)
A
4
5
b1
1
(b)
(c)
FIGURE 14. (a) Symmetries of a digital circle. (b) and (c) Illustrations of the proof of Theorem 3.
We introduce a counterclockwise order on the 2-hexels of
D (r), as follows:
C+
• The 2-hexel h(r, 0) is assigned number 1.
D (r) be numbered by a natural
• Let a 2-hexel h(i, j ) ∈ C+
number k.
D (r), then h(i − 1, j ) is numbered
If h(i − 1, j ) ∈ C+
by k + 1.
D (r) but h(i, j + 1) ∈ C D (r),
Else, if h(i − 1, j ) ∈
/ C+
+
then h(i, j + 1) is numbered by k + 1.
D (r) but
Else, if h(i − 1, j ), h(i, j + 1) ∈
/ C+
D
h(i − 1, j + 1) ∈ C+ (r), then h(i − 1, j + 1) is
numbered by k + 1.
See Figure 14b which provides two examples of enumerD (3) and C D (4). Clearly the proposed ordering
ation for C+
+
D (r).
assigns consecutive numbers to all 2-hexels of C+
We also have the following plain fact. Let a 2-hexel
D (r) be numbered by k > 1. Then the 2-hexel
h(i, j ) ∈ C+
numbered by k − 1 is either h(i, j − 1) or h(i + 1, j − 1)
or h(i + 1, j ), and the 2-hexel numbered by k + 1 is either
h(i, j + 1) or h(i − 1, j + 1) or h(i − 1, j ).
D (r) has a 1-tunnel.
Now assume by contradiction that C+
It is easy to realize that this is possible only if there is
D (r), such that one or both of the
a 2-hexel h(i, j ) ∈ C+
following conditions hold6 :
D (r);
(i) h(i, j − 1), h(i + 1, j − 1), h(i + 1, j ) ∈
/ C+
D (r).
(ii) h(i, j + 1), h(i − 1, j + 1), h(i − 1, j ) ∈
/ C+
We assume that (i) is the case, the case (ii) being analogous. Then, since the radii of the two Euclidean circles
C(r − 21 ) and C(r + 21 ) that determine C D (r) differ by one,
the length of the segment AB must be >1 (see Figure 14c).
Thus we obtain that the distance between the centers of the
hexels h(i, j + 1) and h(i − 1, j + 1) is >1, which is a
contradiction.
Theorem 3 (b) follows from the fact that every point with
integer coordinates in Ob1 b2 falls in between two concentric circles whose radii differ by one. Hence every 2-hexel
belongs to a certain digital circle C D (r).
A circle C D (r) can be thought as having thickness one,
since it is bounded between two circles whose radii differ
by one. As Theorem 3 demonstrates, a circle C D (r) is
6 Since h(r, 0) ∈ C D (r) we have that C D (r) = ∅.
+
+
The Computer Journal Vol. 48 No. 2, 2005
189
Analytical Honeycomb Geometry
x
v’
1
x 3/4 x
v
3/4
u
(a)
(b)
(c)
(d)
FIGURE 15. Extreme cases for which a 2-hexel of a digital line has maximal deviation from the continuous line: (a) in a square grid, (b) in
a regular hexagonal grid, (c) in a regular brick-built grid and (d) in the optimal brick-built grid.
always tunnel-free. Note that in a square grid a digital circle
C(r) defined by Equation (1) has 0-tunnels, in general, as
demonstrated in [29]. In order to ensure tunnel-freedom
larger thickness should be taken.
We notice that a digital circle C D (r) may contain a
type of ‘extra’ 2-hexels (see the circle with radius 3 in
Figures 13b and 14b). Nevertheless, C D (r) provides better
approximation to the Euclidean circle, as we will see in
Section 3.5.
We remark that the digital circle C D (r) can be obtained
by applying the algorithm proposed in [21], with possible
technical simplifications based on our analytical approach.
It is also worth mentioning that in [14] digital circles
in a hexagonal grid have been defined by certain distance
transforms providing approximations to the Euclidean
distance transform. The theoretical considerations and
examples from [14] provide evidence that digital circles
obtained by distance transforms and those obtained
analytically become more and more similar as the radius of
the circle increases. Carrying out a more detailed comparison
is seen as a further task. Here we only mention that, according
to [14], the Euclidean distance transform (that is implicitly
related to our analytical definition) appears to be ‘the ideal
that the other distance transforms try to achieve’.
3.5.
Optimality of the honeycomb model
To have a ground for comparison between models built upon
square or hexagonal grids, we suppose that the square and
the hexagonal tiles have the same area 1. We call such tiles
the unit square and the unit 2-hexel respectively. It is easy to
calculate
√ that√the√length of a side of the unit 2-hexel is equal
to a = 2/( 3 4 3) = 0.62040 . . . .
3.5.1. Quality of approximation
We show next that in certain sense the hexagonal grid
provides better approximation to a continuous line segment
than the square and the brick-built grids. To this end we first
recall the definition of Hausdorff distance. Let E be a metric
space with metric d, and E a family of closed non-empty
subsets of E. For every x ∈ E and every A ∈ E let d(x, A) =
inf{d(x, y) : y ∈ A}. Then, given two sets A, B ∈ E,
Hd (A, B) = max{sup{d(a, B) : a ∈ A}, sup{d(A, b) : b ∈
B}} is called the Hausdorff distance between A and B. We
will suppose that d is the Euclidean metric.
We measure the deviation of a straight line discretization
g D from the corresponding continuous line g by the
Hausdorff distance Hd (g D , g) between them.
We have the following results.
Theorem 4. In the regular hexagonal grid the obtained
approximation to a straight line is optimal over all possible
uniform tilings of the plane, in terms of minimizing the
maximal possible deviation from the continuous object.
Proof. The proof consists of a number of steps in which
different grids are compared.
Let us first compare the hexagonal and the square grids.
As Figure 15a demonstrates, a tunnel-free (standard) digital
line might contain
√ a pixel (or pixels) whose farthest point is
in a distance 2 = 1.41421 . . . from the digitized straight
line. For the hexagonal grid the maximal possible deviation
is reached in the extreme
case illustrated
in√Figure
√ 15b.
√
√
It is equal to u = ( 13/2)a = ( 13/2)( 2/ 4 27) =
1.11844 . . . , which is considerably less than that in the case
of square grid.
Consider now the brick-built grid whose tiles are unit
squares. The maximal possible deviation for this grid is
reached in the case shown in Figure 15c and equals v = 1.25.
This is smaller than the corresponding value for a square grid
but larger than the one for a hexagonal grid.
Note that in a brick-built grid framework unit square tiles
do not provide the best possible approximation. In order
to determine the dimensions of the optimal rectangle, with
reference to Figure 15d, let us denote the length of its
horizontal side by x. Then the length of the other side is
1/x. Now we have to determine x which minimizes the
function f (x) = 1/x 2 + 43 x 2 . Using simple calculus we
find that the solution to the above optimization problem is
x = √2 = 1.15470 . . . . Then the other side has length
3
1/x = 0.86602 . . . . A grid with such a tile will be called the
optimal brick-built grid. For this grid the maximal deviation
is equal to v = ((1/x)2 + ( 43 x)2 )1/2 = 1.22474 . . . .
It is easy to show that the maximal deviation of a digital line
in a regular hexagonal grid is minimal over all quasi-regular
The Computer Journal Vol. 48 No. 2, 2005
190
V. E. Brimkov and R. P. Barneva
hexagonal grids. We also have that the maximal deviation
of a digital line in the optimal brick-built grid is minimal
over all possible brick-built grids. Thus we obtain the result
stated.
y
b1
T
b2
Pl
0
x=y
Q
S
We have seen that the specific geometry of the brick-built
grids is superior to the geometry of the square grid regarding
approximability of continuous linear objects. Together with
the useful topological properties, this makes the brick-built
grids quite an interesting subject of study both from the
theoretical and practical point of view.
Similar results hold true for the maximal possible deviation
of a digital circle from the corresponding Euclidean circle.
We have the following theorem.
Theorem 5. A digital circle C D (r) in a 2D hexagonal
space provides better approximation to the Euclidean circle
than a digital circle defined by Equation (1) in a square grid.
Proof. By |XY | we denote the length of a segment XY with
endpoints X and Y .
Consider first a circle C D (r) in a hexagonal space M
and an arbitrary 2-hexel h(x, y) ∈ C D (r). It is clear
that a maximal deviation is reached in the case depicted in
Figure 16a. Actually, this extreme case is only a hypothetical
one, since the center of a 2-hexel from C D (r) cannot lie on
the outer circle defining C D (r) [see the strict right inequality
in Equation (2)]. Hence the maximal deviation is bounded
by |AB| + |BC| = 0.5 + a = 0.5 + 0.62040 . . . =
1.12040. . . < 1.1205. We will show that the maximal
possible deviation of a digital circle C̄(r) in a square grid
is larger than the above upper bound 1.1205.
Consider the infinite family of Euclidean circles C(r +
1
),
r = 1, 2, . . . , where the radius r is a variable parameter.
2
Every circle C(r + 21 ) is an outer circle in the definition of
a digital circle C̄(r) in the square grid. Consider the points
of the form Pl (l, l), where l is a positive integer. Let ε > 0
be an arbitrary real constant. It is not hard to show that there
is a point Pl0 (l0 , l0 ) and a circle C(k0 + 21 ) with a radius
k0 + 21 for some positive integer k0 , such that the distance
from Pl0 to C(k0 + 21 ) is less than ε. To see this consider the
line with equation x = y and its intersection point Q with a
circle C(k + 21 ). The coordinates of Q satisfy the system of
equations
x 2 + y 2 = (k + 21 )2
x = y,
√
from where we obtain x = (k + 21 )/ 2, i.e. Q =
√
√
((k + 21 )/ 2, (k + 21 )/ 2). We need the Euclidean distance
between the points Q and Pl to be less than ε, i.e.
2 2
k+ 1
+ 21
√ 2 − l + k√
− l < ε,
2
2
√
from where we get |k + 21 − l 2| < ε. Then the existence
of positive integers k and l satisfying the above inequality
A
x
B
C
(a)
(b)
FIGURE 16. Illustrations of the proof of Theorem 5.
immediately follows from the well-known Kronecker’s
theorem.7
Let us now fix an ‘enough small’ ε, for instance ε =√0.01.
Let l0 and k0 be positive integers satisfying |k0 + 21 −l0 2| <
0.01. Then the Hausdorff distance from the point Pl0 (l0 , l0 )
to the circle C(k0 + 21 ) is given by the length |Pl0 Q| of
the segment with endpoints Pl0 and Q (Figure 16b). By
construction, |Pl0 Q| < 0.01. Let pl0 be the pixel with center
at the point Pl0 and T be the farthest point (a vertex) of pl0 to
the circle C(k0 ). The distance from T to C(k0 ) is given by the
length |T S| of the segment with endpoints T and S, where the
point S belongs to C(k0 ) and to the line x = y (Figure 16b).
Then the Hausdorff distance between the digital circle C̄(k0 )
and the Euclidean circle C(k0 ) satisfies
Hd (C̄(k0 ), C(k0 )) ≥ Hd (pl0 , C(k0 ))
√
2
− |Pl0 Q|
=|ST | = 0.5 +
2
√
√
2
2
> 0.5 +
− ε = 0.5 +
− 0.01
2
2
> 0.5 + 0.707 − 0.001 = 1.206.
Thus we showed that for the square grid the maximal
possible deviation is >1.206, while for the 2D hexagonal
grid it is <1.1205. This completes the proof.
3.5.2. Grid cost
We conclude this section with one more observation,
revealing that the hexagonal grid is in a sense ‘more
economic’ than the square grid. For a given grid H , the total
length of all its edges is called the cost of H and denoted
c(H ). One can calculate that a unit 2-hexel has perimeter
6a = 3.72241 . . . , which is less than the perimeter of the
unit square, i.e. 4. This fact may have an advantageous
impact on the cost of the corresponding grid, as illustrated by
the following example.
Example 1. Consider the grid of a screen with a square
shape, consisting of n rows, each of them containing n
unit tiles. Consider the square grid H1 (Figure 17a) and
hexagonal grid H2 (Figure 17b). Both grids cover the same
7 Kronecker theorem states that for any irrational number α and real
number ε > 0 there exist natural numbers m and n for which |mα − n| < ε.
The Computer Journal Vol. 48 No. 2, 2005
Analytical Honeycomb Geometry
(b)
(a)
FIGURE 17. Illustration of Example 1.
area n2 . One can easily find that in the former case the
grid cost is c(H1 ) = 2n2 + 2n, while in the latter case it
is c(H2 ) = (3n2 + 4n − 1)a = (3a)n2 + 4an − a. Since the
coefficient of n2 in c(H2 ) is 3a = 1.86120 . . . <2, we obtain
that c(H2 ) is asymptotically smaller than c(H1 ).
It is easy to show that a unit 2-hexel has a minimal
perimeter among all quasi-regular hexagons with area 1. One
can also show that a tiling by unit 2-hexels has minimal cost
among all possible tilings by quasi-regular hexagons with
area 1. Similarly, a unit square has a minimal perimeter
among all parallelograms with area 1, and a tiling by a
unit square has a minimal cost over all possible tilings by
parallelograms with area 1. Thus, one can conclude that
the tiling with regular hexagons has minimal cost over all
possible uniform tilings.8 This property may be of interest
for possible more economical wire grid fabrication of a novel
computer screen design based, e.g. on liquid-crystals, plasma
panels or electroluminescent technologies. On the other
hand, however, the properties of the square tiling make its
drawing easier and more efficient.
4.
3D HONEYCOMB GEOMETRY
In this section we present two 3D honeycomb models and the
related analytical digital geometry. In both of them the 3D
space is tiled by a right hexagonal prism called 3-hexel. The
base of the prism is a unit 2-hexel and its height has length 1.
Thus the 3-hexel volume equals 1.
As briefly indicated earlier one of our main concerns is
to design 3D honeycomb spaces that provide a basis for
obtaining tunnel-free discretization of surfaces, in particular,
planes and spheres. To support the better understanding of
tunnel-freedom issues in a honeycomb environment, we start
by introducing a ‘naive’ 3D honeycomb model, labeled as
Model 0.
4.1.
Model 0
Consider a discrete space M0 consisting of slices composed
by 3-hexels placed consecutively one on top of the other
(Figure 18).
8 It is based on this fact that bees are known to minimize the quantity of
materials used to build honeycombs.
191
We observe that in this space a connected set of 3-hexels
may have 1-tunnels. For example, two 3-hexels such as those
in Figure 18b are 1-adjacent and there is a 1-tunnel between
them. As another example, consider the configuration of
the four 3-hexels h1 , h2 , h3 and h4 in Figure 18c. It can be
thought of as a portion of a digital surface in M0 . One can
easily see that this set is 2-connected, but there is a 1-tunnel
between the hexels h1 and h4 .
In order to eliminate the possibility for such kind of
undesirable cases, in what follows we introduce more
sophisticated 3D honeycomb spaces in which any connected
set of 3-hexels is always 2-connected and tunnel-free (i.e. no
0- or 1-tunnels may occur).
4.2.
Model I
Let h0 be a 3-hexel in R3 . We assign the coordinates (0, 0, 0)
to its center O and consider it as the origin of a coordinate
system Oe1 e2 e3 , defined as follows.
The direction of the third axis e3 coincides with the
direction of the height of h0 (Figure 19b). The other two
axes are determined as follows. Consider the plane P passing
through the origin and orthogonal to e3 . Consider a hexagonal
tiling T of P generated by the 2-hexel that is an orthogonal
projection of h0 over P (Figure 19a). Starting from h0 we
assign to every 2-hexel of T a 3-hexel, in a way that the third
coordinates of two side-neighboring 3-hexels differ either by
1
2
3 or by 3 . This assures that any two neighboring 3-hexels
are 2-adjacent. In Figure 19a the third coordinates of the 3hexels (i.e. of their centers) are marked on the corresponding
2-hexels from T . The hexels’ centers belong to a plane which
is chosen to be one of the digital coordinate planes. In it we
fix the coordinate axes e1 and e2 , as illustrated in Figure 19.
Figure 19a exposes the projections of e1 and e2 on P. The
origin O and the axes e1 , e2 and e3 determine the coordinate
system Oe1 e2 e3 .
Now for each 3-hexel h from the digital coordinate plane
Oe1 e2 , we tile the space upward and downward by adjacent
copies of h. Thus, we obtain a tiling of R3 , which defines a
digital honeycomb space M1 . In the digital coordinate plane
Oe1 e2 one can consider the four quadrants QuadI, QuadII,
QuadIII and QuadIV, determined by the coordinate axes.
Figure 19b shows the three coordinate axes of M1 , and
Figure 19c exposes a portion of a digital quadrant determined
by the axes e1 and e2 .
By construction, the centers of the 3-hexels form a lattice
H in R3 whose cells are parallelepipeds. Its basis vectors
e1 , e2 and e3 are also basis vectors for the coordinate system
Oe1 e2 e3 . We denote the length of e1 and e2 by α and the
length of e3 by β.√Note that α and β may be different. As in
the 2D case α = 3a, where a is the length of the side of a
hexagonal face. In√general,
√ √β may be an arbitrary positive real
number. If a = 2/( 3 4 3) then the 3-hexel’s hexagonal
face has area 1. If, in addition, β = 1, then the 3-hexel has
volume 1.
In contrast to space M0 , in the so constructed digital
honeycomb space M1 , every two neighboring 3-hexels are
2-adjacent. No 0- or 1-tunnel is possible in a connected
The Computer Journal Vol. 48 No. 2, 2005
192
V. E. Brimkov and R. P. Barneva
h1
h2
h3
h4
(a)
(b)
(c)
FIGURE 18. Illustration of Model 0.
e3
e1
Proj e2
–2
–5/3 –4/3 –1
–5/3 –4/3 –1
–4/3 –1
–1/3
–2/3 –1/3
–2/3 –1/3
–1
–2/3 –1/3
0
0
1/3
0
1/3
2/3
–2/3 –1/3
–2/3 –1/3
0
1
2/3
1
4/3
2/3
1
1/3
2/3
1
4/3
4/3
5/3
0
1/3
1/3
1/3
2/3
0
5/3
2
2/3
1
5/3
2
7/3
e1
1
O
4/3
4/3
e3
O
5/3
2
Proj e1
7/3
8/3
(a)
(b)
e2
(c)
e2
FIGURE 19. Illustration of Model I. (a) The orthogonal projections of the centers of 3-hexels on the plane P. (b) The axes of the coordinate
system Oe1 e2 e3 . The rays e1 and e2 make an angle of 60◦ . (c) Portion of a digital quadrant determined by the axes e1 and e2 .
e3
Proj e2
–5/3 –5/3 –4/3
–5/3 –4/3
–4/3
1
2/3
–2/3 –1/3
–1/3
0
–1
1/3
–1
–2/3 –1/3
1/3
0
0
1/3
2/3
1/3
2/3
1
–1
–2/3 –1/3
–2/3 –1/3
0
1/3
2/3
1
4/3
0
5/3
2/3
1
4/3
1/3
2/3
1
4/3
5/3
2
0
1/3
7/3
4/3
5/3
2
2/3
1
1
4/3
5/3
e1
2
7/3
Proj e1
8/3
–e 2
O
(a)
(b)
FIGURE 20. Illustration of a brick-built version of Model I. The rays e1 and −e2 make an angle of 120◦ .
set of 3-hexels. We also mention that every 3-hexel has
one 2-neighbor for the upper hexagonal face, one for the
lower hexagonal face and two for every one of the six side
rectangular faces, i.e. 14 neighbors overall.
As in 2D, one can consider brick-built versions of the
model, where the hexagonal prisms are substituted by
rectangular prisms (Figure 20).
4.3.
Model II
Consider a regular hexagonal tiling of the plane. On every
tile we build the corresponding 3-hexel. Thus we obtain a
digital 3D plane P0 . We fix one of its hexels and choose its
center O(0, 0) to be the origin of a coordinate system. Then
we define the axes e1 and e2 as in the 2D case. Thus we obtain
a digital coordinate plane Oe1 e2 . We now build on top of P0
The Computer Journal Vol. 48 No. 2, 2005
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Analytical Honeycomb Geometry
e3
e3
e2
e2
e2
e1
Proj e3
e1
e1
O
O
(b)
(a)
(c)
FIGURE 21. Illustration of Model II. (a) Projections of two consecutive slices of 3-hexels on the coordinate plane Oe1 e2 . (b) The axes of
the coordinate system Oe1 e2 e3 . (c) Portion of the digital quadrant determined by the axes e1 and e2 .
e3
e2
e2
Proj e3
e1
O
e1
(b)
(a)
FIGURE 22. Illustration of a brick-built version of Model II.
another equivalent ‘slice’ of 3-hexels, shifted as illustrated in
Figure 21a. Proceeding analogously upward and downward,
we obtain a tiling of R3 . The tiles’ centers constitute a lattice
in R3 whose cells are parallelepipeds. This lattice determines
the third axis e3 as shown in Figure 21b. The basis vectors
of the coordinate system are the basis vectors e1 , e2 and e3
of the√underlying lattice. The length of e1 and e2 is equal to
α = 3a and the length β of e3 is a positive real number
satisfying the condition β > 43 a. (This last inequality implies
that the hexagonal prism tile has positive height.) Thus we
obtain a digital honeycomb space M2 (Figure 21).
We mention that for every 3-hexel of the digital space M2
there are four neighbors for every one of the two hexagonal
faces and one for each of the six rectangular faces, i.e. 14
neighbors overall.
The brick-built version of Model II is illustrated in
Figure 22.
4.4.
Adjunct spaces
Similar to the 2D case one can associate the digital
honeycomb spaces M1 and M2 with the adjunct digital
spaces M̄1 and M̄2 respectively, whose cells are
parallelepipeds. Since the hexels’ centers form a lattice, for
P’
e3
(x,’y,’ z’)
P
e2
(x,y,z)
e1
FIGURE 23. Constructing an adjunct 3-hexel. All vertices of the
parallelepiped P are centers of 3-hexels from M1 .
any 3-hexel h one can define a parallelepiped p, called adjunct
to h, which has the same center as h.
For definiteness consider the space M̄1 .
Given a
3-hexel h(x, y, z) with center (x, y, z), the corresponding
parallelepiped p(x, y, z) can be obtained as follows.
Consider the parallelepiped p (x , y , z ) with center at the
point (x , y , z ) and generated by the basis vectors e1 , e1 , e1
at the point (x, y, z) (Figure 23).
By construction of the space M1 , the other vertices
of p (x , y , z ) are centers of 3-hexels of M1 . Then
The Computer Journal Vol. 48 No. 2, 2005
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V. E. Brimkov and R. P. Barneva
the parallelepiped p(x, y, z) is obtained by translation
of p (x , y , z ) with a translation vector (x − x , y −
y , z − z ). Thus we obtain a 3D digital space M̄1
with the same center and basis as M1 , whose cells are
parallelepipeds.
4.5.
Analytical digital planes in models I and II
Consider a Euclidean plane P with equation a1 x1 + a2 x2 +
a3 x3 = b, a1 , a2 , a3 , b ∈ Z, with respect to the coordinate
system in the space M1 or M2 . Assume, as before, that
gcd(a1 , a2 , a3 ) divides b, i.e. P contains a 2D lattice L
which is a sublattice of H .
We define in M1 (resp. M2 ) a digital plane corresponding
to P , as follows:
P D (a1 , a2 , a3 , b) = {x ∈ H : 0 ≤ a1 x1 + a2 x2 + a3 x3
+ b + t/2 < t},
(3)
where
t=
|a1 | + |a2 | + |a3 |,
if n = (a1 , a2 ) ∈ C1 ,
max(|a1 |, |a2 |, |a3 |), if n = (a1 , a2 ) ∈ C2 .
C1 and C2 are cones in the coordinate plane Oe1 e2 defined
as in the 2D case. The parameter t is called the width of the
plane P D .
One can show that in both spaces M1 and M2 , the
so defined digital plane is 2-connected and tunnel-free.
Moreover, it is the thinnest possible digital plane of this
type. For this one can use the adjunct spaces M̄1 and M̄2
introduced in Section 4.4.
For definiteness consider Model I and the spaces M1 and
M̄1 . Because of the one-to-one correspondence between the
3-hexels of M1 and the parallelepiped cells of M̄1 , there is
a one-to-one correspondence between the 3-hexels and the
parallelepiped cells of a digital plane defined by formula (3)
in M1 and M̄1 respectively. We also take advantage of the
well-known fact [41] that in the cubic grid, a standard plane
(i.e. one of width ω = |a1 | + |a2 | + |a3 |) is always tunnelfree, while a naive plane (of width ω = max(|a1 |, |a2 |, |a3 |))
is 2-tunnel-free, but has 0- or 1-tunnels (unless two of the
coefficients a1 , a2 , a3 equal 0). Moreover, the standard/naive
plane is the thinnest arithmetic plane that is tunnel-free/
2-tunnel-free. Then, using the approach of the proof of
Theorem 2, one can show that a digital plane P D in the
space M1 is tunnel-free, and for the so defined thickness it
is the thinnest possible tunnel-free plane of this type. The
considerations are technically more involved and lengthy,
although the basic steps are analogous to those of the proof
of Theorem 2. Therefore, the details are omitted here and
left for a future exercise.
As mentioned above, in the cubic grid a standard plane is
always tunnel-free, while a naive plane is 2-tunnel-free, but
has 0- or 1-tunnels. In contrast, in models I and II a digital
plane is always tunnel-free, even if its width t equals that of
a naive plane.
4.6.
Digital spheres in models I and II
In the cubic model an analytical digital sphere with center
O(0, 0, 0) and radius r ∈ N is defined as follows [29, 43]:
2
C(r) = (x, y, z) ∈ Z3 : r − 21 ≤ x 2 + y 2 + z2
2 1
.
(4)
< r+2
In order to obtain a reasonable definition within the
considered honeycomb models, we have to make certain
relevant transformations.
Consider first the honeycomb coordinate system Oe1 e2 e3
of Model I. It is not hard to compute that a point (x, y, z)
of the honeycomb
coordinate system has coordinates (α(x +
√
3
1
1
1
y),
α
y,
x
−
2
2
3
3 y +βz) with respect to a cubic coordinate
system Oe1 e2 e3 with unity 1, centered at O(0, 0, 0), and such
that the projection of the basis vector e1 over the coordinate
plane Oe1 e2 has z-coordinate 13 , and the projection of the
basis vector e2 over Oe1 e2 has z-coordinate − 13 . Then the
equation of a Euclidean sphere in the honeycomb coordinate
system is
2
2
α 2 x + 21 y + 43 α 2 y 2 + 13 x − 13 y + βz = r 2 ,
or, equivalently,
α 2 + 19 x 2 + α 2 + 19 y 2 + β 2 z2 + α 2 − 29 xy
+ 23 β xz − 23 β yz = r 2 .
This equation implies a definition of a digital honeycomb
sphere.
In Model II a digital sphere is defined analogously. It is
not hard to calculate that the equation of a sphere with center
O(0, 0, 0) and radius r is
√
α 2 x 2 + α 2 y 2 + 49 a 2 + β 2 z2 + α 2 xy + 3 2 3 αa xz
√
+ 3 2 3 αa yz = r 2 ,
which immediately implies an analytical definition of a
digital sphere.
See Figure 24 for illustrations of spheres in models I
and II.9
A digital honeycomb sphere in models I and II is
2-connected and tunnel-free. Moreover, concentric spheres
with radii 1, 2, 3, . . . fill the whole space. The argument
proving this last fact is the same as in the case of a digital
honeycomb circle. In fact, every lattice point (center of a
3-hexel) falls in between two concentric spheres whose radii
differ by one. Thus we can conclude that every 3-hexel of
M1 (resp. M2 ) belongs to a digital sphere, which completes
the proof. The proof of 2-connectivity and tunnel-freedom is
similar (although technically more involved) to the proof of
Theorem 3, therefore details are omitted. Issues concerning
quality of approximation are addressed in Section 4.8.
9 Simple visualization programs implementing models I and II have been
developed in OpenGL. They allow to scan-convert (discretize) objects in
these models and display them from different view points.
The Computer Journal Vol. 48 No. 2, 2005
195
Analytical Honeycomb Geometry
(a)
(b)
(c)
(d)
(e)
(f)
FIGURE 24. Illustration of digital spheres. (a–c) Spheres in the space M1 with radii r, (a) r = 10, (b) r = 7 and (c) r = 5. (d–f) Spheres
in the space M2 with radii r, (d) r = 10, (e) r = 7 and (f) r = 5.
4.7.
Uniqueness of models I and II
In this section we further study the structure of a 3-hexel
neighborhood in models I and II. We have already seen
that in both models, a connected set of 3-hexels is always
2-connected and does not contain any 0- or 1-tunnels.
Moreover, the adjacency relation µ defined by the nearest
neighbor forms a good pair (µ, µ).
We remark that in models I and II the geometric position
of two neighboring hexels is strictly fixed. In Model I, the
z-coordinates of any two hexels of the digital space M1
differ by an integer multiple of 13 (Figure 19). In Model
II, the digital space M2 consists of ‘slices’, shifted in such
a way that the orthogonal projection of a hexel’s center
over the coordinate plane Oe1 e2 falls exactly in the middle
of a side of a hexagonal hexel’s face from the lower slice
(Figure 21). Note that these models can be changed to a
certain extent, so that the obtained tilings are still uniform
and any two adjacent tiles are 2-adjacent. For instance,
in the framework of Model II, these properties would be
preserved if two consecutive slices are shifted in any arbitrary
manner, unless the projection of a hexel’s center falls on a
diagonal of a hexagonal hexel’s face from the lower slice
(Figure 25a and b). Model I admits various modifications
as well. One can see that all modifications preserving the
required properties must satisfy the following conditions.
(i) In Model I, every 3-hexel of the digital space M1 has
one 2-neighbor for the upper hexagonal face, one for
the lower hexagonal face and two for each one of the
six side rectangular faces.
(ii) In Model II, for every 3-hexel of the digital space
M2 there are four neighbors for every one of the two
hexagonal faces and one for each of the six rectangular
faces.
Modifications of models I and II satisfying the above
conditions will be called feasible. Note that although the
structure of a 3-hexel neighborhood in feasible modifications
of models I and II is quite different, in both models a 3-hexel
has the same number of neighbors (i.e. 14).
We now show that the neighborhood structures relative
to models I and II are the only possible structures, under
the condition that the tiling is uniform and any two adjacent
tiles are 2-adjacent. Consider a 3-hexel h. The following
two cases are possible. Case 1—any hexagonal face of h is
2-adjacent to exactly one hexagonal face of another 3-hexel;
Case 2—any hexagonal face of h is 2-adjacent to exactly four
hexagonal faces of four other 3-hexels. (Clearly, a hexagonal
face of h cannot be covered by two hexels only. If this is
done by three hexels then in the tiling there must be pairs of
hexels which share a single vertex. This would contradict the
condition that any two adjacent tiles are 2-adjacent.)
Consider Case 1. We will show that every one of the
rectangular side faces of h is 2-adjacent to exactly two 3hexels h1 and h2 , as h2 is on top of h1 and the vertical edges
of h, h1 and h2 are aligned (Figure 25c). To see this assume,
The Computer Journal Vol. 48 No. 2, 2005
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V. E. Brimkov and R. P. Barneva
h
(a)
(b)
h2
h2
h1
h1
(c)
h
h
(d)
h1
(e)
FIGURE 25. (a) The orthogonal projection of a hexel’s center over the lower slice belongs to a diagonal of a hexagon. (b) The orthogonal
projection of a hexel’s center over the lower slice in a feasible modification of Model I. (c–e) Illustrations of the proof of Theorem 6.
by contradiction, that h is 2-adjacent only to h1 and shares
a common rectangular face with it. Then h and h2 share
an edge (Figure 25d) and there is a 1-tunnel between h and
h2 —a contradiction. Now assume that the vertical edges of
h and that of its neighbor h1 are not aligned (Figure 25e).
Then clearly R3 cannot be tiled by a 3-hexel, since the space
angle between some faces of h and h1 will equal 60◦ , which
is a contradiction. Similar reasoning shows that in Case 2,
every rectangular side face of h will be shared with another
3-hexel. Thus we have proved the following important
result.
Theorem 6. The two neighborhood structures relative to
feasible modifications of models I and II are the only ones
possible, under the condition that the tiling is uniform and
any two adjacent tiles are 2-adjacent.
Q
A
B
(a)
Optimality issues
Let us define a grid cost in a 3D model as the total area of the
surfels (faces) of the tiles included in certain volume (e.g. in
a cube with a side n). We notice that the area of a 3-hexel is
equal to 2 × 1 + 6 × a = 2 + 3.72241 . . . = 5.72241 . . . ,
which is less than the one of the unit cube, i.e. 6. Similar
to the case of a 2D hexagonal grid, this implies that the grid
cost in a honeycomb model is lower than the one in the cubic
model.
More importantly, the definition of a digital plane
within models I and II ensures better approximation to the
continuous plane (in terms of maximal possible deviation)
compared with the classical cubic model. In fact, a tunnelfree
√ standard plane may contain voxels that are at a distance
3 from the continuous plane (Figure 26a), while within
models I and II this distance is clearly smaller. This can be
immediately deduced from the fact that the longest diagonal
of a 3-hexel √
equals f = (1+(2a)2 )1/2 = 1.59361 . . . , which
is less than 3 = 1.73205 . . . (Figure 26b). It is not hard to
realize that tiling by a right prism with a regular hexagonal
base provides better approximation than tiling by a prism
which is inclined and/or with a quasi-regular hexagonal base.
Similar considerations and conclusions apply to the brickbuilt models. One can show that with respect to the value
of the maximal possible deviation from the continuous
plane, the honeycomb models I and II are superior to
the corresponding brick-built models, while the latter are
superior to the cubic model.
(b)
√
FIGURE 26. (a) The point B is at a distance 3 from the Euclidean
plane passing through the point A and orthogonal to the straight line
AB. (b) The diagonal PQ
√ of a 3-hexel is shorter than the one of the
unit cube which equals 3.
5.
4.8.
P
ALGORITHMIC AND COMPLEXITY ISSUES
The considered analytical discrete primitives (digital straight
lines and line segments, digital circles, planes and spheres)
admit an efficient algorithmic generation, which can be
performed in time that is linear in the number of the generated
hexels. To show this we explain below how every one of the
well-known linear time algorithms can be straightforwardly
adapted to the honeycomb model. We illustrate this in the
problem of digital straight line generation. We have the
following theorem.
Theorem 7. A digital line g D (a1 , a2 , b) in a 2D digital
hexagonal space M can be generated in linear time with
respect to the number of generated 2-hexels.
Proof. Consider the adjunct space M̄ defined in Section 3.2.
Recall that the rhomboidal cells of M̄ are in one-to-one
correspondence with the 2-hexels of M. In M̄ the rhombuses
with centers corresponding to 2-hexels from g D (a1 , a2 , b)
constitute a digital line ḡ(a1 , a2 , b) (Figure 27). In view
of the discussion in Section 3.2, if the normal vector n to
the Euclidean line g: a1 x1 + a2 x2 = b belongs to the cone
C1 , then ḡ(a1 , a2 , b) is a standard line in M̄. Otherwise,
if n ∈ C2 , then ḡ(a1 , a2 , b) is a naive line (Figure 27).
Thus, in order to obtain g D (a1 , a2 , b) in M, it suffices to
obtain a digital line ḡ(a1 , a2 , b) in M̄. It is either standard
or naive depending on the normal vector of g determined
by the coefficients a1 and a2 . Thus the time-complexity of
digital line generation in M matches the time-complexity of
the algorithms for standard/naive line generation in M̄.
The Computer Journal Vol. 48 No. 2, 2005
Analytical Honeycomb Geometry
b2
b1
FIGURE 27. Digital hexagonal space Ob1 b2 with two digital line
segments in it, together with the corresponding rhomboidal digital
space.
In a square or parallelogram grid a standard/naive line can
be generated in linear time, e.g. by the algorithms described
in [36]. Then, because of the one-to-one correspondence
between the rhombuses of M̄ and the 2-hexels of M, we
immediately obtain the desired digital line g D (a1 , a2 , b).
This completes the proof.
As mentioned earlier, within the cubic model, a digital line
is tunnel-free if and only if it is standard. Since a standard
line segment between two points contains as a subset the
corresponding naive line between the same points, generation
of the former is more time-consuming than generation of the
latter. Within the 2D hexagonal model, if n ∈ C2 , we have to
construct a naive line in the space M. Thus in this case the
algorithm efficiency will be superior to the one for standard
line generation within the square model.
The above results straightforwardly extend to digital plane
generation in the spaces M1 and M2 . For this, one can use
their adjunct spaces M̄1 and M̄2 , in which digital planes can
be efficiently generated. (See [7, 28, 44] for more details on
linear-time algorithms for digital plane generation within the
cubic model.)
Similarly, digital circle in M and a digital sphere in
M1 or M2 can be efficiently generated by applying known
algorithms [29] for generation of the corresponding digital
circle/sphere in the adjunct spaces M̄1 or M̄2 respectively.
6.
CONCLUDING REMARKS
In this paper we have developed analytical digital geometry
for raster and volume graphics based on a hexagonal grid
(resp. honeycomb prism tilings). In particular, we have
defined digital lines and circles (in the 2D space), and digital
planes and spheres (in the 3D space). We have also observed
that the honeycomb models ensure discretizations which are
in some respects superior to the classical ones.
One of our main objectives was to ensure tunnel-freedom
of the generated digital objects. With that end in view we
proposed two fundamental 3D honeycomb spaces in which
every connected set of 3-hexels is 2-connected and tunnelfree. In particular, any digital plane or sphere in these spaces
has no tunnels.
197
On the basis of the presented models one can
develop methods for discretizing more sophisticated objects
(especially in R3 ), such as 3D line segments, polygons and
meshes of polygons. For this, one can appropriately modify
certain discretization schemes and algorithms, which have
been developed for square/cubic models [7, 31, 32, 45].
Note that some problems that required long and complicated
solutions within the cubic model [7, 31] can be handled
more easily within the honeycomb models. The research
can also be extended to discretization of non-linear curves
and surfaces. For this one can take advantage of some results
of implicit geometry used in computer graphics.
This paper shows that in various respects the honeycomb
models can serve as a useful alternative to the classical
approaches in raster and volume modeling. One should admit
that at the current stage the proposed 3D spaces are mostly
of theoretical interest and importance, although in some
areas of human practice constructions of this kind are widely
used.10 It is not easy to predict whether or not these or other
alternatives to the cubic grid will find more extensive practical
applications in computer science. We believe that this will
happen at least in relation to certain specific problems whose
solution would be essentially harder within the traditional
cubic model.
ACKNOWLEDGEMENTS
We thank Eric Andres, Claudio Montani, Roberto Scopigno
and Marco Pellegrini for reading early versions of this paper
and making comments and suggestions. We are grateful to
Gabor Herman and Edgar Garduno for a useful discussion
on the possible advantages of using non-cubic models. We
would also like to thank David Bateman for helping us
prepare Figure 24. V. E. B. thanks Reinhard Klette for
a number of interesting discussions. We are indebted to
the three anonymous referees, whose valuable remarks and
suggestions helped us improve the presentation greatly. Part
of this work has been done while V. E. B. was visiting the
Laboratory on Signal, Image and Communication, IRCOM,
Université de Poitiers, Poitiers, France, and Dipartimento di
Elettronica ed Informatica, Università di Padova, Italy. Part
of this work has been done while R. P. B. was visiting the
Multimedia Laboratory of the Dipartimento di Elettronica ed
Informatica, Università di Padova, Italy, and the Computer
Graphics Group of IEI-CNR Pisa, Italy. Some of the results
presented in this paper have been reported at the Eight
International Workshop on Combinatorial Image Analysis,
Philadelphia 2001.
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