[IEEE 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007) - Glamorgan, UK (2007.07.9-2007.07.11)] 4th International Symposium on Voronoi Diagrams

High Order Voronoi Sculpture

James Dean Palmer

Department of Computer Science

Northern Arizona University

Flagstaff, AZ 86011, USA

[email protected].

Abstract

While techniques exist to compute three dimensional

Voronoi diagrams and their higher-order and generalized

cousins, visualizing Voronoi diagrams and their underly-

ing distance functions remains challenging. In this paper

we specifically consider visual representations that take the

form of shapes or sculptures formed from three dimensional

iso-surfaces.

1 Introduction

Informal usage of Voronoi diagrams seemingly extends

into antiquity. Rene Descartes, for example, includes dia-

grams in his works Principia Philosophiæ and Le Monde

which seem to represent the general structure of Voronoi

diagrams. One interesting feature of Descartes’ figures are

that they not only illustrate the boundaries of Voronoi cells

but also the interior space of the Voronoi cells.

Work in the mid 1800s by Lejeune Derelict and in the

late 1800s by Georgy Voronoi formalized what we now call

Voronoi diagrams. But Voronoi diagrams have been re-

invented or re-discovered in many different disciplines and

often masquerade behind other names. In geophysics, for

example, data is often grouped spatially in Thiessen poly-

gons and in condensed-matter physics, the same structures

are called Wigner-Seitz unit cells.

There are also countless variations on Voronoi diagrams

that use different distance metrics, and different generator

combinations and effects. In this paper we are particularly

interested in higher-order Voronoi diagrams. A higher-

order Voronoi diagram is a diagram where each cell is de-

fined by more than one generator site. In this paper we will

assume generator sites are points.

The order-k Voronoi diagram of n sites in R2 is a de-

composition of the plane into convex polygons such that

the points in each region have the same k closest generator

sites.

Figure 1. Rene Descartes uses this figurefrom Principia Philosophiæ (1644) to representrelationships in the heavens.

4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)0-7695-2869-4/07 $25.00 © 2007

More formally, let P = {pi, . . ., pn} be a set of generator

sites and let A(k)(P ) = {{p1,1, . . ., p1,k}, . . ., {pl,1, . . .,

pl,k} define a set of size k subsets of P where pi,j ∈ P

and l =(

nk

)

. Let P(k)i = {pi,1, . . ., pi,k} denote one such

subset. The order-k Voronoi polytope associated with P(k)i

is then described by [12],

V (P(k)i ) =

k⋂

h=1

V (pi,h|[P\P(k)i ] ∪ {pi,h}).

The order-k Voronoi diagram is a generalization of the

classical Voronoi diagram which is obtained if k = 1. The

furthest-site Voronoi diagram is obtained if k = n− 1. An-

other higher order Voronoi diagram we consider is the or-

dered order-k Voronoi diagram. Cells in an ordered order-

k Voronoi diagram have the same ordered set of k closest

sites. Written more formally,

V (P(k)i ) =

⋃

P <k>j

∈A<k>(P <k>i

)

V (P<k>j ),

where P <k>i is an ordered k-tuple of P and A<k>(P ) is

the set of all ordered k-tuples of P [13]. A closely related

diagram, the kth nearest point (or nearest neighbor) diagram

is described by the equation,

V [k](pj) =⋃

P(k−1)i

∈A(k−1)(P\{pj})

V (P (k−1) ∪ {pj}).

2 Related Work

In two dimensions, efficient construction of order-k

Voronoi diagrams has been studied from an algorithmic per-

spective by many researchers including Lee [11], Chazelle

and Edelsbrunner [5], Aurenhammer [2], Clarkson [6], and

Agarwal et al. [1]. For a survey of many related Voronoi

diagram results, see [3] and the book by Okabe et al [13].

From a visualization perspective, Telea and van Wijk

have studied order-k Voronoi diagrams with an emphasis

on illustrating the relationships between different cells us-

ing a series of colored bevels and cushions [16]. Voronoi

diagrams have also been used for creating procedural tex-

tures. The cellular look that Voronoi diagrams achieve can

be used to mimic natural cellular phenomena. Worley was

one of the first to propose a cellular texturing basis function

which uses Voronoi cells in this way [19]. Worley’s method

implicitly computes Voronoi diagrams of points assigned x

and y coordinates by a pseudo-random noise generator. The

resulting textures have proven to be useful in simulating the

look of waves, stone, metal, leather and other organic ma-

terials. Interestingly enough, a similar mechanically-based

technique has also been employed by artist Jonathan Callan

to create organic canyon like sculptures [18].

(a) (b)

(c) (d)

Figure 2. (a) A texture based on a 2nd near-est neighbor contour plot, (b) a pattern basedon symmetric generator sites, (c) a secondsymmetry using a fibonacci based coloringfunction, and (d) a 2nd nearest neighbor plotencorporating the L1 or Manhattan distancemetric.

In previous work we have previously considered tech-

niques based on iso-curves to visualize higher-order

Voronoi diagrams and generate textures and ornamental de-

signs in two dimensions [14]. See Figure 2. The actual

2D tilings obtained from order-k Voronoi diagrams are ex-

tremely interesting. Many of the computational techniques

and the study of tilings that result from Voronoi diagrams

has been pioneered by Kaplan [9].

3 Rendering Techniques

Our approach to rendering 3D sculptures based on high-

order Voronoi diagrams is to render the iso-surfaces of

the underlying distance function associated with generator

sites. That is, we have extended the two dimensional ap-

proach in [14] to three dimensions.


3.1 Volume Rendering & Marching Cubes

One of the simplest volume rendering approaches is to

cast rays through the volume and then color pixels based

on which samples the ray passes through. This is a natu-

ral approach that avails itself to standard ray tracing tech-

niques. But ultimately the quality of a volume render de-

pends on the number of sample points and the efficiency of

a rendering depends on how quickly such samples can be

computed. Similarly, polygonalization techniques such as

marching cubes are based on a discretization of the space

coupled with a sampling function used to discover the de-

sired iso-surface.

A naive brute force volume renderer might calculate the

kth nearest point for each sample. But this ignores the kth-

nearest point diagram with which it is intimately linked.

Several researchers have considered discrete approaches for

Voronoi diagram generation [15, 8]. By using such a dis-

crete algorithm first, one can potentially “preprocess” the

discrete space to avoid kth nearest point lookups.

3.2 Constructive Solid Geometry &Voronoi Based Approximations

A potentially faster rendering technique utilizes con-

structive solid geometry (CSG) and order-k Voronoi dia-

gram construction. If we limit ourselves to the Euclidean

distance metric, it’s clear that the iso-surfaces for any kth

nearest point diagram is a piece of a sphere centered at a

generator site. Specifically, it’s the intersection of a con-

stituent Voronoi polytope with a sphere centered at the cor-

responding generator site. The efficiency of this technique

is dependent on 1) the efficiency of sphere-polyhedra inter-

sections and corresponding rendering and 2) the efficiency

of three dimensional kth nearest point diagram construc-

tion.

High-order Voronoi diagrams in higher dimensions have

gotten less attention than their 2D counter parts both in

the literature and practical implementation. As has been

pointed out in [4] concerning 3D generalized Voronoi dia-

grams, exact computation of such diagrams requires manip-

ulation of high-degree algebraic surfaces which poses many

problems in terms of robustness, complexity and practical

usage.

An obvious alternative is approximation. Lavender et

al [10] and Boada et al [4] have proposed two different

3D generalized Voronoi diagram approximation techniques

based on octrees, while Vleugels et al. [17] have proposed

techniques based on other types of subdivision.

In our own work, we have also considered an alternative

approximation technique based on particle swarms and ge-

netic algorithms. Our technique involves letting particles

search for the polyhedron boundaries. In the case of a kth

Figure 3. Voronoi cell boundaries are discov-ered as particles collide into them. Differentcolors represent the primary cell each parti-cle “belongs” to.

nearest point diagram, at the end of the simulation each par-

ticle points to either one or two kth-closest generator sites.

If the particle points to one one point generator, it may sim-

ply not have hit a boundary yet or it may be floating into

free space in an unbounded cell on the outside of the subdi-

vision. If the particle points to two point generators, it will

likely have “ping ponged” between two neighboring cells

several times getting closer and closer to the planar bound-

ary between the two cells. A third possibility is that if the

particle is near an intersection of two or more Voronoi dia-

gram boundaries it will pass back and forth between these

boundaries - but ultimately we will only associate the parti-

cle with two regions. We then attempt to construct the poly-

hedron boundaries from the final position and state of par-

ticles in the system. Specifically regarding the creation of

CSG cuts on spheres it’s actually unnecessary to construct

a formal Voronoi diagram structure. Rather it is enough to

simply identify the cutting planes discovered by the parti-

cles and the boundaries that they represent.

In future work we hope to do a more formal compar-

ison of this technique to the discrete techniques we have

described. Subjectively, we have found this technique gen-

erates very similar results with the benefit that we can di-

rect particle exploration toward finding Voronoi diagram

boundaries near the actual iso-surfaces they will be “cut-

ting” yielding a Voronoi diagram approximation that tends

to be very accurate in areas of interest and less accurate in

areas that have little effect on the final sculpture.


Figure 4. An iso-surface for three pointsbased on the first nearest point. The dot-ted line represents the Voronoi diagram of thethree points.

4 Results

The system we developed implements both the particle

based system we described and an octree based approxi-

mation system for discovering Voronoi diagram boundaries.

We use CSG intersections with spheres and half-spaces to

represent the iso-surface. The intent in our work is to create

figures that not only convey these spatial concepts but that

are aesthetically pleasing and visually interesting. A sim-

ple cluster of three points may suddenly become something

unusual and mysterious.

We begin by explaining the visual meaning of an iso-

surface in relation to its generator sites in three dimensions.

Consider Figure 4. Here the iso-surface generated is that of

three spheres centered at their respective generator sites. If

the iso-value being plotted is gradually increased these three

spheres will gradually expand until they touch and then in-

tersect and form a single surface. All the points on this sur-

face are equi-distant to a closest generator site. The Voronoi

diagram of the three points represents a discontinuity in the

surface. That is, the underlying distance function changes at

the Voronoi diagram boundary. Conversely, we can look at

the structure of an iso-surface and identify discontinuities

in the surface as intersections of the surface with related

Voronoi diagrams.

This visual interpretation is easily extended to kth near-

est point Voronoi diagram. In Figure 5 an iso-surface is

Figure 5. An iso-surface for three pointsbased on the second nearest point. Thedotted line represents a 2nd nearest pointVoronoi diagram of the three points.

generated based on the 2nd nearest generator point. Here

you can clearly see the discontinuity in the smoothness of

the surface. Again, these discontinuities occur where the

surface intersects the underlying Voronoi diagram. In the

same way that a 2D iso-contour plot of a distance function

reveals the structure of a related Voronoi diagram, the 3D

iso-surface also reveals the structure of related 3D Voronoi

diagrams.

4.1 Geometric Structures and Symmetryin Three Dimensions

We have studied several randomly generated point clus-

ters and while interesting, such contours usually don’t em-

body qualities traditionally associated with being aestheti-

cally pleasing. Thus, we have focused on the examination

of geometric shapes with inherent symmetries and existing

distance related associations.

One of the first structures we studied was the regular do-

decahedron, a platonic solid with 20 vertexes and 12 regu-

lar pentagonal faces. In Figure 6 we render an iso-surface

for the kth-nearest point distance function for k values 2,

3, 4 and 5. It is immediately obvious that the symmetries

that exist between the points continue to invoke symmetries

for higher-order distance functions. Pentagonal symmetries

continue to dominate the iso-surface even for high values of

k.


Next we consider the result for a truncated icosahe-

dron which provides the underlying geometry for a C60

molecule, also known as Buckminsterfullerene or “bucky

ball”. Such an Archimedian solid had 60 vertexes and 20

regular hexagonal faces and 12 regular pentagonal faces.

Figure 7 illustrates iso-surface renderings for k values of

2, 3, 5, and 7. The interaction of the underlying pentagonal

and hexagonal faces induce incredibly complex patterns in

the iso-surfaces generated.

4.2 Molecular Structures

Inspired by the inherent geometries in the Buckminster-

fullerene, we have considered several other molecules. Here

we describe three that we found interesting.

In Figure 8 we have rendered iso-surfaces for a caffeine

molecule with k values 2, 3, 4 and 7. Initially the molecule

shows a number of pentagonal and hexagonal symmetries.

But as k becomes large such symmetries become less ap-

parent. See Figure 8 (d).

Figure 9 illustrates the effects of k values of 2 and 3 for

two historically valuable compounds: vanillin and Tyrian

purple. Vanilla (Figure 9 (a) and (b)) comes from the seeds

of a tropical orchid native to Mexico which are fermented to

yield the vanillin compound (C8H8O3). In the 1600s vanilla

was only available to the very rich as it was truly worth

more than it’s weight in gold. Tyrian purple (See Figures 9

(c) and (d)) was a dye made by the ancient Phoenicians in

the city of Tyre from the secretions of indigenous sea snails.

The snails responsible, Murex brandaris, gave up this purple

compound in very low yield. Approximately 10,000 snails

would yield 1 gram of dye. The primary constituent of the

dye, 6, 6′-dibromoindigo, was discovered by Paul Friedl-

nder in 1909 [7].

Both of these molecules show symmetries (though not

shape) similar to the caffeine molecule, in part because of

commonalities in atomic composition.

4.3 Multiple Iso-Surfaces

Two dimensional distance functions are often plotted

with several different iso-contours for various relevant or

related iso-values. Unfortunately the complexity of such an

exercise in three dimensions is often visually confusing. We

might imagine layers like an onion surrounding a generator

site, but how do we visualize these layers? An obvious so-

lution is to use transparency to represent multiple surfaces

in three dimensions. Figures 10 and 11 utilize this tech-

nique for two iso-values with limited success in conveying

structural meaning but interesting results none the less.

(a) (b)

(c) (d)

Figure 6. Generator sites taken from the ver-texes of a dodecahedron where (a) k = 2, (b)k = 3, (c) k = 4 and (d) k = 5.

(a) (b)

(c) (d)

Figure 7. Generator sites taken from a C60

molecule (commonly referred to as a “buckyball”) where (a) k = 2, (b) k = 3, (c) k = 5, and(d) k = 7.


(a) (b)

(c) (d)

Figure 8. Generator sites taken from PDB po-sitions for a caffeine molecule (C8H10N4O2)(a) k = 2, (b) k = 3, (c) k = 4 and (d) k = 7.

(a) (b)

(c) (d)

Figure 9. Generator sites taken from a vanillinmolecule (C8H8O3) molecule where (a) k = 2and (b) k = 3 and a Tyrian purple moleculewhere (c) k = 2, and (d) k = 3.

Figure 10. A random cluster of points are thebasis for two different iso-surfaces.

Figure 11. A kth nearest point iso-surface fork=3 with two different iso-values on 6 gener-ator sites in the shape of a diamond.


5 Conclusions

In this paper we have explored several techniques for

finding iso-surfaces for 3D distance functions associated

with Voronoi diagrams. These techniques can be used to

construct what we call Voronoi sculptures. These inter-

esting solid forms are closely related to and actually pro-

vide some three-dimensional intuition about the Voronoi di-

agrams that govern their shape.

Some of the most interesting shapes occur when gener-

ator sites are positioned according to two or three dimen-

sional symmetries. Thus, we think some of the most in-

teresting future work (from an artistic perspective) in this

area may focus on designing generator sites that obey three

dimensional isometries.

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[IEEE 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007) - Glamorgan, UK (2007.07.9-2007.07.11)] 4th International Symposium on Voronoi Diagrams