1. Introduction

Computer graphics and visualization deal with preparing data in order to show (“visualize“) these data on a (two-dimensional) computer screen. In computer graphics, the original data typically stem from the three-dimensional Euclidean space R3, whereas in scientific visualization the data might originate from spaces of much higher dimension (e.g., in information visualization or high-dimensional sphere models in statistical mechanics) [Swayne et al. 1998].

In these scenarios, visibility computations play a central role [O'Rourke 1997]. In the simplest case, we are given a fixed viewpoint υ ∈ ℝ n, and the scene consists of a set B of bodies. Now the task is to compute a suitable two-dimensional projection of the scene (“to render the scene“) that reflects which part of the scene is visible from the viewpoint υ;. Ina more dynamic setting, the viewpoint can be moved interactively (see [Bern et al. 1994; Lenhof and Smid 1995], for example). However, in general, after each movement of the viewpoint a new rendering process is necessary. In order to speed up this process, commercial Tenderers apply caching techniques [Wernecke 1994].

1. Introduction

Many interesting problems in combinatorial geometry have remained unsolved or only partially solved for a long time. From time to time breakthroughs are made. In this survey, we shall discuss the known results about some metric and nonmetric problems. In particular, we shall discuss the Sylvester-Gallai problem and the Dirac-Motzkin conjecture on the existence and number of ordinary lines, the Dirac conjecture on the number of connecting lines, and the problem of distinct and repeated distances. The main focus will be on versions of these problems in the Euclidean and real projective plane. The method of allowable sequences will be described as a tool to give purely combinatorial solutions to extremal problems in combinatorial geometry.

2. Sylvester's Problem

Sylvester [1893] posed a question in the Educational Times that was to remain unsolved for 40 years until it was raised again by Erdos [1943]. Then it was soon solved by Gallai [1944], who gave an affine proof. More followed: Steinberg's proof in the projective plane and others by Buck, Grünwald and Steenrod, all collected in [Steinberg et al. 1944]; Kelly's Euclidean proof [1948], and others, including [Motzkin 1951; Lang 1955; Williams 1968].

We give the following definitions before we state the problem and its solutions. Let P be a finite set of 3 or more noncollinear points in the plane. Let F be a finite collection of simple closed curves in the real projective plane which do not separate the plane, every two of which have exactly one point in common, where they cross. F is known as a pseudoline arrangement

The basic idea of multicarrier modulation is to divide the transmitted bitstream into many different substreams and send these over many different subchannels. Typically the subchannels are orthogonal under ideal propagation conditions. The data rate on each of the subchannels is much less than the total data rate, and the corresponding subchannel bandwidth is much less than the total system bandwidth. The number of substreams is chosen to ensure that each subchannel has a bandwidth less than the coherence bandwidth of the channel, so the subchannels experience relatively flat fading. Thus, the intersymbol interference on each subchannel is small. The subchannels in multicarrier modulation need not be contiguous, so a large continuous block of spectrum is not needed for high-rate multicarrier communications. Moreover, multicarrier modulation is efficiently implemented digitally. In this discrete implementation, called orthogonal frequency division multiplexing (OFDM), the ISI can be completely eliminated through the use of a cyclic prefix.

Multicarrier modulation is currently used in many wireless systems. However, it is not a new technique: it was first used for military HF radios in the late 1950s and early 1960s. Starting around 1990, multicarrier modulation has been used in many diverse wired and wireless applications, including digital audio and video broadcasting in Europe, digital subscriber lines (DSL) using discrete multitone, and the most recent generation of wireless LANs.

1. Introduction

In 1933, Esther Klein raised the following question. Is it true that for every n there is a least number — which we will denote by ES(n) — such that among any ES(n) points in general position in the plane there are always n in convex position? This question was answered in the affirmative in a classical paper of Erdős and Szekeres [1935]. In fact, they showed (see also [Erdős and Szekeres 1960/1961]) that

The lower bound, 2n-2 +1, is sharp for n = 2, 3, 4, 5 and has been conjectured to be sharp for all n. However, the upper bound, was not improved for 60 years. Recently, Chung and Graham [1998] managed to improve it by 1.

1. Introduction

In the variational description of surfaces, several functionals are of primary importance:

• • The area A = ∫ dA, where dA is the area element, is preserved by isometries.

• • The total Gaussian curvature g = ∫ K dA, where K is the Gaussian curvature, is a topological invariant.

• • The total mean curvature M = ∫ H dA, where H is the mean curvature, depends on the external geometry of the surface.

• • The Willmore energy W = ∫ H2 dA is invariant with respect to Möbius transformations.

Geometric discretizations of the first three functionals for simplicial surfaces are well known. For the area functional the discretization is obvious. For the local Gaussian curvature the discrete analog at a vertex v is defined as the angle defect.

In multiuser systems the system resources must be divided among multiple users. This chapter develops techniques to allocate resources among multiple users and examines the fundamental capacity limits of multiuser systems. We know from Section 5.1.2 that signals of bandwidth B and time duration T occupy a signal space of dimension 2BT. In order to support multiple users, the signal space dimensions of a multiuser system must be allocated to the different users. Allocation of signaling dimensions to specific users is called multiple access. Multiple access methods perform differently in different multiuser channels, and we will apply these methods to the two basic multiuser channels: downlink channels and uplink channels. Because signaling dimensions can be allocated to different users in an infinite number of different ways, multiuser channel capacity is defined by a rate region rather than a single number. This region describes all user rates that can be simultaneously supported by the channel with arbitrarily small error probability. We will discuss multiuser channel capacity regions for both the uplink and the downlink. We also consider random access techniques, whereby signaling dimensions are allocated only to active users, as well as power control, which ensures that users maintain the SINR required for acceptable performance. The performance benefits of multiuser diversity, which exploits the time-varying nature of the users' channels, is also described. We conclude with a discussion of the performance gains and signaling techniques associated with multiple antennas in multiuser systems.

1. Introduction

The set R of real numbers carries the structure of a semiring if equipped with the tropical addition min﹛ and the tropical multiplication where + is the ordinary addition. We call the triplet the tropical semiring. It is an equally simple and important fact that the operations are continuous with respect to the standard topology of R. So the tropical semiring is, in fact, a topological semiring. Considering the tropical scalar multiplication (and componentwise tropical addition) turns the set [Rd+1 into a semimodule. The study of the linear algebra of the tropical semiring and, more generally, of idempotent semirings, has a long tradition. Applications to combinatorial optimization, discrete event systems, functional analysis etc. abound. For an introduction see [Baccelli et al. 1992]. A recent contribution in the same vein, with many more references, is [Cohen et al. 2004].

Convexity in the tropical world (and even in a more general setting) was first studied by Zimmermann [1977]. Following the approach of Develin and Sturmfels [2004] here we stress the point of view of discrete geometry.

Many signals in communication systems are real bandpass signals with a frequency response that occupies a narrow bandwidth 2B centered around a carrier frequency fc with 2B « fc, as shown in Figure A.1. Since bandpass signals are real, their frequency response has conjugate symmetry: a bandpass signal s(t) has |S(f)| = |S(−f)| and ∠S(f) = −∠S(−f). However, bandpass signals are not necessarily conjugate symmetric within the signal bandwidth about the carrier frequency fc; that is, we may have |S(fc + f)| ≠ |S(fc − f)| or ∠S(fc + f) ≠ −∠S(fc − f) for some f : 0 < f ≤ B. This asymmetry in |S(f)| about fc (i.e., |S(fc + f)| ≠ |S(fc − f)| for some f < B) is illustrated in the figure. Bandpass signals result from modulation of a baseband signal by a carrier, or from filtering a deterministic or random signal with a bandpass filter. The bandwidth 2B of a bandpass signal is roughly equal to the range of frequencies around fc where the signal has nonnegligible amplitude. Bandpass signals are commonly used to model transmitted and received signals in communication systems. These are real signals because the transmitter circuitry can only generate real sinusoids (not complex exponentials), and the channel simply introduces an amplitude and phase change at each frequency of the real transmitted signal.

1. Introduction

The binary space partition tree is a geometric data structure obtained by a recursive partitioning scheme, called binary space partition (for short, BSP) over a set of input objects: The space is partitioned along a hyper plane into two half-spaces, then either half-space is partitioned recursively until every subproblem contains only a trivial fraction of the input objects. The concept of BSP has emerged from the computer graphics community in the seventies. It was originally designed to assist efficient hidden-surface removal algorithms for moving viewpoints, but it has later found widespread applications in many areas of computational and combinatorial geometry.

In many of the applications, the bottle neck of the space complexity is the size of the BSP tree they rely on. Combinatorial research focused on determining the worst case complexity of BSPs for certain classes of inputs.

1. Introduction

A geometric random walk starts at some point in ℝn and at each step, moves to a “neighboring” point chosen according to some distribution that depends only on the current point, e.g., a uniform random point within a fixed distance δ.The sequence of points visited is a random walk. The distribution of the current point, in particular, its convergence to a steady state (or stationary) distribution, turns out to be a very interesting phenomenon. By choosing the one-step distribution appropriately, one can ensure that the steady state distribution is, for example, the uniform distribution over a convex body, or indeed any reasonable distribution in ℝ n.

Geometric random walks are Markov chains, and the study of the existence and uniqueness of and the convergence to a steady state distribution is a classical field of mathematics. In the geometric setting, the dependence on the dimension (called n in this survey) is of particular interest. Polya proved that with probability 1, a random walk on an n-dimensional grid returns to its starting point infinitely often for n ≤ 2, but only a finite number of times for n ≥ 3. Random walks also provide a general approach to sampling a geometric distribution. To sample a given distribution, we set up a random walk whose steady state is the desired distribution. A random (or nearly random) sample is obtained by taking sufficiently many steps of the walk. Basic problems such as optimization and volume computation can be reduced to sampling.

1. Introduction

What is the maximum radius rn of a circular disk which can be covered by n closed unit circles? The determination of rn for n ≤ 4 is an easy task: we have and. The problem of finding r5 has been motivated by a game popular on fairs around the turn of the twentieth century [Neville 1915; Ball and Coxeter 1987, pages 97-99]. The goal of the game was to cover a circular space painted on a cloth by five smaller circles equal to each other. The difficulty consisted in the restriction that an “on-line algorithm” had to be used, that is no circle was allowed to be moved once it had been placed. Neville [1915] conjectured that r5 = 1.64100446… and this has been verified by K. Bezdek [1979; 1983] who also determined the value of r6 = 1.7988… . The proofs of these cases are complicated. The case n = 7 is again easy. We have r7 = 2 and if 7 unit circles cover a circle C7 of radius 2, then one of them is concentric with C7 while the centers of the other circles lie in the vertices of a regular hexagon of side y/3 concentric with C7. In his thesis Denes Nagy [1975] claimed without proof that for n = 8 and n = 9 and that, as for n = 7, the best arrangement has (n — l)-fold rotational symmetry.

1. Introduction

The molecular basis of life rests on the activity of biological macro-molecules, mostly nucleic acids and proteins. A perhaps surprising finding that crystallized over the last handful of decades is that geometric reasoning plays a major role in our attempt to understand these activities. In this paper, we address this connection between biology and geometry, focusing on hard sphere models of biomolecules.

The biomolecular revolution. Most living organisms are complex assemblies of cells, the building blocks for life. Each cell can be seen as a small chemical factory, involving thousands of different players with a large range of size and function. Among them, biological macro-molecules hold a special place.

Although bandwidth is a valuable commodity in wireless systems, increasing the transmit signal bandwidth can sometimes improve performance. Spread spectrum is a technique that increases signal bandwidth beyond the minimum necessary for data communication. There are many reasons for doing this. Spread-spectrum techniques can hide a signal below the noise floor, making it difficult to detect. Spread spectrum also mitigates the performance degradation due to intersymbol and narrowband interference. In conjunction with a RAKE receiver, spread spectrum can provide coherent combining of different multipath components. Spread spectrum also allows multiple users to share the same signal bandwidth, since spread signals can be superimposed on top of each other and demodulated with minimal interference between them. Finally, the wide bandwidth of spread-spectrum signals is useful for location and timing acquisition.

Spread spectrum first achieved widespread use in military applications because of its inherent property of hiding the spread signal below the noise floor during transmission, its resistance to narrowband jamming and interference, and its low probability of detection and interception. For commercial applications, the narrowband interference resistance has made spread spectrum common in cordless phones. The ISI rejection and bandwidth-sharing capabilities of spread spectrum are very desirable in cellular systems and wireless LANs. As a result, spread spectrum is the basis for both second- and third-generation cellular systems as well as second-generation wireless LANs.

The growing demand for wireless communication makes it important to determine the capacity limits of the underlying channels for these systems. These capacity limits dictate the maximum data rates that can be transmitted over wireless channels with asymptotically small error probability, assuming no constraints on delay or complexity of the encoder and decoder. The mathematical theory of communication underlying channel capacity was pioneered by Claude Shannon in the late 1940s. This theory is based on the notion of mutual information between the input and output of a channel. In particular, Shannon defined channel capacity as the channel's mutual information maximized over all possible input distributions. The significance of this mathematical construct was Shannon's coding theorem and its converse. The coding theorem proved that a code did exist that could achieve a data rate close to capacity with negligible probability of error. The converse proved that any data rate higher than capacity could not be achieved without an error probability bounded away from zero. Shannon's ideas were quite revolutionary at the time: the high data rates he predicted for telephone channels, and his notion that coding could reduce error probability without reducing data rate or causing bandwidth expansion. In time, sophisticated modulation and coding technology validated Shannon's theory and so, on telephone lines today, we achieve data rates very close to Shannon capacity with very low probability of error.

The wireless radio channel poses a severe challenge as a medium for reliable high-speed communication. Not only is it susceptible to noise, interference, and other channel impediments, but these impediments change over time in unpredictable ways as a result of user movement and environment dynamics. In this chapter we characterize the variation in received signal power over distance due to path loss and shadowing. Path loss is caused by dissipation of the power radiated by the transmitter as well as by effects of the propagation channel. Path-loss models generally assume that path loss is the same at a given transmit–receive distance (assuming that the path-loss model does not include shadowing effects). Shadowing is caused by obstacles between the transmitter and receiver that attenuate signal power through absorption, reflection, scattering, and diffraction. When the attenuation is strong, the signal is blocked. Received power variation due to path loss occurs over long distances (100–1000 m), whereas variation due to shadowing occurs over distances that are proportional to the length of the obstructing object (10–100 m in outdoor environments and less in indoor environments). Since variations in received power due to path loss and shadowing occur over relatively large distances, these variations are sometimes referred to as large-scale propagation effects. Chapter 3 will deal with received power variations due to the constructive and destructive addition of multipath signal components.