THE PIVOTAL TESSELLATION

The tessellation studied here is motivated by some stereolo gical applications of a new expression for the motion invariant density of straight lines in R3. The term ‘pivotal’ stems from the fact that the tessellatio n is constructed within a plane which is isotropic through a fix ed, ‘pivotal’ origin. Consider either a stationary point process, or a stationary random lattice of points in th at plane. Through each point event draw a straight line which is perpendicular to the axis determined by the ori gin and the point event. The union of all such lines (calledp-lines) constitutes the mentioned tessellation. We concen trate on the pivotal tessellation based on a stationary and isotropic planar Poisson point process; we how that this tessellation is not stationary.


INTRODUCTION
The purpose of this paper is to explore elementary properties of a special planar tessellation stemming from the application of recent stereological results (Cruz-Orive, 2005;2008;Gual-Arnau and Cruz-Orive, 2009).
Consider the equatorial disk B 2,t = B 3 ∩ L 3 2[0] , where B 3 ⊂ R 3 represents a ball of radius R centred at the origin O and L 2[0] denotes an isotropic plane through O with normal direction t ∈ S 2 + .Within the disk B 2,t , generate N independent and identically distributed uniform random points {z 1 , z 2 , ..., z N }, (Fig. 1a).For each i = 1, 2, ..., N, draw a straight line L 1 (z i ) through the point z i and normal to the axis Oz i .Thus L 1 (z i ) is effectively a "point sampled" straight line which will be called a p−line.
The union of all p−lines constitutes a tessellation in the reference disk, (Fig. 1b) which will be called a pivotal tessellation, inasmuch as the containing plane L 2[0] can only rotate around a fixed 'pivot' O.The practical interest of this construction lies in the following fact.Consider a nonvoid compact subset Y ⊂ B 3 of volume ν 3 (Y ) with piecewise smooth boundary ∂Y of area ν 2 (∂Y ).Then, are unbiased estimators of ν 2 (∂Y ) and ν 3 (Y ), respectively, where a := πR 2 , and ν 0 , ν 1 denote number of intersections and chord length, respectively (Cruz-Orive, 2005;2008).
Here we are interested in some properties of the pivotal tessellation constituted by the p−lines associated with a planar Poisson point process.

PRELIMINARIES
Given a point z ∈ R 2 of polar coordinates (ρ, ω), ρ ∈ (0, ∞), ω ∈ (0, 2π), we define a p-line L 1 (z) as a straight line with normal coordinates (ρ, ω), namely, Consider either a stationary planar point process with realizations in R 2 , or a stationary random lattice where Λ 0 ⊂ R 2 is a fixed regular lattice of points and z is a uniform random point in a fundamental tile of Λ 0 , (Fig. 3a).In either case, the pivotal tessellation associated with either Φ or Λ z is namely the corresponding process of p-lines, (Figs. 1b,3b).
In this paper P(dx) represents the probability element of a random variable X , namely P(dx) := P(x < X ≤ x + dx).If X admits a probability density function f (x), then P(dx) = f (x) dx.This notation extends to higher dimensions in a natural manner.

THE PIVOTAL POISSON TESSELLATION IN R 2
When the associated process Φ is a stationary and isotropic planar Poisson point process (Stoyan et al., 1995), then the corresponding pivotal tessellation Ψ will be called the pivotal Poisson tessellation.Let τ denote the fixed intensity of Φ, namely, where B denotes any subset from the Borel σ -algebra in R 2 , and ν q denotes the q-dimensional Hausdorff measure in R 2 , (thus ν 2 represents area, ν 1 curve length, and ν 0 the counting measure).
Next we obtain some properties of the tesselation Ψ.To do this we consider the random intersection (Fig. 1b), where B R ⊂ R 2 is a closed disk of radius R centred at the origin O, whereas {z 1 , z 2 , ..., z N } represent N independent and identically distributed (i.i.d.) uniform random (UR) points in B R , and so that N is a Poisson random variable with mean and variance equal to πR 2 τ.
For a p-line L 1 (z) := L 1 (ρ, ω) such that the point z of polar coordinates (ρ, ω) is UR in B R , it is easy to show that ρ and ω are independent random variables with Then the mean chord length determined in B R by the corresponding p-line is, Proof.Straightforward bearing in mind that ν 1 (L 1 (z) ∩ B R ) = 2 R 2 − ρ 2 and using Eq. 9.

Proposition 1. The mean total length per unit disk area of the straight line segments determined in B R by the p-lines of
Proof.Conditional on the number N of p-lines from Ψ hitting B R , by Eq. 10 we have, Using the premise (Eq.8) and dividing by πR 2 , the result follows.
Consequence.From Eq. 11 we see that λ 2 1 (R) = O (R), which implies that the pivotal Poisson tessellation Ψ associated with Φ is not stationary.
Remark 1.The mean area of the equatorial disk B 2,t considered in the Introduction (see Fig. 2) per unit volume of the corresponding ball B 3 , is λ 3 2 (R) = πR 2 /(4πR 3 /3) = 3/(4R).On the other hand, the mean total chord length of the bounded pivotal Poisson tessellation in B 2,t , per unit area of B 2,t , is given by Eq. 11.Therefore, the mean total chord length of such planar tessellation per unit volume of the reference ball B 3 , is namely a constant.This result is consistent with the fact that p-lines are effectively motion invariant in R 3 .
Lemma 2. Let z 1 , z 2 denote two i.i.d.UR points in B R .Then, that is, the probability that the corresponding two plines intersect inside B R is a known constant equal to 3/8 for any R > 0.
Fig. 2. The probability that a p-line L 1 (z 2 ) associated with a UR point z 2 ∈ B R hits a given p-chord L 1 (z 1 ) ∩ B R (thick straight line segment in the figure) is equal to the probability that z 2 falls in the support set (shaded region) of the given p-chord with respect to O.
Proposition 2. Let λ (0) 0 (R), and λ (2) 0 (R) denote the mean total numbers per unit disk area of the vertices, edges and connected regions constituting the bounded tessellation Ψ ∩ B R , respectively.Then, λ (0) (2) where the terms following the first one in the right hand side of the preceding identities represent the contributions of the disk boundary ∂ B R .
Proof.We use the method of Santaló (1940;1976 p. 51).Conditional on the number N of p-lines from Ψ hitting B R , let V B o (N), V ∂ B (N) denote the mean number of vertices interior to B R and in ∂ B R , respectively, and set Likewise, let E B o (N), E ∂ B (N) denote the mean number of edges interior to B R and in ∂ B R , respectively, and set E B o (N) + E ∂ B (N) = E (N).At each interior vertex there meet 4 edges, but they are counted twice because each edge has two vertices as endpoints.On the other hand, at each boundary vertex For estimation purposes via Eq. 1 it is simpler and more efficient to start with a stationary random lattice of points (Fig. 3a), instead of a Poisson point process.The corresponding pivotal lattice tessellation (Fig. 3b) will enjoy similar properties.An exact study of the latter might be prohibitive, however, because the number of lattice points inside a disk is a complicated oscillating function of the disk diameter.

Fig. 1 .
Fig. 1.(a) Disk centred at O containing 50 independent uniform random point events.(b) Associated pivotal tessellation formed by the corresponding p-lines with respect to O.