Image Anal Stereol 2008;27:29-37 Original Research Paper MEAN VALUES FOR HOMOGENEOUS STIT TESSELLATIONS IN 3D Werner Nagel1 and Viola Weiss2 1Institut fu¨r Stochastik, Friedrich-Schiller-Universita¨t Jena, D-07737 Jena, Germany; 2Fachhochschule Jena, FB Grundlagenwissenschaften, Carl-Zeiss-Promenade 2, D-07745 Jena, Germany e-mail: nagel@minet.uni-jena.de (Accepted January 30, 2008) ABSTRACT Recently (Nagel and Weiss, 2005), the class of homogeneous random tessellations that are stable under the operation of iteration (STIT) was introduced. In the present paper this model is reviewed and new results for the mean values of essential geometric features of STIT tessellations in two and three dimensions are provided and proved. For the isotropic model, these mean values are compared with those ones of the Poisson-Voronoi and of the Poisson plane tessellations, respectively. Keywords: crack pattern, mean values, random tessellations, spatial statistics, stochastic geometry. INTRODUCTION The geometry of several structures that are studied in material science, biology and other sciences can appropriately be modeled by random tessellations. Examples are single-phase polyhedral microstructures, foams, systems of cracks (joints, fissures) in rock, craquele´e of thin layers, systems of cells. There are physical theories explaining the genesis of such structures and resulting in geometric models. Sometimes such models are hardly treatable, as, e.g., in the case of the Gilbert model, see Noble (1967) or the model by Gray et al. (1976), cf. Stoyan et al. (1995). On the other hand, there are idealized mathematical models which are easier to deal with theoretically, but their goodness-of-fit in certain applications has to be checked thoroughly (cf. Nagel et al., 2007). Two well-established mathematical models for random tessellations are the Poisson-Voronoi tessellation and the Poisson plane tessellation (in 3D) or Poisson line tessellation (in 2D), respectively, (cf. Stoyan et al., 1995). A further model is the so-called STIT tessellation, introduced in Nagel and Weiss (2005). Although it arose as a result of purely mathematical investigations, it will enrich the choice of models. The simulations in Figs. 2 and 3 suggest that STIT tessellations can be potential models for crack or fissure structures. In the present paper, after a brief description of the STIT model and a review of some key properties, new results for mean values of important parameters are presented. This is done for the homogeneous (i.e., spatially stationary) but not necessarily isotropic case. These mean values are compared with the corresponding ones for Poisson plane (or line respectively) tessellations and for the Poisson-Voronoi tessellations. Since homogeneous Poisson-Voronoi tessellations are always isotropic this comparison is restricted to the isotropic case. The proofs for the mean value formulas are given in the Appendix. In this paper, a tessellation is assumed to be a partition of the Euclidean space into bounded and convex polytopes – the so-called cells – with the additional condition of local finiteness, i.e., any bounded subset of the space intersects a finite number of cells. Alternatively, a tessellation is given by the set of all boundary points of the cells, i.e., the boundaries between the cells (which is a topologically closed set). STIT TESSELLATIONS The name STIT is an abbreviation for “stable with respect to iteration”, and this will be explained below in this Section. We start with a short description of STIT tessellations that was already given in earlier papers (Nagel and Weiss, 2005; Nagel et al., 2007). DESCRIPTION OF THE MODEL Let Rd denote the d-dimensional Euclidean space. Here we consider the cases d = 2 and d = 3. A hyperplane is a (d -1)-dimensional plane in Rd, i.e., a plane in R3 or a line in R2, respectively. A hyperplane h can be described by the signed distance p of h from the origin and by its unit normal vector u in the upper half-space, i.e., a vector u in the upper unit half sphere S+d-1 in Rd. We define that p > 0 if the intersection of h and the orthogonal line through the origin, h?, is in the upper half-space and p < 0 otherwise. A random hyperplane has a random direction and a random distance to the origin. 29 Nagel W et al: STIT tessellations Let L be the measure on the set of hyperplanes that is given by K{d{p,u))=dpM{du), (1) where dp is the element of the Lebesgue (length) measure on the real axis, and M is a probability distribution on the space of directions. If M is the uniform distribution on ,5^+1 (and thus the measure A is invariant under all rotations of hyperplanes around the origin) then it is called isotropic. Notice that for the following construction it is not necessary that M is isotropic. But in order to generate a tessellation with bounded polytopes as cells it is assumed that not all random hyperplanes are orthogonal to one single plane (in R3) or to one single line (in R2), respectively. For a set C c Rd denote by [C] the set of all hyperplanes h that hit C, i.e., C Ci h / 0/. Thus, if C is bounded, then A( [C] ) is finite and MO 1 MC]) A(-n[C]) defines a probability measure on [C], i.e., the distribution of a random hyperplane that hits C. Now let W C Rd be a d-dimensional compact and convex domain in which the STIT tessellation will be generated. Further assume that (t7j,yj), j = 1,2,... is a sequence of independent and identically distributed (i.i.d.) pairs, T7, yj independent, T7 exponentially distributed with parameter A( [W] ) , and yj a random hyperplane with distribution AW. For a (random) hyperplane y denote by y+1 and y-1 the two half-spaces generated by y. For a > 0 the construction can now be described by the following algorithm. ALGORITHM (a, W, A) 0. j = 0, T = {(0,W)}, R = 0 1. UNTIL T = 0 FOR (t, W) G T DO (i) j=j + 1 (ii)IF T + Tj < a THEN (a) IF yj G [W'] THEN T = (T \ {(t,W>)}) U {(t + T7,W n YfW + TjW'nY]-1)} (b) ELSET = (T\{(T,W')})u{(T+Tj,W')} (iii)ELSE T = T\ {{i,W')}, R END RU{W'} The OUTPUT of the algorithm is R(a,W) = R which is a set of random convex polytopes in W if W itself is a convex polygon. This can be considered as a tessellation within W. We denote Y(a,W) = cl U dW'\dW W'eR(a,W) (2) where dW denotes the boundary of the set W and cl the topological closure of a set. Thus Y(a,W) is the union of all the chords inside W which are generated by the yj during the construction. Roughly, this construction can be understood as follows. The set W is subsequently divided at random times. After the first division at time T1 by y1 the two ’cells’ WP\y+1 and WP\y-1 arise. These two cells are treated separately and independently. Each cell has a life time until it is divided by a random hyperplane. The condition ’IF yj G [W] THEN’ in the algorithm defines a rejection method (i.e., not all the yj do divide a cell) which yields that the life time of a cell W is exponentially distributed with the parameter A([W/]), and hence in the average small cells live longer than larger ones. At the fixed time a this procedure stops and the state of the construction defines Y (a, W). Fig. 1 illustrates the result of the construction of a non-isotropic tessellation with a small number of horizontal and vertical edges. In Figs. 2 and 3 simulations of isotropic STIT tessellations are shown. Fig. 1. Illustration of the construction with horizontal and vertical segments only. SOME KEY PROPERTIES Here we summarize important properties of the constructed tessellation. The proofs were already given in earlier papers (Nagel and Weiss, 2003; 2005). If Y is a random tessellation of Rd then it is meant that Y is the random closed set (RACS) of all cell boundaries. By C(Y) we denote the set of all cells of Y. This is a random and space-filling ensemble of convex polytopes. (i) Existence: There exists a homogeneous (i.e., spatially stationary) tessellation Y of the whole Rd such that Y(a,W) = Y C\W , (3) _ 30 Image Anal Stereol 2008;27:29-37 Fig. 2. Simulation of a 2D STIT tessellation (kindly provided by Joachim Ohser). Fig. 3. Simulation of a 3D STIT tessellation (kindly provided by Joachim Ohser, Claudia Lautensack, and Tatyana Sych). where = stands for “has the same distribution as.” The tessellation Y does not depend on W, and one can show that this formula holds for all compact and convex d-dimensional sets W C Rd. One possibility to construct also Y is described in Mecke et al. (2007). The directional distribution of the faces in Y coincides with the distribution M that was introduced in Eq. 1. The intensities or densities of other parameters of the random tessellation depend on the value of a in the construction above. For d = 2 let La be the mean total edge length per unit area and, respectively, for d = 3 denote by SV the mean total face area per unit volume. Their values are La = a if the construction is done in R 2 , SV = a if the construction is done in R3. (ii) STIT: This tessellation is stochastically stable with respect to the iteration of tessellations - STIT for short. For tessellations, the operation of iteration (also referred to as nesting) is defined as follows. Let Y1, Y2,... be a sequence of i.i.d. homogeneous random tessellations and denote W = {Y1,Y2,...}. Further assume that Y0 is a homogeneous random tessellation which is independent of W. For this definition it is useful to consider the set C(Y) of the cells (which are convex polygons) of Y. Assume that these cells are numbered and that C (Y 0) = {p1ip 2-, • • •}. The iteration of the tessellation Y0 and the sequence W is defined as I(Y0,^)=Y0u\J(pknYk) (4) k1 This definition means that a cell pk of the so called “frame” tessellation Y0 is - independently of all other cells - subdivided by the cells pki, i = 1,2,... of the tessellation Yk which intersect the interior of pk. For a real number r > 0 the tessellation rY is generated by transforming all points (x,y) G Y into (rx,ry). Accordingly, rty means that this transformation is applied to all tessellations of the sequence ^. Let Y0 be a homogeneous random tessellation and ^1,^2,... a sequence of sequences of tessellations such that all the occurring tessellations (including Y0) are i.i.d. Then the sequence I2(Y0)5I3(Y0)5 ••• of rescaled iterations is defined as (Nagel and Weiss, 2003; 2005). I2(Y0) =I(2Y 0, 2^1), I3(Y0) =I(I(3Y 05 3^1), 3^2) (3 \ "I2(Y 0) 5 3 2^2 2 J Im(Y0) m m 1 Im—1 ( Y0) 1 m '"m—1 m = 3,4,... Here, m is the rescaling factor which is chosen such that the results of tessellations do not degenerate for m —? 00. We use the abbreviation Im(Y0) since it is assumed that all the other tessellations in the sequences ^1,^2,... are independent and have the same distribution as Y0. The tessellation Im(Y0) is generated by an m-fold iteration where all the used tessellations have the same distribution as mY0. I 31 Nagel W et al: STIT tessellations Definition 1 A homogeneous random tessellation Y is said to be stable with respect to iteration (STIT) if Y =Im(Y) for all m = 2,3,... , i.e., if its distribution is not changed by repeated rescaled iteration with sequences of tessellations with the same distribution. It was shown in Nagel and Weiss (2005) that the tessellation Y introduced in Eq. 3 is STIT. It should be emphasized that the STIT property uniquely determines the distribution of a homogeneous random tessellation if the edge length (or surface area) intensity and the directional distribution R are fixed. Roughly, this means that the above given construction yields the only STIT tessellations for R. (iii) Sections are STIT: If a homogeneous STIT tessellation Y in Rd is intersected by a k-dimensional plane h, with 1 < k < d — 1, then the section profile Y n h is again a homogeneous tessellation on this plane. The operations of iteration and of intersection commute, i. e., I 2(Y0(~ih, Y 1 C\h) =I2(Y0, Y 1)V\h where Y 1 n h = {Y1 n h, Y2 n h,...}. Hence it is evident that Y n h is a STIT tessellation if Y is STIT. (iv) Poisson typical cell: For homogeneous random tessellations the notion of the distribution of the random typical cell is defined via Palm distributions, see Stoyan et al. (1995) or Schneider and Weil (2000). Intuitively, the typical cell can be understood as a randomly chosen cell out of a finite number of cells, e.g., out of the set of all cells which have their centroid in a large ball around the origin. In such a choice all these finitely many cells have an equal probability to be selected; i.e.,no weighting with respect to their size. Now, let YP denote a homogeneous Poisson hyperplane tessellation in Rd with intensity La (if d = 2) or SV (if d = 3) respectively and directional distribution R. And Y is assumed to be a homogeneous STIT tessellation with the same parameter and the same R. If one considers the interior of the cells or, more intuitively, the single isolated cells neglecting additional nodes or edges on their boundaries (cf. the following paragraph (v)), one can show (Nagel and Weiss, 2003) that the distributions of the interiors of the typical cell of YP and of Y are identical. (v) Cells are not face-to-face: In Voronoi tessellations as well as in hyperplane tessellations and in several other tessellation models the cells are in a face-to-face position (in Schneider and Weil (2000), the German word “seitentreu” is used). This means that for any two cells C1, C2 the intersection C1 n C2 is either empty or is an r-dimensional face, 0 < r < d — 1, which is a face of the polytope C1 and a face of C2. An example of two cells that are not in a face-to-face position is shown in Fig. 4. One can derive it from the construction that was described by the algorithm above, and one can observe it in the simulation example in Figs. 2 and 3 that not all the cells of STIT tessellations are in a face-to-face position. Thus on the boundary of a cell can appear additional nodes and edges. Therefore, it is essential to define carefully the notions of nodes, edges and faces for such tessellations. \ \ C1 / C2 \ Fig. 4. Two cells C1 and C2 that are not in a face-to-face position. Their intersection is the bold line, and this is not a face of C1 or of C2. MEAN VALUES FOR STIT TESSELLATIONS THE PLANAR CASE (d = 2) For the sake of completeness we recall here results from Nagel and Weiss (2004) for the planar case. In a planar tessellation where the cells are compact convex polygons the set of the nodes is given as the set of all vertices of these polygons. All nodes of a STIT tessellation are so-called T-shaped nodes with exactly three emanating edges. In order to define the edges of the tessellation consider the network of all cell boundaries, i.e., the union of all segments which are faces of cells. Obviously, all the nodes are located on the network. An edge of the tessellation is a linear segmentin this network between two nodes but with no node in between (i.e., no node is in the relative interior of this segment). Thus, an edge of the tessellation is always an intersection of two cells. In Fig. 4 the bold line illustrates one such edge of the tessellation. 32 Image Anal Stereol 2008;27:29-37 We will use the following notation for mean values. LA – mean total edge length per unit area, edge length intensity, U1 – mean length of the typical edge, U2,A2 – mean perimeter and mean area, resp., of the typical cell, N0 – mean number of nodes per unit area, N1 – mean number of edge midpoints per unit area, N2 – mean number of cell centroids per unit area, N01 = N02 – mean number of edges emanating from the typical node = mean number of cells which contain the typical node, N20 = N21 – mean number of nodes = mean number of edges on the boundary of the typical cell. For two directions u1, u2 ? S+1 denote by [u1,u2]= |sin ?(u1,u2)| the area of the parallelogram which is spanned by these two unit vectors. For the directional distribution R as in Eq. 1 we denote z [u1,u2]^(du1)^(du2) In the isotropic case, i.e., when M is the uniform distribution, we have z = p 2. Mecke (1984) showed that all the mean values introduced above for planar homogeneous random tessellations can be expressed by three parameters, namely N0, N2 and La. For the particular case of homogeneous STIT tessellations with edge length intensity La and directional distribution M it is N0=Laz and N2 = —LAz ¦ Thus the well-known mean value formulas yield for the remaining parameters U1 3LAz , U2 = 4 z, LA A2 = L 2 A 2 z, N1 = 3Laz , 2A N01 =N02 = 3, N20 =N21 =6. THE SPATIAL CASE (d = 3) In a tessellation in R3 where the cells are compact convex polytopes the set of the nodes is given as the set of all vertices of these polytopes. In order to define the faces and edges of the tessellation consider the 2-network of all cell boundaries, i.e., the union of all 2-faces of cells. Further, the 1-network is defined as the union of all 1-faces (edges) of the cells. Obviously, the set of nodes is a subset of the 1-network which is itself a subset of the 2-network. An edge of the tessellation is a linear segment in the 1-network between two nodes but with no intermediate nodes (i.e., no node is in the relative interior of this segment). A face of the tessellation is a two-dimensional convex polygon which is a subset of the 2-network and is bounded by edges of the 1-network and has no edges in its relative interior. Hence, a face of the tessellation is always the intersection of two cells. We consider the following mean values. Mean ... LV – total edge length per unit volume, edge length intensity, SV – total face area per unit volume, face area intensity, U1 – length of the typical edge, U2,A2 – perimeter and area, resp., of the typical face, U3,B3,A3,V3 – total edge length, mean width, surface area and volume, resp., of the typical cell, N0,N1,N2,N3 – number of nodes, of edge midpoints, of face centroids, of cell centroids, resp., per unit volume, N01,N02,N03 – number of edges, of faces, of cells, resp., that meet in the typical node, N12 = N13 – number of faces, of cells, resp., that meet in the typical edge, N20 = N21 – number of nodes, of edges, resp., on the boundary of the typical face, N30,N31,N32 – number of nodes, of edges, of faces, resp., on the boundary of the typical cell. Also for the three-dimensional case, Mecke (1984) showed that all these mean values can be expressed by a system of 7 parameters, namely N0, N3, N=N1 +N2, LV, SV , T =N03N0, Z=N2U2 =N3U3 . The parameter T can be interpreted as the mean multiple number of nodes per unit volume, and the parameter Z as the mean total multiple edge length per unit volume. The multiplicity of a node or of an edge is given by the number of the adjacent cells. A plane h in R3 can be parametrized by p and h?, as it was already used in Eq. 1. Denote by S+2 the _ 33 Nagel W et al: STIT tessellations upper unit half-sphere in R3, and identify the lines h? with the unit vector that is defined by the intersection u = h? S 2. Thus the directional distribution R can also be understood as a distribution on S?\. We abbreviate z2 = f f[u1,u2]&(du1)&(du2) , z3 = f f [[u1,u2,u3]&(du1)&(du2)&(du3) , where the integration is over <5f\ and [u 1,u2] denotes the area of the parallelogram spanned by u1 and u2 and [u1,u2,u3] denotes the volume of the parallelepiped spanned by u1, u2 and u3. In the isotropic case we have z2 = p/4 and z3 = p/8. For a homogeneous STIT tessellation with face area intensity SV and directional distribution M the mean values are as follows. N0 =SV3 z3 , (5) N3 = 6 SV 3 z3 , (6) N=169SV 3z3, (7) LV =SV2 z2 , (8) T =4SV3 z3 , (9) Z =3SV2 z2 . (10) The proof is given in the Appendix. Hence, the known formulas for mean values of homogeneous tessellations (see Mecke, 1984) yield for STIT tessellations N1 U1 U2 U3 A3 N01 2SV z 3, 1 z2 2 SV z3 , 18 z2 7 SV z3 , 18 z2 SV z3 , 12 1 N2 7 SV3 z3, A2 B3 V3 4, N 02 = 6, N 03 6 1 7 SV 2 z3 , 3 z2 2 SV z3 , 6 1 4, N12=N13 =3, N20 =N21 = 376, N30=24, N31 = 36, Finally, denote by R1 the directional distribution of the edges in a STIT tessellation Y, more precisely, the length-weighted directional distribution of edges or directional distribution in a typical edge point, respectively. The corresponding directional distribution for a Poisson plane tessellation is denoted by R1P. If Y and YP are homogeneous STIT and Poisson plane tessellations respectively which have identical directional distributions R of their faces then R1 =R1P . (11) A proof is given in the Appendix. COMPARISON WITH OTHER TESSELLATIONS Now the mean values of different random homogeneous tessellations are compared: the STIT tessellations, Poisson-Voronoi tessellations and the Poisson line (in R2) or Poisson plane (in R3) tessellations, respectively. This will be restricted to the isotropic case. The notation for STIT will be as above, the values for the Poisson-Voronoi tessellation are indicated by an upper index V and those ones for the Poisson line or plane tessellations by the upper index P, respectively. THE PLANAR CASE (d = 2) In the planar and isotropic case we have the parameter z = 2/p. In order to make the tessellations comparable we assume that N2 = N2P = N2V which also implies that the mean areas of the typical cells of all three tessellations are the same. Then we obtain with the results in Mecke (1984) (cf. the references to Miles and Santalo´ therein) 2N0P -- N0 — N0V, 3 AP 2 1 -- N1 — N1V, AP2 -- A2 — AV2 , U2P -- U2 — p U V 2 2 , 2 1 -- U1 — 2 1 , LP -- LA — pLV 2 A , N0P1 = NP 2 = 4, N 01 =N02 N 0V N2P0 = NPl = 4, N20 = N21 N 2V — N 2 1 3, 6. N32 = 14. Since the distribution of the typical cell of a STIT tessellation is the same as that one of a Poisson _ _ 6 _ _ _ _ _ _ _ 34 Image Anal Stereol 2008;27:29-37 line tessellation, also the respective mean values U2 and LA coincide. On the other hand. The numbers N0, N1, Nij, i, j = 0,1,2, that express some features of the mutual arrangement of the cells are conform for STIT and Poisson-Voronoi. THE SPATIAL CASE (d = 3) In the three-dimensional and isotropic case the parameter values are z2 = p/4 and z3 = p/8. In analogy to the planar case we assume that N3 = NP P 3 N3V which also implies that the mean areas of the typical cells of all three tessellations are the same. The formulas given in Mecke (1984) (partially with references to Miles and to Santalo´) yield the following results. Per unit volume: 6 NP N0 pu 0.887 N0 4 N 1 P = N1 ps 0.887 N1 V 37 N 2 P = N2 ~ 0.901 N2 V SP = SV PÜ 0.853 S 2 LVP = LV ˜ 0.829 LVV Typical cell: V3P AP3 BP3 U3P Typical face: AP2 9 14 V3 A3 B3 U3 A2 V 3 V ps 0.853 AV 3 f« 0.829 BV 3 pä 0.829 U 3 V pä 0.946 A2 U2P = U2 ~ 0.920 U 2 V Typical edge: 0.5 U1P U1 f« 0.935 U 1 V Adjacent to the typical node: N0P1 =6 N0P2 = 12 N0P3 =8 N01 N02 N03 v01 N 02 N 03 4 6 4 In the typical edge: N1P3 =4, N13 For the typical face: N 13 N = 4 , N20 = 36 ~ 5.143 For the typical cell: P 8, N30 N3P0 = A31 NP 2 12 6 N31 N32 24 36 14 V 30 f« 27.071 V 31 f« 40.606 V 32 pä 15.535 As pointed out in (iv) above, the distributions of the interiors of the typical cells of the STIT tessellation and of the Poisson plane tessellation coincide if their intensities SV = SVP and their directional distributions are the same. Hence it is obvious that also the other ’metrical’ (or scale dependent) values V3, A3, B3, U3 coincide. On the other hand, the cells of the STIT tessellation are not ’face-to-face’. Hence on the faces of the STIT cells are further edges and nodes. The mean values Nij with i > j, i = 2, 3, are some quantitative expression for that, and thus they provide some information about the mutual arrangement of the cells within a tessellation. The results clearly show that in this respect isotropic STIT and Poisson Voronoi tessellations are rather close together and that significant differences appear to the mean values for isotropic Poisson plane tessellations. These results can be generalized also for some homogeneous but not necessarily isotropic tessellations which are generated by an affine transformation of an isotropic one. APPENDIX Now the proofs are given for the Eqs. 5-11 for homogeneous (but not necessarily isotropic) STIT tessellations. This will be based on the following facts. i) A STIT tessellation has a Poisson typical cell, see (iv) in the section on key properties. ii) If Y is a STIT tessellation, then it is identically distributed as I(2Y, 2Y ). The calculation of mean values of I(2Y, 2Y ), where the effect of iteration is taken into account, provides equations for the mean values of Y. iii) If N0, LV, SV are the mean values of Y, then the respective mean values of 2Y are N0/8, LV/4, SV/2. N3P, and thus N2 V0 ~ 5.228 The Poisson plane tessellation with SV and directional distribution R has N3P = N0P = SV3 z3/6. Due to (i) it is V3 = V3P and hence N3 Eq. 6. In order to show Eq. 8, observe that the network of edges of I(2Y, 2Y ) consists of edges of the _ _ _ _ _ _ _ _ _ _ _ _ _ 3 _ _ 35 Nagel W et al: STIT tessellations frame tessellation 2Y and edges of (2pkC\2Yk) nested into the frame cell 2pk, k = 1,2,..., cf. Eq. 4, and, additionally, further edges which are generated by the intersection of the two-dimensional faces of 2pk with two-dimensional faces of 2Yk. In order to calculate the length intensity of those edges that are newly generated by iteration consider independent and stationary processes O1, O2 of two-dimensional faces in R3 with distributions P1, P2, directional distributions R1, R2 and intensities SV(1), SV(2) (with 0 < SV(1), S 0) (2) (1) (2) V < ¥), respectively. Analogously to Theorem 1.1. and Lemma 3.2 which were shown for fibre processes by Mecke (1981), one can assert also for processes of faces in R3 that IB(y)m (p1(l(p2 (dy)P1(dj1)P2(dj2) S <2) / / / IBiy) I sin(uTyÇ1 — v) I x S ^2(dv)ßq>1(dy)P1(dc1) 1 SV 2 f[\sin(u-v)\&1(du)&2(dv) , (12) where Tyç = ç —y, uç denotes the normal direction to ç in the origin 0 (given that 0 G ç), ß(? the area measure jointed by