INTERSECTIONS OF RANDOM LINES

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1 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 65 (2000) pp INTERSECTIONS OF RANDOM LINES Rodney Coleman Imperial College of Science Technology and Medicine, University of London Abstract We have three overlapping planar sets within a bounded window. These sets do not have to be convex, or even connected domains. We find a simple expression for the probability that random straight lines across two of the sets intersect within the third. The lines are taken uniformly at random from beams of parallel lines, with the orientations of the beams from an arbitrary joint distribution. That is to say, each of the lines is taken with the probability measure invariant under translations, restricted to those lines that intersect the set. This generalises a celebrated result of Sylvester ( ) for the intersection within a convex planar domain of two random lines. The proof uses only elementary probability theory, expressing the invariance through uniform probability distributions. It extends results for the isotropic secant case presented in Coleman (1997a). IUR SECANTS; FUR SECANTS; STEREOLOGY; RANDOM SETS; GEOMETRIC PROBABILITY; STOCHASTIC GEOMETRY; INVARIANT MEASURE; SYLVESTER. AMS 1991 Subject Classification: Primary 60D05 Secondary 52A10; 52A22

2 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 65 (2000) pp INTERSECTIONS OF RANDOM LINES Rodney Coleman Abstract We have three overlapping planar sets within a bounded window. These sets do not have to be convex, or even connected domains. We find a simple expression for the probability that random straight lines across two of the sets intersect within the third. The lines are taken uniformly at random from beams of parallel lines, with the orientations of the beams from an arbitrary joint distribution. That is to say, each of the lines is taken with the probability measure invariant under translations, restricted to those lines that intersect the set. This generalises a celebrated result of Sylvester ( ) for the intersection within a convex planar domain of two random lines. The proof uses only elementary probability theory, expressing the invariance through uniform probability distributions. It extends results for the isotropic secant case presented in Coleman (1997a). 1. Introduction The invariant measure of all straight lines in the plane, restricted to those that intersect a compact convex set, is proportional to its perimeter length (Cauchy, 1850). This gives the remarkable result (Barbier, 1860, Crofton, 1869) that, for two such sets, one interior to the other, the probability that a random line through the outer set cuts the inner one is the ratio of their perimeter lengths. This heralded modern geometrical probability. Without a practical application beyond the

3 68 RODNEY COLEMAN computation of odds in games of chance, such as Buffon s needle problem (Buffon, 1777; see also Coleman, 1997b), there was almost no development until recent times. In image processing, and particularly in stereology, features seen in the image often cannot be modelled as convex bodies. Understanding and measuring irregular set processes is essential in modern science. However, extensions to R n, with lengths, areas, etc., replaced by quermassintegrale and kinematic formulae (Santaló, 1977), and in particular combinatorial integral geometry (Ambartzumian, 1982) and geometric measure theory, hide the simplicity inherent in the Barbier-Crofton result. Representations of lines and planes, etc., as points over a phase-space, allow an invariant distribution to be modelled as a Poisson point process, and those that intersect the body by points uniformly at random on a domain D in the phasespace. This technique is a simplistic early application of the methods of stochastic geometry (Stoyan, Kendall, Mecke, 1995). The probability structure is thus given by a probability density function which is a constant over D, and zero outside. This simple description continues to apply if we relax the conditions that the lines and planes be invariant under Euclidean motions across compact convex sets. We can treat invariance under translation only, with arbitrary anisotropic orientations, across not-necessarily-convex, not-necessarily-connected, sets. This is illustrated by extrapolating from the celebrated result of Sylvester ( ) giving the probability that two independent invariant lines in the plane, which each hit a compact convex set, cross in the interior of the set. In Coleman (1977a) a formula was derived for the probability of isotropic secants crossing when we have one secant through each of two planar sets. These results are obtained as a special case of the generalisations shown below. The structure of the paper is as follows. In Section 2 is explained what is meant by a random secant of a set. In Section 3 some results for a single secant across a set are given. In Section 4 the probability of intersection of two independent random secants across a set is found. In Section 5 this is extended to the case of two overlapping sets with one random secant across each.

4 INTERSECTIONS OF RANDOM LINES Random secants Let Z be a planar set which lies in a bounded window. This avoids the problem of potentially infinitely long secants. We demand of the set only that it has a rectifiable area and boundary, that is to say, it has an area, projections and perimeter length in the natural sense. In the following, a random secant of Z is the intercept in Z of a straight line having the probability measure invariant under translations, restricted to those lines intersecting Z. The measure used by Crofton, Sylvester and other 19th century mathematicians for lines uniformly at random in the plane, gives these lines isotropic orientations. The lines are then termed IUR by stereologists, where IUR stands for Isotropic Uniform Random. Their intercepts in Z will be called IUR secants. If we restrict the lines to a single orientation, the lines are called FUR (Fixed orientation Uniform Random). If the FUR lines are normal to an axis making an angle θ to the x-axis, the FUR secants will be termed FUR θ-secants. If the orientation is from an arbitrary distribution, the secants will be called AUR (Anisotropic Uniform Random). In each case the line is taken uniformly from a beam of parallel rays striking the set Z. Every undirected straight line in the plane can be given the polar coordinates (x, θ) of its closest point to an origin of coordinates. Let D(Z) be the set of coordinates of those straight lines T (x, θ) which intersect Z, and let Q(x, θ) be the intercept in Z of T (x, θ). 2.1 IUR secants The IUR secants correspond to taking the points (x, θ) uniformly at random over D(Z) in (x, θ) phase space. The coordinates (x, θ) of an IUR secant Q(x, θ) of Z therefore have the probability density function (pdf) f(x, θ) = 1 I{(x, θ) D(Z)} (1) H(Z) where H(Z) is a constant, the area of D(Z) in the phase space: 1 = R 2 f(x, θ) dx dθ = 1 dx dθ. (2) H(Z) D(Z)

5 70 RODNEY COLEMAN 2.2 FUR secants Let Z θ be the projection of Z onto an axis at angle θ to the axis of polar coordinates, and let L(Z θ ) denote its length. The intercept of a line T (x, θ) in Z and normal to Z θ, we shall term the FUR θ-secant Q θ (x). It corresponds to taking the x-coordinate uniformly over Z θ, ie over D θ (Z) = {x : T (x, θ) Z} = {x : T (x, θ) Z }, where is read as hits. The x-coordinate has the uniform probability density function 1 f(x θ) = H θ (Z) I{x D θ(z)}. (3) Clearly H θ (Z) = L(Z θ ) (4) and H(Z) = D(Z) dx dθ = π θ=0 { x D θ (Z) } dx dθ = π 0 L(Z θ ) dθ = π π 0 L(Z θ ) dθ π, (5) so H(Z) is π times the mean length of the projection of Z with respect to a uniform distribution for θ. For planar domains this mean projection is sometimes referred to as the mean thickness, but this is not appropriate when applied to our more general sets. Since f(x, θ) = f(x θ) f(θ), (6) for IUR secants we have f(θ) = H θ(z) H(Z) I{0 < θ π}. (7) 2.3 AUR secants The orientations of AUR secants have a generalised probability density function f(θ), ie one which can be a convex combination of delta functions if the distribution is discrete, or a mixture of the delta functions with a pdf if the distribution is a

6 INTERSECTIONS OF RANDOM LINES 71 Figure 1: Planar set Z containing subset Z 0 with FUR θ-secant, Q θ, showing the projections Z θ and Z 0θ of the sets onto an axis at angle θ to the axis of polar coordinates. mixture of discrete and continuous distributions. conditional pdf f(x θ) given by (3). The x-coordinate has the 3. A single secant 3.1 FUR secants Lemma If Z 0 Z, an FUR θ-secant Q θ of Z intersects Z 0 with probability H θ (Z 0 ) H θ (Z). Given that Q θ intersects Z 0, it is an FUR θ-secant of Z 0. Proof. The coordinate x is uniformly at random on D θ (Z), and, given that x is on D θ (Z 0 ), a subset of D θ (Z), it is uniformly at random on D θ (Z 0 ). We note that this result does not depend on the location or shape of Z 0 within Z, or even whether Z or Z 0 are connected domains.

7 72 RODNEY COLEMAN Example If Z is (possibly disconnected segments of) a straight line of length L(Z) making angle ψ to a straight line T (x, θ) (angle φ to the axis of polar coordinates), then, from (4), H θ (Z) = L(Z θ ) = L(Z) sin ψ = L(Z) cos(φ θ). (8) Lemma If Q θ is an FUR θ-secant of Z, then where A(Z) is the area of Z. Proof. EL(Q θ ) = A(Z) H θ (Z) = A(Z) L(Z θ ), (9) EL(Q θ ) = E X θ L(Q θ (X)) = x D θ (Z) L(Q θ (x)) 1 H θ (Z) dx = 1 H θ (Z) x D θ (Z) L(Q θ (x)) dx = A(Z) H θ (Z), since Q θ (x) sweeps out the area A(Z) as x sweeps over the projection Z θ. Lemma Suppose that Z 1 and Z 2 are planar sets and that Q θ is an FUR θ-secant of Z 1 Z 2. Then P (Q θ Z 2 Q θ Z 1 ) = P (Q θ Z 1 Z 2 Q θ Z 1 ) = H θ(z 1 Z 2 ) H θ (Z 1 ) by Lemma IUR secants Similar arguments give the corresponding results for IUR secants. Lemma If Z 0 Z, an IUR secant of Z intersects Z 0 with probability H(Z 0 ) H(Z). Given that an IUR secant intersects Z 0, it is an IUR secant of Z 0.

8 INTERSECTIONS OF RANDOM LINES 73 Example If Z is (possibly disconnected segments of) a straight line of length L(Z), then, from (5) and Example 3.1.2, H(Z) = π L(Z θ ) dθ = π 0 0 L(Z) cos(φ θ) dθ = 2 L(Z). (10) Lemma If Q is an IUR secant of Z, then EL(Q) = πa(z) H(Z) (11) The formula for EL, π A(Z)/B(Z), where B(Z) is the perimeter length of a convex planar domain, and its 3-d analogue for the expected length of an IUR secant of a convex 3-d domain, 4 V (Z)/S(Z), where V (Z) is its volume and S(Z) is its surface area, have been widely used in acoustics and radiation studies, and also misused when applied to secants not having the IUR property. Lemma Suppose that Q is an IUR secant of Z 1 Z 2, then P (Q Z 2 Q Z 1 ) = H(Z 1 Z 2 ) H(Z 1 ). (12) For Z 1 and Z 2 convex planar domains, Lemma was proved by Crofton (Crofton, 1869). 4. Two random secants Lemma 4.1. Let C θ,φ be the event that independent FUR secants, Q θ and Q φ, of a planar set Z intersect, where Q θ is an FUR θ-secant and Q φ is an FUR φ-secant. Then sin(φ θ) A(Z) P (C θ,φ ) =. (13) H φ (Z) H θ (Z) Proof. P (C θ,φ Q θ ) = P (Q φ Q θ Q θ ) = H φ(q θ ) H φ (Z) by Lemma = L(Q θ) sin(φ θ) H φ (Z),

9 74 RODNEY COLEMAN by setting ψ = φ θ in Example We see that, if θ = φ, with probability one there will be no intersection. P (C θ,φ ) = E Qθ P (C θ,φ Q θ ) = sin(φ θ) H φ (Z) EL(Q θ ) = sin(φ θ) H φ (Z) A(Z) H θ (Z) by Lemma Lemma 4.2. Let C θ,a be the event that independent secants, Q θ and Q φ, of a planar set Z intersect, where Q θ is an FUR θ-secant and Q φ is an AUR secant with orientation φ taken from the (generalised) pdf f Φ (φ). Then P (C θ,a ) = E Φ P (C θ,φ ) = A(Z) H θ (Z) π φ=0 sin(φ θ) H φ (Z) f Φ (φ) dφ. Lemma 4.3. Let C θ,i be the event that independent secants, Q θ and Q, of a planar set Z intersect, where Q θ is an FUR θ-secant and Q is IUR. Then P (C θ,i ) = 2 A(Z) H(Z) H θ (Z). Lemma 4.4. intersect, then Let C I,I be the event that two independent IUR secants of Z P (C I,I ) = 2π A(Z) H 2 (Z). If Z were a convex domain, this is the celebrated formula of Sylvester referred to in the abstract (see also Crofton, 1869). 5. Secants crossing overlapping sets An elementary conditioning argument extends the lemmas to the crossing of two secants within Z 0, when one is in Z 1 and the other is in Z 2, where Z 0, Z 1 and Z 2 are planar sets. Clearly, for any intersection, the intersection must take place in the overlapping region Z = Z 0 Z 1 Z 2 of the three sets.

10 INTERSECTIONS OF RANDOM LINES 75 Figure 2: A secant crossing a two-piece planar set Z 1, another crossing a square set Z 2, and intersecting within a circular set Z 0. Theorem 5.1. Let Q θ and Q φ be independent FUR θ- and φ-secants of Z 1 and Z 2 respectively, and let C θ,φ (Z 0 ) be the event that Q θ and Q φ intersect within a planar set Z 0. Then if Z = Z 0 Z 1 Z 2 P (C θ,φ (Z 0 )) = sin(φ θ) A(Z) H θ (Z 1 ) H φ (Z 2 ). (14) Proof. For C θ,φ (Z 0 ), the secants must intersect in Z. Furthermore, if they do, their intercepts in Z are independent FUR θ- and φ-secants of Z by Lemma Let K θ, K φ be the events Q θ Z, Q φ Z, respectively. Then P (K θ ) and P (K φ ) are given by Lemma 3.1.1, and P (Q θ Q φ K θ K φ ) by Lemma 4.1. Then P (C θ,φ (Z 0 )) = P (Q θ Q φ K θ K φ ) P (K θ ) P (K φ ) = sin(φ θ) A(Z) H φ (Z) H θ (Z) H θ (Z) H θ (Z 1 ) H φ (Z) H φ (Z 2 ). We note that { 0 (Qθ and Q φ parallel), sin(φ θ) = 1 (Q θ and Q φ perpendicular).

11 76 RODNEY COLEMAN To prove the corresponding result for the case in which one secant is IUR and the other is FUR we take expectations with respect to one of the secants over the IUR pdf (7). Theorem 5.2. Let Q be an IUR secant of Z 1, and Q θ be an independent FUR θ-secant of Z 2, and let C I,θ (Z 0 ) be the event that Q and Q θ intersect in Z 0. If Z = Z 0 Z 1 Z 2, 2 A(Z) P (C I,θ (Z 0 )) = H θ (Z 1 ) H(Z 2 ). Similarly we obtain the following result for the case in which both secants are IUR. Theorem 5.3. Let Q 1 and Q 2 be independent IUR secants of Z 1 and Z 2 respectively, and let C I,I (Z 0 ) be the event that Q 1 and Q 2 intersect within a planar set Z 0. Then if Z = Z 0 Z 1 Z 2 P (C I,I (Z 0 )) = 2 π A(Z) H(Z 1 ) H(Z 2 ). This Theorem 5.3 was first proved in Coleman (1997a). We note the following unsurprising corollary to the above theorems. Corollary 5.4. If {Z 0,1,..., Z 0,m } is a partition of Z 1 Z 2, then, for the three cases covered by the theorems, for k = 1, 2,..., m, P (C θ,φ (Z 0,k )) A(Z 0,k ), P (C I,θ (Z 0,k )) A(Z 0,k ), P (C I,I (Z 0,k )) A(Z 0,k ). References Ambartzumian, R. V. (1982). Combinatorial Integral Geometry. Wiley, Chichester. Barbier, E. (1860). Note sur le problème de l aiguille et le jeu du joint couvert. J. Math. pures et appl. (2) 5, Buffon, G. L. L. (1777). Essai d arithmétique morale. Chapter XXIII in Suppl. à l Histoire Naturelle, Vol. 4. Imprimerie Royale, Paris.

12 INTERSECTIONS OF RANDOM LINES 77 Cauchy, A. (1850). Mémoire sur la rectification des courbes et la quadrature des surfaces courbes. Mém. Acad. Sci. Paris 22, 3. (Pages in Oeuvres complètes. Series 1, Vol. 2 (1908)) Coleman, R. (1997a). Crossings of random secants. Rendiconti del Circolo Matematico di Palermo Serie II Suppl 50, Coleman, R. (1997b). Extending Buffon s needle. Acta Stereol. 16, Crofton, M. V. (1869). On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus. Phil. Trans. Roy. Soc. London 158, Santaló, L. A. (1976). Integral Geometry and Geometrical Probability. (Encyclopedia of Mathematics and its Applications. Vol. 1) Addison-Wesley, Reading, Mass. (Cambridge U. P., London, 1984). Stoyan, D., Kendall, W. S., & Mecke, J. (1995). Stochastic Geometry and its Applications. 2nd edition. Wiley, Chichester. Sylvester, J. J. (1890 1). On a funicular solution of Buffon s Problem of the needle in its most general form. Acta Mathematica 14, (Pages in Vol. 4 of The Collected Mathematical Papers of James Joseph Sylvester. Cambridge (1912)). Dr Rodney Coleman Department of Mathematics Imperial College 180 Queen s Gate London SW7 2BZ, UK r.coleman@ic.ac.uk Web: rcoleman/

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