Geometric Probability Problems Of Applications And Generalizability

In this paper. we discuss the history origin of integral geometry, its product anddevelopment in the introduction. Also. the kinematic in-variant density formulas forpairs of needles and lines in R~2 and R~3 the concept and properties of chord-powerintegrals of a convex body are introduced in brief. On this foundation, the followingfew problems are resolved.Firstly, we have a kinematic density formula for pairs of needles by using thekinematic density formula for pairs of oriented lines. The geometric probability thatthe random pairs of needles intersect at a point in convex body K is considered. Andwe get two results that kinematic measure of the random pairs of needles intersect inconvex body K and the random pairs of needles intersect at a point in convex body Kby calculation. Thereby, the result of geometric probability problem and its inferencesare gained.Secondly, we give rise to the new concept of double chord-power integrals of aconvex body, which considered the conclusion of geometric probability problem andthe concept and theory of chord-power integral. So we mention another geometricprobability problem which is the geometric probability of the intersection of pairs ofintersecting lines belonzs to convex body K in R~2 by using the concept of doublechord-power integrals and a kinematic invariant density formula for pairs of lines. Inthe paper, we study the relation of pairs of lines and pairs of needles, and two results ofintegral formulas are gained. Thus, the geometric probability problem is solved. Also.we give a solution to the problem: pick two points randomly in a convex body K anddraw two lines passing through two points. what is the probability that the intersectingof two lines belongs to K?Finally, in this paper, we introduce the concept of double chord-power integrals ofa convex body in R~3, and some properties of double chord-power integrals of a convex body in R~3 are got. At last, the approximate probability that the intersection of pairsof lines belongs to a ball is given.