The Kinematic Density For Pairs Of Intersecting Lines And Geometry Probability Applications

In this paper, we get a kinematic density formula for pairs of intersecting lines in R~d, which has not yet got in integral geometry, by using the moving orthogonal frames method. And the writer got some different formula expressions of the kinematic density for the intersecting lines. At the same time some important properties and some relative geometric formulas are given. At last, a application in geometric probability is given, which is solved by using the properties and concept of the pairs of intersecting lines.In integral geometry, we did not do a particularity research of conditional probability. We always come into contact with some problems puzzling ourselves, such as the following situation.In R~d space, we consider the situation of the intersecting of pairs of random lines with convex bodies. We known that there are three relative relationships: parallel, intersecting, and in different flats. Therefore when we are considering their kinematic density, at first, we must consider their relative relationship that they are parallel, intersecting or in different flats. In fact, it is easy to see, compared with the kinematic set of in different flats, the measurement for the kinematic set of parallel and intersecting is zero. It is complex to do reaserch about the situation of in different flats. The author only discussed problems in the situation which the pairs of random lines were in the same flat.Moving orthogonal frames method is the method that is often used to get the kinematic density. In this paper using the method. I got some expect results, but not as good as we thought. In high dimensional space, the dimension of their parameter is too high. It is difficulty to find a proper formula. We should make great efforts to seek proper theoretic tools to solve the problems in order to abtain useful application.In the paper, the probality of which pairs of random lines are intersecting with convex bodies and the points of intersection are in convex bodies is given. And a geometry probability in R~2 is extended.