Another Form Of The Formula For The Kinematic Measure Of A Segment Contained In A Convex

Convex bodies are the research objects in the paper. In this paper we research the formula for the kinematic measure of a segment contained in a convex domain. The kinematic measure of a segment contained in a convex domain belongs to the problem in the theory of geometric probability. Buffon's needle problem is considered the first problem in the theory of geometric probability. At 1980's, the kinematic density, the Poincare's formula, which is the measure of the set of motions that carries a moving curve to intersect a fixed curve and the fundamental kinematic formula of Blaschke, which is the measure of the set of motions that carries a moving domain to intersect a fixed domain are applied in《Topics in Integral Geometry》by De-lin Ren. What we obtain is the kinematic measure to solve the problem in the theory of geometric probability.In the《Topics in Integral Geometry》, the two new definitions: the generalized support function and the restricted chord function are introduced, and using them to set up the popular formula for the kinematic measure of a segment contained in a convex domain. The measure of m (l ) for a rectangle is calculated. In the base of this, the paper introduces the definition of the radial function. Through the radial function rather than the generalized support function in the old measure formula, we can obtain another form of the formula for the kinematic measure of a segment contained in a convex domain, and give strict mathematic proof. When the convex domain is given by the radial function, the formula supplies the way which immediately calculates the contained measure.On the other hand, we give an example in a circle by the new form of the formula.