Kinematic Measure Of A Rectangle With Fixed Length And Width Within A Convex Domain

Kinematic formulas in integral geometry are integral formulas that represent integrals of geometric functions on the intersection of fixed and moving domains. These formulas can be viewed as integral formulas for various intersection measures. They are useful for solving problems in geometric probabilities. Some problems in geometric probability require more tools other than intersection measures. For instance, solutions to the Buffon needle problem of lattices need to compute the measure of the needle that is contained in a fundamental region of the lattice. The kinematic measure of a moving domain that is contained in a fixed domain is called a containment measure.Ren-Delin introduced the notions of generalized support function and restricted chord function of a convex body in the plane, and used them to establish a kinematic measure formula for a segment of fixed length within a convex domain. He then applied this formula to solve generalized Buffon needle problems of lattices.In this paper, we search the rectangle with fixed length and width in the base of above results. We establish a formula for the kinematic measure of a rectangle with fixed length and width and contained in a convex domain. Using it, we calculate the kinematic measure for such a rectangle contained in a circle and a rectangle. Then the later is applied into the geometric probability.