Geometric Probability Of Beam Of Planes Intersected With Convex Body

This thesis reviews the historical origin of integral geometry and its development.Besides, we brie?y introduce constant movement formulas of the couple of linear space,definition and property of homogeneous integral formula and the mean curvature inte-gral, and measure formula on the plane intersecting with the convex body in the linearspace. In this paper, we solve the following problems.Firstly, we discuss the problem of the dual of linear space intersecting with con-vex body in the linear space, and solve further the problem of geometric probabilityabout the dual of linear space intersecting with convex body and also the part of theintersection of the dual of linear space intersecting with convex body. We simplify theformulation of the movement density about which is the dual of linear space intersect-ing with convex body and also the part of the intersection of the dual of linear spaceintersecting with convex body as the computing form of the measure of only one planeintersecting with convex body, by means of the measurement formula of plane inter-secting with convex body. We obtain again the measurement results of which is thedual of linear space intersecting with convex body and also the part of the intersectionof the dual linear space intersecting with convex body by the relevant theory of theformula of Grassmann manifold sizes and the homogeneous integral. Finally,we calcu-late the measure of the dual of linear space intersecting with convex body, which cansolve the problem of geometric probability, also getting some of the relevant inferencesfurther.Then we promote the above conclusions to the geometric problem of the ?at beamintersecting with convex body. We mainly discuss the problem of three plane inter-secting with convex body. First of all, we disintegrate this problem into the geometricproblem of the dual of linear space and plane intersecting with convex body. We firstlycalculate the measure of the dual of linear space intersecting with convex body by in- tegration step by step,which have been discussed in the previous. Finally, we can solvethis problem by the same way. And we obtain some result in the special conditions.