Geodesic Density And Some Integral Formulae On The Surface Of Constant Curvature

In this paper,the geodesic density and some integral formulae on the surface of constant curvature in integral geometry are discussed.Firstly,some basic properties of geodesic density on the surface of constant curvature are derived.Other forms of geodesic density in special circumstances are also discussed.Finally,some inte-gral formulae of the surface of constant curvature are obtained from the density of geodesics.This paper is divided into two parts.The first part is mainly related to the geodesic density on the surface of constant curvature.Firstly,the properties of geodesic density on the surface of constant curvature are derived by using some uniform basic trigonometric formulae.Secondly,other forms of geodesic density under special circumstances are discussed,and two commonly used geodesic density on the unit sphere are studied,and the relationship between them is found.In the second part,the classical Crofton formulae on the plane are generalized by the density of geodesics on the surface of constant curvature,and another integral formula related to Crofton formula on the unit sphere is obtained,which is a special case of the Poincaré formula on the unit sphere.