本文利用活动标架法,得到了积分几何中至今还没有的相交直线偶的运动密度公式,并根据此公式获得到了相交直线偶的交点落入凸体$K$的运动公式.

Journal of Mathematical Research&Exposition

Feb.,2008,Vol.28,No.1,pp.207–210

DOI:10.3770/j.issn:1000-341X.2008.01.026

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The Kinematic Density for Pairs of Intersecting Lines

XIE Peng,XIE Ke-yao,CHENG Meng-liang

(Department of Mathematics,Huazhong University of Science and Technology,Hubei430074,China)

(E-mail:xiepeng@mail.hust.edu.cn)

Abstract In this paper,we get a kinematic density formula for pairs of intersecting lines which has not yet been gotten in integral geometry by using the moving orthogonal frames method.

And we obtain a kinematic formula of the intersection of the pairs of intersecting lines belonging to convex body K by using it.

Keywords pairs of intesecting lines;kinematic density;kinematic formula.

Document code A

MR(2000)Subject Classi cation53C65;60D05

Chinese Library Classi cation O186.5

1.Introduction

The kinematic density for pairs of intersecting lines in R2is the same as the kinematic density for pairs of lines in R2.Let G and L be two lines which intersect at the point P,θbe the angle between G and L,andαbe the angle G with the x-axis.We can get the formula[1,2]

d G d L=sinθd P dθdα.(1)

The kinematic density for pairs of intersecting lines in R3is di erent from the kinematic density for pairs of lines in R3.The kinematic density for pairs of lines in R3is[3]

d G d L=sin2θd P N d Q N d G N d L N d N.(2)

Where N is the common perpendicular of lines G and L,P=N∩G,Q=N∩L,θdenotes the angle between G and L,d P N and d Q N are the densities of P and Q on the common perpendicular N,respectively,and d G N and d L N are the densities of G and L for the rotations around N, respectively.

The kinematic density for pairs in R n is[4,5]

d G d L=t n 3sin2θd P N d Q N d G N d L N d N,(3)

where t denotes the length of P Q,θdenotes the angle between G and L,d P N and d Q N are the densities of P and Q on the common perpendicular N,respectively,and d G N and d L N are the densities of G and L for the rotations around N,respectively.