两种根管弯曲度测量法间数学关系的推导

A mathematical analysis of the relationship between Schneider's method and Weine's method for the measurement of root canal curvature

目的:研究2种根管弯曲度测量法测值(Schneider法的测值为θS,Weine法的测值为θW)间的数学关系。方法:将弯曲根管长轴简化为依次由相切的线段(长度为l1)、圆弧(半径为r、圆心角为θ)、线段(长度为l2)组成的连线,根据2测量法确定简化弯曲根管长轴测量角时不同的定点、连线方式,推导θS、θW间的函数方程以及与l1、l2、r、θ间的数学关系,分析方程与函数图形的特征。结果:θS与θW间存在以下数学关系:(1)tanθS=(1-cosθW+ksinθW)/(sinθW+kcosθW),k=l2/r;(2)θW/2≤θS≤θW。k取不同值时函数曲线的形态具有不同的特点。θW在0,π的区间内,θS与θW间呈近似的线性关系;k越大,θS越接近θW,k越小,θS越接近θW/2。结论:Schneider法与Weine法测值间存在着复杂的函数关系,根管根尖部直段的长度(l2)与弯曲段曲率半径(r)之比值是影响2种方法测值间差异的重要因素。

Objective：To study the mathematical relation between the measurements of root canal curvature obtained by Schneider＇s method（ θs ） and Weine＇s method（θw）. Methods：The axis of a curved root canal was simplified by an arc （ whose radius was r and central angle θ） and 2 tangent line segments （ whose lengths were 11 and 12 ） ,the relation of θs and θw was studied with the help of mathematical analysis according to the different ways of determining the measured angles on the simplified axis. The graphs of the function were analysed. Results：The following formulas were proved：（ 1 ） tanθs = （1 - cosθw ＋ ksinθw） / （ sinθw, ＋ kcosθw ） ,k =l2/r ; （ 2 ） θw/2 〈= θs 〈=θw. If θw was in the interval [ 0, π] , a proximate linear correlation existed between θs and θw. Conclusion ：There is a complex function between the 2 angles（θs and θw） measured by Schneiderg method and Weine＇s method, the ratio of the length of the apical straight part to the radius of the canal curvature is an important factor determining the different values of Os and θw.