广义正则平面曲线的顶点

<正> 引言.四顶点定理.自印度加尔各答大学教授 Mukhopadhyaya 证明后,继而论之者颇多.Blaschke 证明:一正则卵形线若与任一圆之交点至多有四点时,则此卵形线至多有四顶点.另一人证明:一正则卵形线若与一圆有2n个交点时,则此卵形线至少有2n个顶点.Fog 及 Graustein 曾推广四顶点定理至简单闭曲线.Jackson 从研究二顶点曲线的特性出发,以研究四顶点曲线,指出了更广泛的四顶点曲线族.因不具二顶点曲线的特性就为四顶点曲线.以上这些曲线是指正则曲线,即曲线上的曲率不仅存

VERTICES OF GENERALIZED REGULAR PLANE CURVES

A point P of a plane arc(?)is called regular if the curvature of(?)iscontinuous at P.Otherwise,P is called non-regular.The arc(?)is calledregular if every point of(?)is regular.This definition may be generalizedin the following:Let(?)be a differential plane arc,there are only n non-regular points on(?),n being positive integer or zero.Let P be a regular point of(?),and P_0a non-regular point of(?),denote the length of(?)by l and (?)=S,(?)=S_0,0K(S)).Where δ be any sufficient small positive number.Then P is called a vertexof(?)and having the minimum(or maximum)of curvature.Let(?)be a sub-arc of(?)(C,D≠A,B),let the curvature of(?)beconstant and greater(or less)than the curvatures of P and Q,where P andQ are respectively regular points on(?)and(?)and near C and D.Then(?)is also called a vertex of AB.We have the following generalized theorems.1.Let(?)be an arc of generalized regular oval,O_1 and O_2 be the anglesformed by the chord AB and the tangents at A and B respectively and in- terior to the region bounded by the chord AB and the arc(?).Let P by anyregular point on(?),(?)=S,K(S)the curvature of P,(?)=L,Then wehave the following properties:(a)If(?)have no vertex and O_1>O_2.Then K(S)is decreasing in(O,l).(b)If O_1=O_2.Then AB has at least one vertex.(c)If(?)has no vertex and K(S)decreasing in(O,l).Then O_1>O_2.2.Every generalized regular oval has at least four vertices.3.Let A_1,A_2,…,A_(2n)be the vertices of a generalized regular oval(not acircle),where A_1,A_3,…,A_(2n-1)have the minimum of curvatures and A_2,A_4,…,A_(2n),the maximum of curvatures.Then the angles of the inscribed poly-gon A_1 A_2…A_(2n),have the following property:∠A_1+∠A_3+…+∠A_(2n-1)>∠A_2+∠A_4+…+∠A_(2n).4.The vertices of a generalized regular oval(not a circle)are not all ona circle.5.Every generalized regular simple closed curve has at least four vertices.6.If any circle cut a generalized regular oval at most four points.Thenthe oval has at most four vertices.7.If a generalized regular oval and a circle have 2n points in common.Then the oval has at least 2n vertices.8.The two vertex-curve C(generalized regular closed curve)has the fol-lowing propertices:(a)The curve C has nodal points and every nodal point is simple.(b)If the curve C touch itself at a point A.Then the directed tangentsat A coincide.(c)Let A,B(≠A)be two points of inflection of C and the arc(?)hasno point(≠A,B)in common with the tangents at A and B.Then the arcs(?)and(?)have no common point other than A and B.(d)Let A,B be two points of inflection of C and the tangents at A andB are parallel to each other but not coincide.Then the arcs(?)and(?)haveno common points other than A and B.