An inequality for double tangents
An inequality for double tangents
复制标题
二重切线的不等式
DOI：
10.1090/s0002-9939-1979-0534404-2
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发表时间：
1979
期刊：
影响因子：
--
通讯作者：
B. Halpern
中科院分区：
文献类型：
--
作者：
B. Halpern
对于平面上的正则闭曲线，已知E = I + X +2F，其中E，I，X和F分别是外双切线，内双切线，自交点和双曲线点的个数。这里证明了如果F = 0，则I是偶数且I <（2X + 1）（X1）。此外，还举例说明了如果四个非负整数元组（E，I，X，F）满足（a）F偶，（B）E = I + X + F，（c）如果F = 0则I偶且I < X（X1），则存在实现这些值的正则闭平面曲线.
For a regular closed curve on the plane it is known that E = I + X + 2 F where E, I, X and F are the numbers of external double tangents, internal double tangents, self-intersections, and inflexion points respectively. It is proven here that if F = 0 then I is even and I < (2X + 1)(X 1). Furthermore, examples are given which show that if the four tuplet (E, I, X, F) of nonnegative integers satisfies (a) F even, (b) E = I + X + F, and (c) if F = 0 then I is even and I < X(X 1), then there is a regular closed plane curve which realizes these values.