Anti-orthotomics of frontals and their applications
Anti-orthotomics of frontals and their applications
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DOI：
10.1016/j.jmaa.2020.124019
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发表时间：
2019-06
影响因子：
1.3
通讯作者：
S. Janeczko;T. Nishimura
中科院分区：
文献类型：
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作者：
S. Janeczko;T. Nishimura
令 f: N n→ R n+ 1 为高斯映射 ν: N→ S n 的正面，并令 P∈ R n+ 1 为一个点，使得 (f (x)− P)⋅ ν (x)≠ 0 对于任何 x∈ N。在本研究中，对于映射 f~: N→ R n+ 1 定义为： f~(x)= f (x)−|| f (x)− P|| 2 2 (f (x)− P)⋅ ν (x) ν (x)，则以下四种说法成立。 (1) f～ 是高斯映射的额波 ν～(x)= f (x)− P|| f (x)− P||在 f~(x).(2) f~ 是 f 相对于 P 的唯一反正交。(3) (f~(x)− P)⋅ ν~(x)≠ 0 对于任何 x∈ N 都成立。(4) || 的等式f~(x)− P||=|| f~(x)− f (x)||对于任何 x∈ N 都成立。此外，给出了主要结果的三个应用。 Cahn-Hoffman 向量公式的推广作为第一个应用给出。第二个应用涉及澄清反正交组学的光学含义。第三个应用程序提供了用于确定给定锋面的锋面的标准。
Let f: N n→ R n+ 1 be a frontal with the Gauss mapping ν: N→ S n and let P∈ R n+ 1 be a point such that (f (x)− P)⋅ ν (x)≠ 0 for any x∈ N. In this study, for the mapping f˜: N→ R n+ 1 defined by: f˜(x)= f (x)−|| f (x)− P|| 2 2 (f (x)− P)⋅ ν (x) ν (x), the following four statements hold.(1) f˜ is a frontal with the Gauss mapping ν˜(x)= f (x)− P|| f (x)− P|| at f˜(x).(2) f˜ is the unique anti-orthotomic of f relative to P.(3) The property that (f˜(x)− P)⋅ ν˜(x)≠ 0 holds for any x∈ N.(4) The equality of|| f˜(x)− P||=|| f˜(x)− f (x)|| holds for any x∈ N. Moreover, three applications of the main result are given. A generalization of the Cahn–Hoffman vector formula is given as the first application. The second application involves clarifying an optical meaning of anti-orthotomics. The third application provides a criterion for determining a front for a given frontal.