CAUSTICS OF SURFACES IN THE MINKOWSKI 3-SPACE
CAUSTICS OF SURFACES IN THE MINKOWSKI 3-SPACE
复制标题
MINKOWSKI 3 空间中的表面焦散
DOI：
10.1093/qmath/haq030
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发表时间：
2012
影响因子：
0.7
通讯作者：
F. Tari
中科院分区：
文献类型：
--
作者：
F. Tari
欧几里德 3 空间中光滑表面的焦散是表面法线光线的包络线。它也是表面曲率中心（焦点）的轨迹。这就是为什么它也被称为表面的焦点集。它具有拉格朗日奇点，其通用模型在 [1] 中给出（见图 2）。本文的目的是定义嵌入 Minkowski 3 空间中的光滑表面 M 的焦散 C(M) 并研究其几何形状。我们用 LD 表示 M 上度量退化的点的轨迹。如果 M 是闭曲面，则其 LD 不为空。在 LD 上的一点处，M 的“法线”线是类光线并且与 M 相切。此外，M 的焦点集未在 LD 上的点处定义。我们将 M 的焦散定义为 M 上距离平方函数族的分叉集。那么C(M)与M\LD的焦点集重合，并提供焦点集到LD的扩展。我们研究 C(M) 上度量的局部行为。
The caustic of a smooth surface in the Euclidean 3-space is the envelope of the normal rays to the surface. It is also the locus of the centres of curvature (the focal points) of the surface. This is why it is also referred to as the focal set of the surface. It has Lagrangian singularities and its generic models are given in [1] (see Figure 2). The aim of this paper is to define the caustic C(M) of a smooth surface M embedded in the Minkowski 3-space and to study its geometry. We denote by the LD the locus of points on M where the metric is degenerate. If M is a closed surface then its LD is not empty. At a point on the LD the “normal” line to M is lightlike and is tangent to M . Also, the focal set of M is not defined at points on the LD. We define the caustic of M as the bifurcation set of the family of distance squared functions on M . Then C(M) coincides with the focal set of M \ LD and provides an extension of the focal set to the LD. We study the local behaviour of the metric on C(M).