Projections of Surfaces with Boundary
Projections of Surfaces with Boundary
复制标题
带边界的曲面的投影
DOI：
10.1112/plms/s3-60.2.392
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发表时间：
1990
影响因子：
1.8
通讯作者：
P. Giblin
中科院分区：
文献类型：
--
作者：
John W. Bruce;P. Giblin
本文研究了有边界曲面投影的奇异性。我们的工作是由下面的几何问题的动机。给定一个有边界的“一般”表面，从各个方向观察这个表面，描述表面和边界的表观轮廓的局部外观（换句话说，描述人们实际看到的东西）。发生在内部点的表观轮廓的奇异性已经被分类，例如在[2，11]中（另见[3，8]），因此我们将只关心边界点处投影的性质。因此，我们需要对从平面到平面的映射芽进行分类，其源包含一条通过坐标变化而保持不变的可区分的线，我们有两种分类技术，我们在§§ 1和§ 2中介绍。第一种方法遵循标准的方法，根据所涉及的映射的程度进行归纳分类，但使用了[6]中的新技术，以及最近对计算确定性程度的方法的改进。第二种方法的目的是减少我们的地图芽的分类的解析曲线（即图像的边界连同明显的轮廓）的芽的分类。由于平面曲线芽的分类比原来的问题容易得多（人们必须处理^"-等价而不是^-等价的变体），这是相当富有成效的，并且比相当盲目的算法归纳方法更具几何性。在第3节中，我们实际上进行了分类，把必要的确定性计算留到第4节。最后一节解释的结果几何，指出其后果的几何表面的边界，并包含图片的轮廓和边界在各种有趣的情况下。我们注意到，我们的工作可以被看作是映射芽的分类：[R2，0-» R2，0，它们关于源中的反射（x，y）>-+（x，-y）是等变的。详见下文（定理1.2之后）。
In this paper we consider singularities of projections of surfaces with boundary. Our work is motivated by the following geometric question. Given a'general'surface with boundary and viewing this surface from every direction describe the local appearance of the apparent contour of the surface and the boundary (in other words describe what one actually sees). The singularities of apparent contours occurring at interior points have already been classified, for example in [2, 11](see also [3, 8]), so we shall be concerned solely with the nature of the projections at boundary points. We need consequently to classify map germs from the plane to the plane with the source containing a distinguished line which is preserved by co-ordinate changes.We have two classification techniques which we introduce in §§ 1 and 2. The first follows the standard approach of classifying inductively on the degree of the mappings involved, but uses a new technique, from [6], as well as recent improvements in methods for calculating degrees of determinacy. The second method aims to reduce the classification of our map germs to that of classifying germs of analytic curves (namely the image of the boundary together with the apparent contour). Since the classification of germs of plane curves is a much easier problem than the original (one has to deal with^"-equivalence instead of a variant of^-equivalence) this is quite fruitful, and has the advantage of being more geometric than the rather blind algorithmic inductive approach. In § 3 we actually carry out our classification, leaving the requisite determinacy calculations until § 4. The final section interprets the results geometrically, points out their consequences for the geometry of surfaces with boundary, and contains pictures of the contour and boundary in various interesting cases. We note that our work can be viewed as a classification of map germs/:[R2, 0—» R2, 0 which are equivariant with respect to a reflection (x, y)>-+(x,—y) in the source. See below (after Theorem 1.2) for details.