Some Investigations in the Geometry of Curve and Surface Elements
Some Investigations in the Geometry of Curve and Surface Elements
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曲线和曲面单元几何的一些研究
DOI：
10.1112/plms/s3-4.1.24
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发表时间：
1954
影响因子：
1.8
通讯作者：
J. Semple
中科院分区：
文献类型：
--
作者：
J. Semple
一个我^。，或k阶曲线元素，在代数几何中可以最方便地描述为一个简单的序列O {= 0), Olt…，投影空间Sr或代数流形Vr中k-\-1个连续点的Ok。Ek的原点0是Sr的一个实际点（或Vr的一个简单点），对于i^ 1，每个O{都是一个虚拟点，近似于（在）它的直接前身Oi_v的第一个邻域。如果Ek包含在以0为原点的线性曲线分支中，则它是线性的；否则它是单数。对于给定的k， Ek的最简单的分类是基于它们的邻近格式，Ek的邻近格式表示对于每个i^ 1， Oi（包括0f _2）的前导，其中Oi是近似值，f在空间变换上是不变的，在0是解析的。在这个分类中，线性Ek构成了一个维数为（k-\-\） r-tc的类，更一般地说，任何允许的邻近方案都表征了一个维数为（k-\-l） r-k ‘的Ek的类，其中k’^ k是该方案中包含的邻近条件的个数。随着k的增加，不同类别的Ek（即可容许接近方案）的数量迅速增加。因此，对于平面（或曲面）上的Ek， E1形成单一类，E2有两类（线性E2和倒轴E2）， E3有五类>等。
AN JE^., or curvilinear element of order k, in algebraic geometry can be described most conveniently as a simple sequence O {= O0), Olt..., Ok of k-\-1 consecutive points in a projective space Sr or on an algebraic manifold Vr. The origin 0 of the Ek is an actual point of Sr (or a simple point of Vr), and for i^ 1 each O {is a fictitious point proximate to (in the first neighbourhood of) its immediate predecessor Oi_v The Ek is linear if it is contained in a linear curve branch with 0 as origin; otherwise it is singular. The simplest classification of Ek for given k is that based on their proximity schemes, the proximity scheme of an Ek being that which indicates for each i^ 1 the predecessors of Oi (including 0f _2) to which Oi is proximate, f and being invariant over space transformations which are analytic at 0. In this classification the linear Ek constitute one class of dimension (k-\-\) r—tc\and, more generally, any admissible proximity scheme characterizes a class of Ek of dimension (k-\-l) r—k', where k'^ k is the number of proximity conditions contained in the scheme. As k increases, the number of distinct classes of Ek, ie of admissible proximity schemes, increases rapidly. Thus, for Ek in the plane (or on a surface), the E1 form a single class, there are two classes of E2 (the linear E2 and the cuspidal E2), there are five classes of E3> and so on.