开曲面凸性判别条件(一)

ON THE CRITERIONS FOR CONVEX SURFACE(Ⅰ)

给定一曲面片,问:在什么条件下它是凸的?如果一曲面称为“凸曲面”是以它能安装在某一个凸体的表面上作为定义的话,那么,进一步要问:它要满足什么样的条件才能安装在某个凸体的表面上呢? 如果曲面π是封闭曲面,问题早已解决,即封闭曲面π是凸曲面的充要条件是:π对其所包围的有界域D而言是点点局部凸的。

First, we formulate the definition of local convexity. Let π and π be a surface and its boundary respectively. For a point P ∈ int π=π\π, if there is a small neighborhood about P in the topology of π such that it can be considered as part of a surface of some convex body, then we say π is locally convex at point P. If every point of int π is locally convex, we say that π has the property of local convexity. Evidently, if π processes the property of local convexity only, it may not be a convex surface, We are going to discuss the conditions under which π will be a convex surface. In this paper we stipulate that π is a n-connected surface, it means that πeorresponds topologically to a n-conneeted domain Ω in E_2 plane, where the boundary of π consists of n-closed Jordan curves. The main result of this paper is the followingTheorem 1. Let π be a n-connected surface belonging to C^o, λ=π (consisting of finite closed curves). If1° π is locally convex;2° For every point x∈γ, there is a plane S_x, x ∈S_x S_x is an entirely supportingplant for π, i.e. the points of π lie on S_x or in one half of the space divided by S_x.Then π is a convex surface. Conversely,if π is a convex surface then 1°and 2° arevalid.