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开曲面凸性判别条件(一) 被引量:2

ON THE CRITERIONS FOR CONVEX SURFACE(Ⅰ)
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摘要 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.
出处 《数学学报(中文版)》 1979年第4期495-501,共7页 Acta Mathematica Sinica:Chinese Series
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同被引文献15

  • 1唐妥,方逵,李芸.参数曲面保凸的一个定理[J].数学理论与应用,2004,24(3):88-90. 被引量:2
  • 2欧新良,陈松乔.关于凸曲面的几个定义的关系[J].数学理论与应用,2005,25(3):87-89. 被引量:4
  • 3陈发来.双二次Bezier曲面的正性与凸性[J].高校应用数学学报(A辑),1996,11(4):467-476. 被引量:3
  • 4陈翰麟 等.开曲面凸性判别条件(二).数学学报,1979,22(5):579-583.
  • 5陈翰麟 等.开曲面凸性判别条件(三).数学学报,1980,23(2):265-279.
  • 6邝志全.外形设计中检验凸曲面的判别条件.应用数学学报,1983,6(2):205-214.
  • 7Barnhill, R. E. Surfaces in computeraided geometric design: a survey with new results[J]. Computer Aided Geometric Design, 1985, (2) : 1 - 17.
  • 8Kuijt, et al. Convexity Interpolation - Stationary Nonlinear Subdivision and Splines [ EB/OL ] 1998, http://utwente.nl/ fid/1487.
  • 9华宣积,等.Bezier曲面的凸性定理[C].计算几何讨论会论文集,青岛:中国青岛,1982:83-127.
  • 10Chang,G.Z. et al. The convexity of Bemstein polynomial over triangle[J]. Approx. Theory, 1984, (40) : 11 - 28.

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