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normal projection： deterministic and probabilistic algorithms

Frontiers of Mathematics in China 2014年第1期9卷 93-99页

作者： Dongmei LI Jinwang LIU Weijun LIU Department of Mathematics and Computing Sciences Hunan University of Scienceand Technology Xiangtan 411201 China Department of Mathematics and Statistics Central South University Changsha 410083 China

We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.

关键词： normal projection primary decomposition of ideal deterministic algorithm

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Constructing G^2 Continuous Curve on Freeform Surface with normal projection

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Chinese Journal of Aeronautics 2010年第1期23卷 137-144页

作者： Wang Xiaoping,An Luling,Zhou Laishui,Zhang Liyan Jiangsu Key Laboratory of Precision and Macro-manufacturing Technology,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China [a] Jiangsu Key Laboratory of Precision Macro-manufacturing Technology Nanjing University of Aeronautics Astronautics Nanjing 210016 China

This article presents a new method for G2 continuous interpolation of an arbitrary sequence of points on an implicit or parametric surfaee with prescribed tangent direction and curvature vector, respectively, at every point. First, a G2 continuous curve is constructed in three-dimensional space. Then the curve is projected normally onto the given surface. The desired interpolation curve is just the projection curve, which can be obtained by numerieally solving the initialvalue problems for a system of first-order ordinary differential equations in the parametric domain for parametric case or in three-dimensional space for implicit ease. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification so that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface. Numerical experiments demonstrate that it is effective and potentially useful in numerical control （NC） machining, path planning for robotic fibre placement, patterns design on surface and other industrial and research fields.

关键词： Hermite interpolation normal projection freeform surface G2 continuity ordinary differential equations

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