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...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.展开更多
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...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.展开更多
基金National Natural Science Foundation of China(60673026,50875130,50805075 and 50875126)
文摘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.
基金Acknowledgements the article and made The authors are thankful to numerous helpful suggestions the referees for their carefully reading This work was partially supported by the National Natural Science Foundation of China （Nos. 11071062, 11271208）, the Natural Science Foundation of Hunan Province （14JJ6027）, and the Scientific Research ~nd of Education Department of Hunan Province （10A033, 12C0130）.
文摘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.
