Lagrange Interpolation On The Conical Surface

Multivariate polynomial interpolation always is an important research subject in computational mathematics. One of the basic questions of the research about the theory and method of multivariate polynomial interpolation is the well-posedness of multivariate interpolation. So far, the research about this topic is mainly divided into two categories here and abroad. The first category is the space of the given interpolation polynomial, which is to construct the corresponding properly posed set of nodes of polynomial space. The second category is the given posed set of nodes for interpolation, which is to construct the corresponding properly interpolation polynomial space and require the index of polynomial space as small as possible. For the first category, currently, the discuss results of the interpolation in the whole space and the interpolation of normal algebraic manifold which follows the space are not particular relatively. And the result about the specific manifold which has an important practical value is less. Conical surface is an important conic. It has an important application in the engineer design. For example, the appearances of many mechanical components and structures use conical surface. The protective housing of the Soviet Union’s second man-made satellite rocket also used the conical surface. Therefore, the research of Lagrange interpolation on conical surface has an important application value.In this paper, we apply the superposition interpolation process and the factorization methods to discussing the well-posedness of Lagrange interpolation on the conical surface. Several ways to construct the properly posed set of nodes for Lagrange interpolation on the conical surface are proposed by choosing proper points along generatrices, conic curve, two and three circles and a number of circles on the conical surface.The paper is made up of four chapters :In chapter one, we introduce the research background and the structure of this paper.In chapter two, firstly, basing on the method of adding properly posed set of nodes on conical surface, the interpolation polynomial of degree n + 1 is configured through the method of adding generating line. Some conclusions of configuring the interpolation polynomial of degree n + 1 is given through the method of adding plane curve. Secondly, the sufficient and necessary condition of properly posed set of nodes for interpolation of degree n along conical surface and on conical surface set of conical curve is given.In the chapter three, referring to configure the properly posed set of nodes for interpolation polynomials of degree n+λ on conical surface and basing on the properly posed set of nodes for interpolation polynomials of degree n along conical surface is known, the method of configuring the properly posed set of nodes for interpolation polynomials of any degree along conical surface is given according to the method of adding generating line and irreducible conic. It is worth to point out that adding generating line and irreducible conic can be used alternately. At the same time, this article configures these two conditions of interpolation form. And referring to the condition of adding irreducible conic which is a circle, the detail configure form of interpolation polynomial is given.In the chapter four, enlightened by the configuration method of properly posed set of nodes on spherical surface and utilizing superposition interpolation method and factorization method for interpolation, the method of configuring properly posed set of nodes for interpolation polynomial space of degree n + λ along conical surface by adding set of conical curve is given in the article. Especially the method of adding double circumferences and three circumferences to configure properly posed set of nodes for interpolation of degree n is given. In these situations, double circumferences can be the symmetrical circumferences about 0xy plane curve; it also can be any twocircumferences of sampling point by equal distance. And three circumferences is used to configure that the degree at most n+3 than double circumferences of degree at most n + 2, and similarly point must be sampled by equal distance. In order to expand the result to more common situations, basing on above methods, we can select the plane curve crossing with set of circumferences to configure properly posed set of nodes for interpolation polynomial space of any degree.In the chapter five, we summarizes the paper comprehensively and puts forward some expectations. Finally, we give an example to explain the difference between Lagrange interpolation on sphere and spherical surface to point out that the methods on conical surface aren’t the simple application of the methods on spherical surface.