Hermite Interpolation Along The Sphere Surface

Interpolation is an important method for function approximation in computationalmathematics and a numerical calculation of the basic issues. As early as1000years ago,China’s calendar has been documented the applications of the linear interpolation and thequadratic interpolation. Widely used in the computer today, the interpolation method and theMATLAB software program combines to play a broader role, so the interpolation method hasbeen an unprecedented development. After years of research and practice, the elementinterpolation problem has been a growing tendency to improve. With the interpolationproblem widely used in the production and living (such as meteorology, oceanography,pollution and other aspects of the application), multivariate interpolation has become aresearch center of gravity of the interpolation problem. Literature [1] has carried out theresearch of the Lagrange interpolation problem on the spherical surface. In many practicalinterpolation problems, in order to make the interpolation function and the primary functionfit better, not only requires function values equal in the nodes, but also requires tangent to thecorresponding derivative values equal, even requires higher order derivatives equal, this typeof the interpolation problem is called Hermite interpolation. As same as the Lagrangeinterpolation methods, when researches on the multivariate Hermite interpolation problems,the first problem to be solved is the well-posedness of Hermite interpolation, in other words,what kind of geometric structure and characteristics of the posed set of nodes can ensure theexistence and uniqueness of the interpolation function.The introduction of this paper first describes the development and practical applicationof the interpolation problem. The first chapter introduces a reference to bivariate interpolationproblem and the Gasca Maeztumethod of bivariate Hermite interpolation. The secondchapter expounds the theory of the classical Hermite interpolation and generalizedHermite interpolation. On the basis of previous research, the third chapter gives a Hermiteinterpolation method in ternary polynomial space, namely, the sphere superposition method.And researches its geometric structure and basic characteristics. The fourth chapter gives thebasic concepts of the Hermite interpolation along the spatial algebraic curve. Given thebasic concepts of interpolation along the space algebraic curve, and has been related to theoryand method of construction. Concluded with the method, solves the structural problems posedfunctional groups along the spherical Hermite interpolation, and gives a specific instance.