Abstract: The interpolating nodes are sliced and Hermite interpolating polynomial is constructed respectively in this paper. Algebra polynomials which coefficient of the highest order terms is unit are constructed with the remaining nodes. The rational function expression satisfying interpolating conditions is obtained with linear combination method. By introducing parameters, it proves that the osculatory rational interpolation function is existent and unique. And the error estimation formula is produced. Examples show that the proposed mothod is intuitive, flexible, efficient and easy to facilitate to practical application with smaller volume of the calculation.