Uniform Two-order Trigonometric B-spline Function

The study of Spline Functions is a very important component of the functional approximation. Spline functions has the advantage of polynomial functions which are convenient in operation,meanwhile overcome the weakness of polynomial functions in poor convergence and shock. Therefore the spline functions are used widely in production practice,especially in curves and surfaces modeling.The research about algebraic polynomial spline functions is almost perfectly completed.In recent years,more and more researchers change their focus to trigonometric spline functions,hope they can have advantages of both algebraic polynomial and trigonometric polynomial.This paper tries to contribute a kind of spline function with quadratic trigonometric polynomials and use it to produce spline curves.First we contribute a group of spline basis functions which have properties we call partly supporting and unite division,then we use them to produce spline functions and spline curves in the two or three dimensional space which shape is determined by several control points.We name them two-order trigonometric B-spline curves.This paper contains four parts:First,reviews main results of the functional approximation theory and the relationship between functional approximation and spline functions,relates the development from spline functions to B-spline to spline curves,then reviews main results of trigonometric spline functions.Second,de?nes two-order trigonometric B-spline basis functions in the case of uniform knots. Their construction process and main properties are discussed.The de?nition of basis functions in the case of multiple knots are given too.Then, uses two-order trigonometric B-spline basis functions to construct two-order trigonometric B-spline functions. Two-order trigonometric B-spline functions are essentially piecewise functions which have certain continuity property. This kind of functions are better in convergence than algebraic polynomial.As the application of functional approximation, this paper discusses the interpolation problem with two-order trigonometric B-spline functions,and gives the computational formula.Finally,constructs two-order trigonometric B-spline curves. Their properties,continuity and approximation are discussed.This paper also gives the process to compute two-order trigonometric B-spline interpolation curves and curves tangent to a given polygon.