Several Properties And Applications Of B-spline With Multiple Knots

In this paper ,firstly,we introduce the concept of B-spline with multiple knots on the basis of difference quotient with multiple knots and summarize three kinds of definition forms of B-spline with multiple knots on the basis of past literatures :M_i,n(x),N_i,n(x),Ω_i,n(x) The past literatures have not given the general explicit expression formula of B-spline with multiple knots ,so we will provide the general explicit expression formula of B-spline with multiple knots according to the launched form of difference quotient with multiple knots , and then give an example. Secondly,we summarize several main properties of three forms of B-spline with multiple knots: recurrence relations, positive and local support property, N_i,n(x) form apartition of unity, standard property of Ω_i,n(x), the continuity of B-spline with multiple knots and fundus property;then we summarize several main properties of the B-spline vision with multiple knots, 2 order 3 order and 4order B-spline visions with multiple knots are discussed explicitly. Finally,we discuss the properties and applications of B-spline curves and rational B-spline curves with multiple knots, summarize the impact of multiple knots on B-spline curves and rational B-spline curves and give the explicit provements, analyse several twice and cubic B-spline curves with multiple knots explicitly.