Explicit Expression Of Hyperbolic B-spline Basis

Hyperbolic polynomial B-spline curve based on space span{1,t,t2,…,tn-3y,sinh t,cosh t} has many exactly the same properties as polynomial B-spline curve, and it also can represent helix, cycloid, catenary, etc., which have been popularly used in many design but can not been represented by polynomial B-spline. In this paper, the explicit expression of hyperbolic polynomial B-spline basis and computation of its area is discussed both on simple knot sequence and multiple knot sequence. As a special case of simple knot sequence, we get the uniform explicit expression of hyperbolic polynomial B-spline basis which can be further simplified and its transition matrix to hyperbolic function and Power-Basis. The calculation of the transition matrix is also given.In the case of simple knot sequence, by using some generating functions which are firstly constructed by certain original functions, we find out a basis to a space of finite-support spline functions in term of determinant. Considering the dimension of this space, the relevant hyperbolic polynomial B-spline Basis can be explicitly presented except for a constant coefficient. After investigating the properties of determinant we just defined, it can be expressed in a much more simplified form Hyperbolic-Vandermonde determinant. Finally, by using the properties of Hyperbolic-Vandermonde determinant, the constant coefficient can be determined, so the explicit expression of hyperbolic polynomial B-spline Basis is obtained. After all, we also get an explicit expression of its area.As a special case of simple knot sequence, the explicit expression of uniform hyperbolic polynomial B-spline Basis can be easily obtained by applying the acquired results of simple knot sequence. Its expression can be further simplified. And its transition matrix to hyperbolic function and Power-Basis is also given, by using the explicit expression. The calculation of the transition matrix can be determined after investigating the further properties of Hyperbolic-Vandermonde determinant in the case of uniform knot sequence. In the case of multiple knot sequence, we also first construct some generating functions like the case of simple knot sequence, by which we can found a basis to a space of finite-support spline functions in term of determinant. Due to the dimension and continuity of this space, the relevant hyperbolic polynomial B-spline Basis can be explicitly presented by the determinant functions except for a constant coefficient. By similar analysis as simple knot sequence, the determinant can be transformed to a simplified form Generalized-Hyperbolic-Vandermonde determinant. After investigating the properties of the determinant, wo can obtain the constant coefficient. Finally, we also give the formula of its area.