Explict Expression Of NUAT B-spline Basis And Its Properties

NUAT B-spline curve based on spacespan {1,t,...,t^k-3,cost,sin t} has exactly the same properties as B-spline curve,and it also can represent circular arc,cycloid,helix etc.,which have been frequently used in many CAD/CAM systems.In this paper,we discuss the explicit expression of NUAT B-spline basis and computation of its area both on simple knot sequence and multiple knot sequence respectively.In the case of simple knot sequence,by using a firstly constructed truncated function, we find out a basis of a space of finite-support spline functions in the term of determinant. Therefore,the relevant NUAT B-spline Basis can be explicitly presented except a coefficient.After investigating the property of determinant,NUAT B-spline Basis can be expressed in a much more beautiful form-Vandermonde-like determinant.Finally, through the properties of Vandermonde-like determinant,we can determine the coefficient and obtain the accurate explicit expression of NUAT B-spline Basis.In addition, we also discover an easy algorithm for the computation of its area.Applying the acquired results of simple knot sequence to special case of uniform sequence,we get the explicit expression of UAT B-spline Basis.And we also obtain the transition matrix into Power-Basis using the explicit expression of uniform case.After delving into the further property of Vandermonde-like determinant,we gave an efficient algorithm for computing thetransition matrix.When given a complex knot sequence,first we also construct a determinant of truncated function first.Due to its continuity,we assert it also consist a basis of a space of finite-support spline functions.Using similar analysis as simple knot sequence,the determinant can be simplified into an easy form- so-called generalized Vandermonde-like determinant.We investigated the determinant,too,which eventually helps us determine the coefficient.And we also obtain an explicit expression of its area.