Explicit Representation And Extension Of Definition Interval For Algebraic Trigonometric Twice-Combined Base

The limitations caused by NURBS curves and surfaces motivate the extensive research on the new representation of curves and surfaces in different functional spaces,so that they can represent transcending curves and surfaces.Some of them such as the cycloid,the conical solenoid and the involutes of circle widely used in mechanical areas are defined in span｛sin t,cos t,t sin t,t cost,t^n-4,…,1｝,so it is important to study the blending model in the space.The extension of the defined intervals for Twice-AT Bernstein base and uniform Twice-B-splines base is very important.The reason lies in two folds:first,we can spread the length of a curve,for example,a circle or a periods of the cycloid,the conical solenoid or the involutes of circle,second,we can modify a curve by shape parameters.The extension of intervals of shape parameters means that we can modify a curve more widely.So in this paper,we discuss these problems,In the first part,Explicit representations of Twice-AT Bernstein base and uniform Twice-AT-B spline base are given.As to Twice-AT-Bernstein bases,we extendαfrom[0,π]to[0,2π].Then we get the transfer matrices from(sin t,cos t,t sin t,t cost,t^n-4,…,t,1) to｛u_i,n｝_i=0ⁿ and take n=5 as an example,the advantage of the extending of the defined intervals is shown.So as to uniform Twice-AT-spline base,we give a solution to the extension of the defined intervals.