Unified and extended form of three types of splines

The three types refer to polynomial, trigonometric and hyperbolic splines. In this paper, we unify and extend them by a new kind of spline (UE-spline for short) defined over the space {cosωt,sinωt,1,t,…,tl,…}{cosωt,sinωt,1,t,…,tl,…}, where l   is an arbitrary nonnegative integer. ωω is a frequency sequence {ωi=αi}-∞+∞,αi∈R. Existing splines, such as usual polynomial B-splines, CB-splines, HB-splines, NUAT splines, AH splines, FB-splines and the third form FB-splines etc., are all special cases of UE-splines. UE-splines inherit most properties of usual polynomial B-splines and enjoy some other advantageous properties for modelling. They can exactly represent classical conics, the catenary, the helix, and even the eight curve, a kind of snake-like curves etc.