The B-spline provides a free control of the parametric polynomial, but it cannot deal with some transcendent curves. Therefore, lots of research works present all kinds of new models. However, these models can encompass neither high order curves nor conical solenoids and involutes of the circle. Thus a new kind of splines generated over the space spanned by cost,sint,tcost,tsint,1,t,t\+2,…,t\+\k-5\(k≥5) are presented, which are called non-uniform algebraic-trigonometric splines of order k with regard to the given node sequence T. The algebraic-trigonometric splines have most of their properties similar to that of the B-splines in the polynomial space. After inserting new nodes to the node sequence, the sequence of the control polygons converts to the spline. Apparently, algebraic-trigonometric splines can encompass conical solenoids, involutes of circles and some other transcendent curves.