A spline curve is a parametric curve in n-dimensional space for which the coordinate functions are spline functions with the same degree and knot vector.
A polynomial spline curve is a spline curve where each coordinate function is a polynomial spline function from the same spline space. It can be written on the form
C(t) = c1B1(t) + c2B2(t) + ... + cnBn(t)
where the Bk are the B-spline functions, and the n-dimensional points ck are called the control points of C. The polygon defined by the lines joining the control points ck and ck+1 is called the control net of C.
A NURBS curve is a spline curve where the coordinate functions are rational spline functions with the same denominator. It can be written as
C(t) = (c1B1(t) + ... + cnBn(t)) / (w1B1(t) + ... + wnBn(t))
where the ck are the control points, and the real numbers wk are the weights. It is normal to let all the weights be positive, then the denominator function will be positive everywhere.