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) + ... + c_nB_n(t)

where the B_k are the B-spline functions, and the n-dimensional points c_k are called the control points of C. The polygon defined by the lines joining the control points c_k and c_k+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) + ... + c_nB_n(t)) / (w₁B₁(t) + ... + w_nB_n(t))

where the c_k are the control points, and the real numbers w_k are the weights. It is normal to let all the weights be positive, then the denominator function will be positive everywhere.