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) = c_{1}B_{1}(t) + c_{2}B_{2}(t) + ... + c_{n}B_{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) = (c_{1}B_{1}(t) + ... + c_{n}B_{n}(t)) / (w_{1}B_{1}(t) + ... + w_{n}B_{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.

Published October 19, 2010

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