Design of three-order cubic non-uniform B-spline curve with multi-parameters

We present a kind of third-order cubic non-uniform B-spline parametric curve, and give out the relationship between its de Boor control points and piecewise cubic Bézier control points. The curve has a number of characteristics similar to the second non-uniform B-spline curve such as: C¹ continuity on the parameter variables, expression by a linear combination of three de Boor control points on each spline interval, affine invariance, and embracement of the secondary non-uniform B-spline curves. Its blending functions contain several shape parameters, with a clear geometric meaning, which can be used to control the shape or deformation of the curve. Some properties and conditions like convex hull and shape-preserving of the de Boor control polygon, etc., are discussed, and the impact of shape parameter to the curve shape is also described.