Alpha-spline: a C2 continuous spline with weights and tension control

The β-spline provides bins and tension control facilities for creating geometrically continuous curves and surfaces. Although geometric continuity is a more appropriate geometric measurement of smoothness than parametric continuity, parametric continuity is still necessary in some applications. The paper proposes a new C² continuous spline scheme called the α-spline which provides weights and tension control. The new scheme is based on blending a sequence of singular reparametrized line segments with a piecewise NURBS curve. The idea is extended to produce α-spline surfaces