Interpolating curves with gradual changes in curvature

Let R ( t ) = [ x ( t ) y(t) z(t)] T denote a parametric curve where the parameter t represents the normalized arc length. This curve is to meet a set of interpolating conditions at given knots ti in the interval [0, 1]. It is well known that this problem may be solved by cubic spline interpolation, achieving continuous curvature, and that the cubic spline, interpolant has the property that it minimizes the norm [Holladay '57]: