A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

提出了一种新方法来确定曲线和曲面生成数据点的参数值（结）。如果四个点形成凸多边形，则在四个相邻数据点的情况下，可以唯一确定二次多项式曲线。当四个数据点不形成凸多边形时，使用具有一个自由度的三次多项式曲线对四个点进行插值，以使插值具有更好的形状，近似由四个数据点形成的多边形。通过最小化三次多项式曲线的三次系数来确定自由度。新方法的优势在于，首先，计算出的结具有二次多项式精度，即，如果从二次多项式曲线中采样数据点，然后使用结来构造二次多项式，它复制原始的二次曲线。其次，新方法是仿射不变的，这很重要，因为大多数参数化方法都不具有此属性。第三，它使用局部方法计算结。实验表明，使用新方法计算出的结所构造的曲线比现有方法具有更好的插值精度。