Details

We would like to use a bicubic nonuniform B-spline surface to interpolate a given set of matrix nets . We use the progressive-iterative approximation (PIA) algorithm rather than solving the control nets of a B-spline surface by a linear system. There are three main steps for the PIA algorithm.

1. Calculating the knot vectors and in two directions:

where

.

Let .

Then the knot vector is:

,

where

1.2 The knot vector :

In a similar way, the knot vector can be calculated.

.

2. The iterative process:

At the start of the iteration process, let

A bicubic nonuniform B-spline surface can be generated via the control nets by

.

Denote the first adjustment of the control net by

Then let

Again, a bicubic nonuniform B-spline surface can be defined by the control nets :

.

Generally, if the bicubic nonuniform B-spline surface was defined by iterations, denoting the adjustment of the control net as , then

In addition, let

So we could generate the bicubic nonuniform B-spline curve via the control nets .

Ultimately, the surface set can be generated, and H. Lin [1] has proved that this surface iteration format is convergent. Namely,

3. The error is given by

.

Reference

[1] 蔺宏伟, 王国瑾, 董辰世. 用迭代非均匀 B-spline 曲线(曲面)拟合给定点集[J]. 中国科学, 2003, 33(10), pp. 912–923.

H. Lin et al., "Use Iterative Non-Uniform B-Spline Curve (Surface) to Fitting Given Point Set [J]." China Science, 33(10), 2003 pp. 912–923 (in Chinese).