Details

There are three main steps in the PIA algorithm.

1. Compute the knot vector via the chord-length parametrization

where

.

Then define the knot vector

,

where

2. Do the iteration.

At the beginning of the iteration, let

First, generate a cubic nonuniform B-spline curve by the control points :

.

The first adjustment of the control point is

,

then let

Again, this gives a cubic nonuniform B-spline curve with the control points , that is

.

In general, if the cubic nonuniform B-spline results from iterating times, let the adjustment of the control point be

.

Also, let

This gives the cubic nonuniform B-spline curve via the control points .

Finally, the set of curves is defined. Lin [1] has proved that these curves converge to the points, namely that

,

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).