# Algorithm for Cubic Nonuniform B-Spline Curve Interpolation

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This Demonstration shows how to interpolate a set of points in the plane via a cubic nonuniform B-spline curve and progressive-iterative approximation (PIA) technique. See Details for full implementation details.

Contributed by: Shutao Tang (December 2015)

(Northwestern Polytechnical University, Xi'an, China)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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).

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