11453

# Algorithm for Bicubic Nonuniform B-Spline Surface Interpolation

This Demonstration shows how to interpolate a set of matrix nets via a bicubic nonuniform B-spline surface and progressive-iterative approximation (PIA) technique. See Details for full implementation details.

### 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
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).

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.