# Cubic Uniform BSpline surface interpolation

I want to understand cubic BSpline surface( very hard for me to figure out). I prefer matrix form which presented here. Equation 4.12 in page 33, describes how data point should be presented U*M*D*M'*V' and since I am looking for uniform presentation of Bspline just I assume u and v values are [ 0, .1,.2,...,.9,1]; but in page 32. figure 4.3 it shows 3D model which has (x,y and z) values. I am confused now, based on equation we have D as a set of control points (4x4). but based on that figure each point should have 3 values( x,y,z). is anybody can help me based on that figure and numerical example?

I am confused more than 3 weeks and still I cannot figure out cubic uniform BSpline interpolation. I need a numerical example based on matrix form step by step if anyone can help me.

Thanks a lot

The $S(u,v)$ is a vector function which has 3 components: $x(u,v)$, $y(u,v)$ and $z(u,v)$. So, you can consider equation 4.12 as 3 equatons:

$x(u,v)=UMD_xM^TV^T$
$y(u,v)=UMD_yM^TV^T$
$z(u,v)=UMD_zM^TV^T$

where $D_x$, $D_y$ and $D_z$ are 4x4 matrices consist of the $x$, $y$ and $z$ components from the 3D control points.

Hopefully this way, it is easier to understand the equation.

• So what is Dx for example? it should be 16 points consist of x,y,z components. In that pdf, it mentioned with i and j indexes. Can you give me an example? Thanks – Ehsan Jun 18 '15 at 4:32
• i mean why there is not k index for Dx if it has x,y and z components? – Ehsan Jun 18 '15 at 4:56
• For a bicubic B-spline, each surface patch is controlled by 4x4 control points that form a grid (see Fig 4.3 in the pdf file of your link). The indices i and j are used to denote individual control points. Matrix Dx is composed of x values from those 4x4 control points. Therefore, there is no k index. Similarly, matrix Dy is composed of y values from those 4x4 control points. – fang Jun 18 '15 at 6:46