Tensor product surfaces:
The Bezier surfaces may also be illustrated in Tensor product form. If we arrange the control points into a matrix of m + 1 rows and n + 1 columns
![2269_Tensor product surfaces.png](https://www.expertsmind.com/CMSImages/2269_Tensor%20product%20surfaces.png)
and the basic functions of the v-direction Bézier curve in a column matrix of n + 1 rows
![2064_Tensor product surfaces1.png](https://www.expertsmind.com/CMSImages/2064_Tensor%20product%20surfaces1.png)
then the result of the parenthesis may be rewritten as a matrix product :
![1452_Tensor product surfaces2.png](https://www.expertsmind.com/CMSImages/1452_Tensor%20product%20surfaces2.png)
In this matrix form, the consequence is a column matrix of m + 1 entry. If further we make the basic functions of the u-direction Bézier curve in a row matrix of m + 1 entries :
[Bm,0 (u), Bm,1 (u),... , Bm,m (u)]
The equation of Bézier curve becomes the product of the three matrices as follows:
![309_Tensor product surfaces3.png](https://www.expertsmind.com/CMSImages/309_Tensor%20product%20surfaces3.png)
Thus, we have successfully converted the definition of a Bézier surface to a product form. As the entities of the control points matrix are matrices (that means each control point may be considered as a vector and therefore a matrix), it is the tensor product form in mathematics. As a consequence, Bézier surfaces are tensor product surfaces.