Computing points of curves and surfaces

We have no problems to compute points of curves constructed in previous section. We have counted B₂-spline vertexes of the curve. From equation (2) we can count B-spline control vertexes of the curve and then well-known DeBoor algorithm can be used. Using blending functions we can also construct a tensor-product surface (approximating or interpolating). Surface G₁₂ from definition of Gordon surface is such tensor product. We can write:

$$G_{12}(u,v)= \sum_{j=1}^{n}[\sum_{i=1}^{m}G(u_{i},v_{j})L^{m}_{i}(u)]L^{n}_{j}(v).$$

In brackets we have got n spline curves. We can compute point of each of them in parameter u. Then we get n control points of other curve. If we compute point of this curve in parameter v, we get a point of the tensor-product surface in parameters u and v. So we can compute point of tensor-product surface using DeBoor algorithm n+1 (or m+1) times. Now it is easy to compute a point of Gordon surface. We have already got algorithm to compute G₁₂(u,v). To compute point G₁(u,v) we just use DeBoor Algorithm, because parameter v is fixed and G(u_i,v) are control points of a curve. We compute point G₂(u,v) the same way. Finally we just count G(u,v)=G₁(u,v)+G₂(u,v)-G₁₂(u,v).