The u-direction and v-direction:

We know that the number of control points is organized in m + 1 row & n + 1 column. What is the relationship among rows & columns and the u-direction & v-direction? Recall that the v-direction is the curve along with u fixed to a constant. In this case, the definition may be rewritten as the following:

Thus, as v changes, the expression in the parenthesis defines a Bézier curve along with control points p_i,0, p_i,1, . . . , p_i,n. These are precisely the control points on row i. Thus, the v-direction curve runs horizontally and by a same argument we know that the u-direction curve runs vertically. As instance, boundary curves p(0, v) & p(1, v) are in the v-direction and are defined respectively by row 0 and row n, , and boundary curves p(u, 0) & p(u, 1) are in the u-direction and are described by column 0 and column m, respectively.