Triangular Bezier surfaces

We study Bezier surfaces defined over triangular domains. We have seen how spaces of bivariate polynomials can be constructed as tensor-products of the univariate ones. An alternative choice of polynomial space is, for each n ≥ 0, the space of polynomials of the form p(x, y) = 0≤i+j≤n a i,j x i y j , where we understand that i ≥ 0 and j ≥ 0 in the summation. As in the univariate case, we denote this space by π n. Such a polynomial has degree ≤ n, its degree (sometimes called the total degree) being the largest value of i + j over all non-zero a i,j in the summation. The monomials x i y j in the sum are linearly independent and therefore form a basis of π n. Since the number of such polynomials is (n + 1) + n + · · · + 1 = n + 2 2 , this is also the dimension of π n. For example, the monomial basis of π 2 is which has six elements.