Triangular Bézier patch ($S(u,v,w)={\sum }_{i+j+k=n}\frac{n!}{i!j!k!}{u}^i{v}^j{w}^k{P}_{ijk},u+v+w=1$)

For nonnegative barycentric parameters, these total-degree Bernstein basis functions define a surface over a triangle. The triangular control net has $(n+1)(n+2)/2$ points. Its index constraint couples the three exponents, so it is not a tensor product of two independent univariate bases. The multinomial theorem gives nonnegativity and partition of unity.