Algoritmus( de Casteljau, pro trojúhelníkové Bezierovy segmenty) Vstup Bezierovy body b_{i,j,k},i, j, k ≥ 0, i + j + k = n, parametrická hodnota (u*,v*). Výstup: Bod křivky b(u*,v*) Postup: Begin FOR i:=0 TO n DO b_{i}^{0}:=b_{i}; FOR j:=1 TO n-i DO Begin k:=n-i-j; b_{i,j,k}^{0}:=b_{i,j,k} END; FOR l:=1 TO n DO FOR i:=1 TO n-l DO FOR j:=1 TO n-l-i DO Begin k:=n-l-i-j; b_{i,j,k}^{l}:= u*b_{i+1,j,k}^{l−1} + v*b_{i,j+1,k}^{l−1} +(1-u*-v*)b_{i,j,k+1}^{l−1} END b(u*,v*):= b_{n}^{0,0,0} END. Pro pomocné body b_{i,j,k}^{l} platí: b_{i,j,k}^{l} =

∑ ∑ ∑_{i^{∗}+j^{∗}+k^{∗}=l} b_{i+i^{∗},j+j^{∗},k+k^{∗}}^{0} B_{i^{∗},j^{∗},k^{∗}}^{l}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})

. Důkaz indukcí vzhledem k l.

1. l = 0: b_{i,j,k}^{0}=b_{i,j,k}^{0} \frac{0!}{0!0!0!}(u^{∗})^{0}(v^{∗})^{0}(1 − u^{∗} − v^{∗})^{0} = b_{i,j,k}^{0}B_{0,0,0}^{0}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗}). i^{∗} + j^{∗} + k^{∗} = 0

2. předpokládejme, že vzorec platí pro (l-1) platí:

b_{i,j,k}^{l−1} = ∑_{i^{∗}+j^{∗}+k^{∗}=l−1} b_{i+i^{∗},j+j^{∗},k+k^{∗}}^{0} B_{i^{∗},j^{∗},k^{∗}}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗}).

b_{i,j,k}^{l} = u^{∗}b_{i+1,j,k}^{l−1} + v^{∗}b_{i,j+1,k}^{l−1} + (1 − u^{∗} − v^{∗})b_{i,j,k+1}^{l−1} =

= u^{∗}( ∑_{i^{∗}+j^{∗}+k^{∗}=l−1} b_{i+1+i^{∗},j+j^{∗},k+k^{∗}}^{0} B_{i^{∗},j^{∗},k^{∗}}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})

v^{∗}( ∑_{i^{∗}+j^{∗}+k^{∗}=l−1} b_{i+i^{∗},j+1+j^{∗},k+k^{∗}}^{0} B_{i^{∗},j^{∗},k^{∗}}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})+

+ (1 − u^{∗} − v^{∗})( ∑_{i^{∗}+j^{∗}+k^{∗}=l−1} b_{i+i^{∗},j+j^{∗},k+1+k^{∗}}^{0} B_{i^{∗},j^{∗},k^{∗}}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗}) =

= ∑_{i′+j′+k′=l} u^{∗}b_{i+i′,j+j′,k+k′}^{0} B_{i′−1,j′,k′}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})+

+ v^{∗}b_{i+i′,j+j′,k+k′}^{0} B_{i′,j′−1,k′}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})+

+ (1 − u^{∗} − v^{∗})b_{i+i′,j+j′,k+k′}^{0} B_{i′,j′,k′−1}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} u^{∗}B_{i′−1,j′,k′}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})+

+ v^{∗}B_{i′,j′−1,k′}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗}) + (1 − u^{∗} − v^{∗})B_{i′,j′,k′−1}^{l−1}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} [u^{∗} \frac{(l − 1)!}{(i′ − 1)!j′!k′!}(u^{∗})^{i′−1}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′}+

+ v^{∗} \frac{(l − 1)!}{i′!(j′ − 1)!k′!}(u^{∗})^{i′}(v^{∗})^{j′−1}(1 − u^{∗} − v^{∗})^{k′}+

+ (1 − u^{∗} − v^{∗})\frac{(l − 1)!}{i′!j′!(k′ − 1)!}(u^{∗})^{i′}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′−1}

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} (u^{∗})^{i′}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′} [\frac{(l − 1)!}{(i′ − 1)!j′!k′!} + \frac{(l − 1)!}{i′!(j′ − 1)!k′!} + \frac{(l − 1)!}{i′!j′!(k′ − 1)!}]

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} (u^{∗})^{i′}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′} \frac{(l − 1)!}{(i′ − 1)!(j′ − 1)!(k′ − 1)!}[\frac{1}{j′k′} + \frac{1}{i′k′} + \frac{1}{i′j′} ]

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} (u^{∗})^{i′}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′} \frac{(l − 1)!}{(i′ − 1)!(j′ − 1)!(k′ − 1)!}\frac{i′ + j′ + k′}{i′j′k′}

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} (u^{∗})^{i′}(v^{∗})^{j′}(1 − u^{∗} − v^{∗})^{k′} \frac{l!}{i′!j′!k′!}

= ∑_{i′+j′+k′=l} b_{i+i′,j+j′,k+k′}^{0} B_{i′,j′,k′}^{l}(u^{∗}, v^{∗}, 1 − u^{∗} − v^{∗})