Lemma

$$\binom{n-1}{i-1} + \binom{n-1}{i} = \binom{n}{i}$$

$\binom{n}{i}$는 binomial coefficient이다.

Proof

$$\begin{aligned} \binom{n-1}{i-1} + \binom{n-1}{i} &= \frac{(n-1)!}{(i-1)!(n-i)!}+\frac{(n-1)!}{i!(n-i-1)!} \\\\ &=(n-1)!\frac{(n-i)+i}{i!(n-i)!} \\\\ &=\frac{n!}{i!(n-i)!} \\\\ &=\binom{n}{i} \quad \square \end{aligned}$$

Explicit form of Bézier curve

다음 두 식은 동치이다.

$$\begin{aligned} B_{P_0}^{(0)}(t) &= P_0,\ t\in[0,1] \\\\ B_{P_0,P_1,...,P_n}^{(n)}(t) &= (1-t)B_{P_0,...,P_{n-1}}(t)+tB_{P_1,...,P_n}(t),\ t\in[0,1] \qquad if \quad n > 0 \qquad -(1) \end{aligned}$$ $$B_{P_0,...,P_n}^{(n)}(t) = \sum^n_{i=0} \binom{n}{i}(1-t)^{n-i}t^iP_i \qquad -(2)$$

Proof

간단하게 $B_{P_0,...,P_n}(t)$을 $B_{0...n}(t)$이라고 쓰자.

By induction.

i) When n = 0: trivial

ii) When n > 0:

$$\begin{aligned} B_{0...n}^{(n)}(t) &= (1-t)B_{0...(n-1)}+tB_{1...n} \\\\ &= [(1-t)\sum^{n-1}_{i=0} \binom{n-1}{i} (1-t)^{n-1-i}t^iP_i]+[t\sum^{n-1}_{i=0} \binom{n-1}{i} (1-t)^{n-1-i}t^iP_{i+1}] \\\\ &= \sum^{n-1}_{i=0} \binom{n-1}{i} (1-t)^{n-i}t^iP_i+\sum^{n}_{i=1} \binom{n-1}{i-1} (1-t)^{n-i}t^{i}P_{i} \\\\ &= \binom{n-1}{0}(1-t)^nP_0+\sum^{n-1}_{i=1} \left[ \binom{n-1}{i}+\binom{n-1}{i-1} \right] (1-t)^{n-i}t^iP_i+\binom{n-1}{n-1}t^nP_n \\\\ &= (1-t)^nP_0+\sum^{n-1}_{i=1}\binom{n}{i}(1-t)^{n-i}t^iP_i+t^nP_n \qquad by\ above\ lemma\\\\ &= \sum^{n}_{i=0}\binom{n}{i}(1-t)^{n-i}t^iP_i \qquad \square \end{aligned}$$