Suppose a cubic Bézier polynomial is placed through (u0, v0) and (u3, v3) with guidepoin...

Suppose a cubic Bézier polynomial is placed through (u₀, v₀) and (u₃, v₃) with guidepoints (u₁, v₁) and (u₂, v₂), respectively.

a. Derive the parametric equations for u(t) and v(t) assuming that

b. Let f (i/3) = u_i , for i = 0, 1, 2, 3 and g(i/3) = v_i , for i = 0, 1, 2, 3. Show that the Bernstein polynomial of degree 3 in t for f is u(t) and the Bernstein polynomial of degree three in t for g is v(t). (See Exercise 23 of Section 3.1.)

Reference: Exercise 23 of Section 3.1

The Bernstein polynomial of degree n for f ∈ C[0, 1] is given by

for each x ∈ [0, 1].

a. Find B₃(x) for the functions i. f (x) = x ii. f (x) = 1

b. Show that for each k ≤ n

c. Use part (b) and the fact, from (ii) in part (a), that

d. Use part (c) to estimate the value of n necessary for to hold for all x in [0, 1].