Suppose a cubic Bézier polynomial is placed through (u0, v0) and (u3, v3) with guide points (u1, v1) and (u2, v2), respectively.
a. Derive the parametric equations for u(t) and v(t) assuming that
u(0) = u0, u(1) = u3, u'(0) = u1 − u0, u'(1) = u3 − u2
and
v(0) = v0, v(1) = v3, v'(0) = v1 − v0, v'(1) = v3 − v2.
b. Let f (i/3) = ui , for i = 0, 1, 2, 3 and g(i/3) = vi , 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).