Question

Approximate sin t by a four point Bezier curve over the range 0<=t<=1 such that the derivates at the end are exact.

Answer 1

In a Bézier curve, xx and yy are polynomials in the parameter tt. Note that you can't just have "a part of the sine function": if y(t)=sin(x(t))y(t)=sin⁡(x(t)) for tt in some interval, since both sides of that equation are analytic functions on the complex plane the equation would be true for all complex numbers tt. Since y(t)y(t) is a polynomial, for any given value of yy (unless yy is constant) there are only finitely many tt and thus finitely many xx. But this is not the case for the sine function: sin(nπ)=0sin⁡(nπ)=0 for all integers nn. So the sine curve can't be given exactly by a Bézier curve of any degree.