An order n Bezier curve (a vector function) can represent smooth curves not possible with an order n polynomial. Are they a practical choice for function approximation? Consider the order 3 Bezier curve, defined by two points and the tangent vectors at those points:

The points and direction of the tangents match f(x). The tangent lengths (weights) have been left variable.
Formally,
Assume f(x) and its first derivative are known at the tabular points. Let f(x) be represented as , with tangent .
Let points ,
tangents , and
.
For the Bezier curve, constrain the tangents to the interval :
Then the order 3 Bezier curve through points A, B, with respective tangents is
Choose some x_ξ, . To approximate f(x_ξ), by I must

Set x(t) = x_ξ.

Solve the cubic equation for t.

Plug this t-value into y(t).

The approximation can indeed be made quite good. Try different functions. In what range should k1, k2 fall? What relationships between k1, k2 can we reasonably choose? How might the solution of x(t) be simplified?
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