# Bezier Approximation, 1

- Author:
- Ryan Hirst

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).