Bézier curves

Bézier curves are parametric polynomial⁴ curves that are widely used in graphical packages. They are often used to approximate another curve, the match being perfect at both endpoints. In order to match position, slope and curvature, a third degree polynomial is needed, these are the classical Bézier curves or cubic Bézier curves. If only position and slope need to match, a second degree polynomial is sufficient, the corresponding curves are quadratic Bézier curves. As an extension, we will also consider line segments to be linear Bézier curves defined by a first degree polynomial and matching only position of endpoints.

The polynomials underlying Bézier curves are very simple when their constants are defined in terms of control points. These polynomials are called Bernshtein polynomials. The range of the $t$ parameter is between 0 and 1. When $t=0$, we are at the start point of the Bézier curve which should match the start point of the approximated curve. When $t=1$, we are at the end point of the Bézier curve which should match the end point of the approximated curve. All graphical packages that support Bézier curves define them by the control points only, the Bernshtein polynomials are used internally and are not available at user level.