Bezierkäyrien

Bézier curves are mathematical representations of smooth curves commonly used in computer graphics and related fields. They are defined by a set of control points. The curve itself does not necessarily pass through all of these control points, but they influence the shape of the curve.

The simplest form is the linear Bézier curve, which is a straight line segment between two control points. Quadratic Bézier curves involve three control points, creating a parabolic arc. Cubic Bézier curves, the most common type, use four control points. The first and last control points define the endpoints of the curve, while the intermediate control points determine the curve's tangents at the endpoints, effectively pulling the curve towards them.

The mathematical definition of a Bézier curve involves Bernstein polynomials. For a curve of degree n, there are n+1 control points. The position of any point on the curve is calculated as a weighted sum of these control points, where the weights are given by the Bernstein basis polynomials. These polynomials ensure that the curve is always contained within the convex hull of its control points, a property known as the convex hull property.

Bézier curves are widely used in vector graphics software, font design, animation, and computer-aided design (CAD) because they offer intuitive control over curve shape. They are easily scalable without loss of quality, making them ideal for vector graphics. Their smoothness and predictable behavior also make them suitable for generating realistic trajectories and shapes in various applications.