Bézierkõveraid

Bézierkõveraid, named after French engineer Pierre Bézier, are parametric curves widely 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 rather they influence its shape. The simplest Bézier curve is a linear Bézier curve, which is a straight line segment connecting two endpoints.

Quadratic Bézier curves are defined by three control points: a start point, an intermediate control point that pulls the curve towards it, and an end point. Cubic Bézier curves, the most commonly used type, are defined by four control points: a start point, two intermediate control points, and an end point. The first and last control points are always on the curve, while the intermediate points dictate the curve's tangent at the endpoints and its overall curvature.

The mathematical definition of a Bézier curve involves Bernstein polynomials. For a Bézier curve of degree n with control points P0, P1, ..., Pn, any point on the curve B(t) can be calculated as the sum of the product of each control point and its corresponding Bernstein basis polynomial, where t is a parameter ranging from 0 to 1. As t varies from 0 to 1, the point B(t) traces the curve from the start point to the end point.

Bézier curves offer several advantages. They are easy to understand and manipulate, providing intuitive control over shape. They are also well-behaved, meaning they are continuous and smooth, and their derivatives are also continuous. This makes them suitable for drawing smooth outlines of objects, designing fonts, and creating animations.