Bezierkõvera

A Bézier curve is a parametric curve used in computer graphics and related fields. It is defined by a set of control points. The curve is not required to pass through any of the control points except for the first and last ones, which are called endpoints. The position of the intermediate control points determines the shape of the curve.

Bézier curves are widely used in vector graphics, allowing for smooth and scalable curves. They are instrumental in defining shapes for fonts, illustrations, and user interface elements. The mathematical properties of Bézier curves make them easy to manipulate and interpolate.

There are different types of Bézier curves, with the most common being quadratic and cubic Bézier curves. A quadratic Bézier curve is defined by three control points, while a cubic Bézier curve is defined by four control points. As the number of control points increases, the complexity and flexibility of the curve also increase.

The mathematical formulation of a Bézier curve involves Bernstein polynomials. For a curve with n+1 control points P0, P1, ..., Pn, the curve B(t) is given by the formula: B(t) = sum(i=0 to n) [Bi,n(t) * Pi], where Bi,n(t) are the Bernstein basis polynomials. The parameter t typically ranges from 0 to 1.

Bézier curves offer advantages such as ease of definition and predictable behavior. They can be easily combined to create more complex shapes, and their derivatives can be computed analytically, which is useful for tasks like calculating curvature or tangent vectors.