Bézierkõveroiden

Bézierkõveroiden are mathematical curves used in computer graphics and related fields to represent smooth shapes. They are defined by a set of control points, which influence the curve's form without necessarily lying on the curve itself. The simplest Bézier curve is a linear Bézier curve, defined by two endpoints, which results in a straight line segment. Quadratic Bézier curves are defined by three control points: a start point, an end point, and one intermediate control point that pulls the curve towards it. Cubic Bézier curves are the most common type, defined by four control points: two endpoints and two intermediate control points that dictate the curve's tangent and curvature at the endpoints.

The mathematical basis for Bézier curves involves Bernstein polynomials. For a curve of degree n, there are n+1 control points. The curve is a weighted average of these control points, where the weights are given by the Bernstein polynomials. As the parameter along the curve varies from 0 to 1, the curve smoothly interpolates between the control points. This property makes them highly predictable and controllable.

Bézier curves are widely used in vector graphics software like Adobe Illustrator and Inkscape, where they allow designers to draw and manipulate complex shapes. They are also employed in font design to define the outlines of characters, in animation for smooth motion paths, and in computer-aided design (CAD) for creating precise curves in engineering and manufacturing. Their ability to generate smooth, scalable curves with a relatively small number of control points makes them a fundamental tool in digital design and geometry.