Beziérkäyrät

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, where the curve is drawn by interpolating between these points. The simplest form is the linear Bézier curve, which is a straight line segment between two control points. Quadratic Bézier curves use three control points, with the middle point influencing the curvature. Cubic Bézier curves, the most commonly used type, employ four control points to define more complex and flexible shapes.

The mathematical foundation of Bézier curves lies in Bernstein polynomials. The position of a point on a Bézier curve is determined by a weighted average of the control points, where the weights are given by the Bernstein basis polynomials. As the parameter, typically denoted by 't' and ranging from 0 to 1, changes, the curve smoothly transitions from the first control point to the last.

A key characteristic of Bézier curves is their intuitive control. Moving a control point directly influences the shape of the curve, making them easy to manipulate for artists and designers. They are widely used in vector graphics software like Adobe Illustrator and Inkscape for drawing shapes, defining paths, and creating smooth transitions. Bézier curves are also employed in typography for defining character outlines and in animation for interpolating motion. Their piecewise nature allows for the construction of complex shapes by joining multiple Bézier segments together.