kuupbezierkõverad

Bézier curves are parametric curves used in computer graphics and related fields to model smooth curves. A Bézier curve is defined by a set of control points. The degree of the curve is one less than the number of control points. For example, three control points define a quadratic Bézier curve, and four control points define a cubic Bézier curve.

The kuupbezierkõverad, or cubic Bézier curves, are particularly common and useful. They are defined by four control points: a start point, two intermediate control points, and an end point. The curve starts at the first control point and ends at the last control point. The intermediate control points, often called "handles," dictate the shape and curvature of the line. The curve does not necessarily pass through the intermediate control points but is "pulled" towards them.

The mathematical definition of a cubic Bézier curve involves Bernstein polynomials. For control points P0, P1, P2, and P3, a point on the curve P(t) for t ranging from 0 to 1 is given by the formula: P(t) = (1-t)³P0 + 3(1-t)²tP1 + 3(1-t)t²P2 + t³P3. This formula allows for precise calculation of any point along the curve.

Cubic Bézier curves are widely used in vector graphics software like Adobe Illustrator and Inkscape, in font design (e.g., TrueType fonts), and in animation for creating smooth motion paths. Their intuitive control mechanism, where users manipulate handles to shape curves, makes them a powerful tool for designers and artists. The ability to create complex, smooth shapes with a relatively small number of control points contributes to their popularity.