Béziergörbe

Bézier curves are a type of parametric curve used in computer graphics and related fields. They are defined by a set of control points. The curve passes through the first and last control points, but generally does not pass through the intermediate control points. Instead, the intermediate control points influence the shape of the curve, acting like magnets pulling it towards them.

The mathematical definition of a Bézier curve involves Bernstein polynomials. For a curve with n+1 control points P0, P1, ..., Pn, the curve B(t) for t ranging from 0 to 1 is given by the formula: B(t) = sum_{i=0}^{n} B_i^n(t) P_i. Here, B_i^n(t) are the Bernstein basis polynomials, defined as B_i^n(t) = C(n, i) * t^i * (1-t)^{n-i}, where C(n, i) is the binomial coefficient.

The simplest and most common Bézier curves are quadratic (3 control points) and cubic (4 control points). Quadratic Bézier curves are smooth and have a single inflection point. Cubic Bézier curves offer more flexibility in shaping and are widely used for drawing smooth, curved lines in vector graphics software. The degree of the Bézier curve is one less than the number of control points.

Bézier curves are fundamental in vector graphics because they can be scaled and manipulated without losing resolution. They are used in font design (TrueType and OpenType fonts), animation, CAD software, and various graphical user interfaces. Their intuitive control point system makes them relatively easy for designers to understand and use for creating complex shapes.