bezierkurver

Bezierkurver (Bezier curves) are parametric curves widely used in computer graphics and geometric design. A Bezier curve of degree n is defined by n+1 control points P0, P1, ..., Pn in a Euclidean space. The curve is given by B(t) = sum_{i=0}^n B_i^n(t) P_i for t in [0,1], where B_i^n(t) = binom(n,i) t^i (1−t)^{n−i} are Bernstein polynomials. The curve starts at P0 and ends at Pn, and its initial and final tangents are directed along P1−P0 and Pn−P_{n−1}.

Two central properties are the convex hull property—B(t) always lies inside the convex hull of the control points—and affine invariance—applying an affine transformation to all control points yields the corresponding transformed curve. A practical method to evaluate B(t) and to subdivide the curve is de Casteljau’s algorithm, which performs repeated linear interpolations between neighboring control points. The same process also yields exact subdivisions, producing new control points for the pieces.

Historically, Bezier curves were developed for car design and CAD applications in the mid-20th century, with Pierre Bézier and Paul de Casteljau contributing foundational work. In modern graphics, Bezier curves provide intuitive direct manipulation of shapes and form the basis of many spline-based systems. They are common in vector graphics formats such as SVG and PostScript, where cubic Bezier segments are especially prevalent; font outlines in some formats use quadratic Bezier curves.

Limitations include that a single Bezier curve may require high degree to represent complex shapes, and curvature behavior is not guaranteed to be monotone. For more complex design tasks, designers often use piecewise Bezier curves with continuity constraints or extend to splines such as B-splines or NURBS.