Bézierkäyriä

Bézierkäyriä, or Bézier curves in English, are a method of representing smooth curves mathematically. They are widely used in computer graphics, computer-aided design (CAD), and typography. Invented by Pierre Bézier in the 1960s, these curves are defined by a set of control points. The curve itself does not necessarily pass through all of these control points, but rather they influence the shape of the curve.

The simplest Bézier curve is a linear Bézier curve, defined by two endpoints. A quadratic Bézier curve is 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, using four control points. The first and last points are typically the start and end of the curve, while the two middle points dictate the curve's tangent at the start and end, respectively.

The mathematical foundation of Bézier curves involves Bernstein polynomials. By varying the positions of the control points, designers and artists can create a vast range of shapes, from simple straight lines to complex, flowing arcs. Their predictable nature and ease of manipulation make them a fundamental tool in digital design, allowing for smooth and scalable curves that can be rendered at any resolution.