Rekcurve

Rekcurve is a term used in geometry and computer graphics to describe a broad class of curves generated by recursive rules. It denotes a family of curves rather than a single fixed object, with definitions varying across sources while sharing a common idea: the curve is produced from a recurrence relation applied to points or segments.

The basic construction starts with a seed sequence of points, P0, P1, and a recurrence rule F that defines each subsequent point via Pn+1 = F(Pn, Pn-1, n, parameters). The rekcurve is then obtained by interpolating between successive points, typically using linear, cubic, or Hermite schemes. Variants may generate new points by recursive subdivision, iterative function systems, or affine recurrences, all aimed at creating a controllable curve from simple building blocks.

Mathematical properties of rekcurves depend on the chosen recurrence. When F is contractive, the generating process tends to produce convergent sequences and predictable limiting behavior. Recurrences can yield smooth curves, piecewise-linear paths, or self-similar (fractal-like) structures. The construction is generally invariant under affine transformations, meaning basic geometric changes preserve the curve’s character. Sensitivity to parameters allows a single framework to produce a range of shapes, from regular to highly intricate.

Applications of rekcurves appear in procedural modeling, font and vector-graphics design, animation path generation, and visual effects. Variants include linear rekcurves (emphasizing straightforward recurrence), fractal rekcurves (showing self-similarity), and circular rekcurves (employing rotational components).

See also fractal curves, subdivision curves, and L-systems. The term remains a flexible descriptor in contemporary literature, applied wherever recursive construction of curves is useful.