kurvradius

Kurvradius, or radius of curvature, is a measure of how sharply a curve bends at a given point. In differential geometry, the curvature κ describes the instantaneous rate of change of the curve’s unit tangent vector with respect to arc length, and the radius of curvature is defined as R = 1/κ. A small radius corresponds to a sharp bend, while a large radius indicates a gentle bend; infinite radius corresponds to a straight line.

For a plane curve given parametrically as r(t) = (x(t), y(t)), the curvature is κ = |x' y'' − y' x''| / (x'^2 + y'^2)^{3/2}, and the radius of curvature is R = (x'^2 + y'^2)^{3/2} / |x' y'' − y' x''|. If the curve is described as y = f(x) with well-behaved derivatives, κ = |f''(x)| / (1 + f'(x)^2)^{3/2}, and R = 1/κ.

The center of curvature is the center of the osculating circle, the circle that best approximates the curve at that point and shares the same tangent and curvature to second order.

For a circle of radius R, κ = 1/R at all points, so its kurvradius is constant. For a straight line, κ = 0 and the radius of curvature is considered infinite.

In broader contexts, the kurvradius concept extends to space curves, with curvature defined similarly using arc length and the unit tangent, and it plays a key role in fields such as road design, optics, and orbital mechanics.