Krümmungsradius

Krümmungsradius is a concept used in geometry and physics to describe how sharply a curve or surface bends. It is the reciprocal of the curvature. For a curve, the curvature measures how quickly the direction of the tangent vector changes as one moves along the curve. A larger Krümmungsradius indicates a gentler curve, while a smaller Krümmungsradius signifies a sharper bend. For a straight line, the curvature is zero, and therefore the Krümmungsradius is infinite.

In two dimensions, for a curve defined by the function y = f(x), the Krümmungsradius at a point (x, y) is given by the formula R = (1 + (dy/dx)^2)^(3/2) / |d^2y/dx^2|. Here, dy/dx represents the first derivative of the function, indicating the slope, and d^2y/dx^2 represents the second derivative, related to the concavity of the curve.

In three dimensions, the Krümmungsradius applies to surfaces as well. For a surface, there are principal curvatures at each point, which represent the maximum and minimum curvatures. The Krümmungsradius is then related to these principal curvatures. Understanding Krümmungsradius is crucial in fields such as mechanical engineering for designing parts that experience stress, in optics for shaping lenses, and in cartography for map projections.