Hochberührungspunkte

Hochberührungspunkte refers to a concept in geometry, specifically related to curves and surfaces. It describes a situation where two geometric objects, such as curves or surfaces, meet or touch at a point in a way that their tangent lines or planes coincide. This implies a higher order of contact than a simple intersection. If two curves have a Hochberührungspunkt at a point, they not only intersect at that point but also share the same tangent line at that point. For surfaces, this means they share the same tangent plane.

The concept can be extended to higher orders of contact. A first-order Hochberührungspunkt means the objects share the same position and the same tangent direction. A second-order Hochberührungspunkt implies they also share the same curvature at that point. In general, an n-th order Hochberührungspunkt means that the first n derivatives of the functions defining the objects are equal at the point of contact, while the (n+1)-th derivatives differ.

This notion is important in various fields, including differential geometry, where it helps classify the nature of intersections and understand the local behavior of curves and surfaces. It also finds applications in areas like computer graphics and mechanical engineering for designing smooth transitions and avoiding abrupt changes in shape or direction. The term is less commonly used in English mathematical literature, where phrases like "osculating" or "higher-order contact" are more prevalent.