Positionsebene

Positionsebene, literally "position plane" in German, is a term used in technical contexts to denote a plane that serves as a reference for specifying the position of points, objects, or features in space. Practically, it is a two-dimensional surface embedded in three-dimensional space, typically defined by a point p0 and a normal vector n, such that any location x on the plane satisfies n · (x − p0) = 0. The plane can also be described by the standard plane equation a x + b y + c z + d = 0.

In use, a positionsebene provides a local coordinate system on the plane through two in-plane basis vectors e1 and e2, allowing a 3D point to be represented in plane coordinates as (u, v) = (e1 · (x − p0), e2 · (x − p0)). Transformations between world coordinates and plane coordinates employ affine or homogeneous transformation matrices. The concept is closely tied to coordinate frames and is often integrated with sensor data and calibration to ensure accurate plane definition.

Applications span robotics, computer-aided design and manufacturing, computer graphics, and geospatial tasks. They include constraining end-effector motion to a work plane, sketch planes and machining on a plane, planar projections and texture mapping, and planar approximations in map projections. Plane estimation from data—via least squares, RANSAC, or other fitting methods—is a common problem, while accuracy depends on the precise definition of p0, n, and the chosen in-plane coordinate system. Related concepts include world planes, sensor planes, and image planes, all within the broader framework of coordinate transformations.