xyEbene

XYEbene, commonly written XY-Ebene, denotes the two-dimensional plane defined by the x- and y-axes in a Cartesian coordinate system. It is a fundamental construct in analytic geometry and is widely used in mathematics, physics, and computer graphics.

In Euclidean space R^3, the XY-Ebene is the subset z = 0, consisting of all points (x, y, 0). It is a flat, infinite surface endowed with the standard Euclidean metric. Its origin is (0, 0, 0), and its directions are given by the x- and y-axis basis vectors. Distances on the plane follow the Pythagorean formula: d((x1, y1, 0), (x2, y2, 0)) = sqrt((x1 − x2)^2 + (y1 − y2)^2).

The XY-Ebene is the natural setting for graphing functions y = f(x) and for visualizing 2D vector fields. It also serves as a reference plane in three-dimensional applications, with perpendicular reference planes such as the XZ- and YZ-planes forming part of the standard coordinate framework used in engineering, graphics, and physics.

In linear algebra and analytic geometry, the XY-Ebene supports transformations including translations within the plane, rotations about the z-axis, and projections from 3D space onto the plane. It is closely related to other coordinate-planes and provides a convenient stage for planar geometry and coordinate computations.

Etymology: in German, Ebene means plane, and XY refers to the two primary coordinate axes. The concept has its roots in the development of analytic geometry in the 17th century by Descartes and others.