quadraten

Quadraten is a term used in some mathematical texts to denote a configuration consisting of four related elements arranged in a closed loop. The concept serves as a generalization of familiar four-vertex figures and is used in geometry and graph theory to study four-point relations without committing to a specific shape.

In planar geometry, a quadraten typically refers to four distinct points connected by four segments in a closed cycle, i.e., a four-vertex cycle. When these points lie in the plane, the figure formed can be any quadrilateral, including squares, rectangles, rhombi, or trapezoids; in graph theory, the corresponding structure is the 4-cycle, or C4.

Classification and properties: A nondegenerate planar quadraten has four vertices and four edges, with interior angles that sum to 360 degrees. If all four vertices lie on a common circle, the quadraten is cyclic, and opposite angles are supplementary. Degenerate quadraten occur when the points are collinear or when two points coincide, in which case the cycle is not a proper quadrilateral.

Applications and usage: The quadraten model is used in computational geometry and computer graphics to describe quadrilateral patches and perspective transforms. In mesh generation and geometric reasoning, it serves as a basic unit for assembling larger structures and for analyzing four-point relations.

See also: Quadrilateral, Tetragon, 4-cycle, Polygon; Notation and terminology may vary by source, and some authors use quadraten interchangeably with quadrilateral or tetragon.