Planar

Planar is an adjective derived from plane, describing things related to or arranged on a flat, two‑dimensional surface.

In mathematics and geometry, planarity refers to the property of being drawable on a plane without crossings or to shapes that lie entirely within a single plane. A planar graph is one that can be drawn on the plane with no edge intersections except at shared endpoints. For connected planar graphs, Euler's formula states that V − E + F = 2, where V, E, and F are the numbers of vertices, edges, and faces, respectively. Planarity can be tested algorithmically, and Kuratowski's theorem provides a characterization: a graph is nonplanar if and only if it contains a subdivision of K5 or K3,3.

In chemistry, planar describes molecules or ions whose non-hydrogen atoms lie in the same geometric plane. Planarity is common in systems with sp2 hybridization, such as benzene, where delocalized electrons enforce a flat ring. Non-planar conformations occur when atoms adopt three-dimensional arrangements, influencing properties like aromaticity and reactivity. In transition metal chemistry, certain geometries are described as planar, including some square planar complexes.

In cartography and computer graphics, planar projections map curved surfaces to a plane. No projection preserves all geometric properties, so trade-offs are made between area, shape, and distance. Planar representations underpin two-dimensional rendering and graph layouts used in circuit design and data visualization.