Planele

Planele is the Romanian plural noun for plană, used in mathematics to refer to planes. A plane is a flat, infinite, two‑dimensional surface that extends without bound in three‑dimensional space. In Euclidean geometry, a plane can be defined implicitly by a linear equation ax + by + cz + d = 0 with at least one of a, b, c nonzero, or parametrically as p0 + s u + t v, where p0 is a point on the plane and u, v are non‑collinear direction vectors. The coefficients a, b, c determine the plane’s normal direction, and d shifts it in space. The standard coordinate planes are the xy‑plane (z = 0), xz‑plane (y = 0), and yz‑plane (x = 0).

Two planes in three‑dimensional space either intersect along a line, are parallel and distinct, or are coincident. The distance from a point P(x0, y0, z0) to the plane ax + by + cz + d = 0 is |a x0 + b y0 + c z0 + d| / sqrt(a^2 + b^2 + c^2). The angle between two planes is the angle between their normal vectors, and is computed from their coefficients.

Planele are fundamental in geometry, computer graphics, CAD, architecture, and physics, where they model flat surfaces, borders between regions, and the constraints of three‑dimensional space. In Romanian mathematical texts, planele are encountered as the plural form of plane. See also: line, space, affine geometry.