2plane

2plane is a term used in mathematics to denote a two-dimensional flat subset of a vector or affine space. In Euclidean space R^n, a 2-plane can be viewed in two related ways: as a linear (through the origin) subspace of dimension 2, or as an affine subspace of dimension 2 (a plane that may be offset from the origin).

A linear 2-plane is the set of all linear combinations of two independent direction vectors v and w: {s v + t w | s, t in R}. This plane passes through the origin and is a 2-dimensional subspace of R^n. An affine 2-plane is obtained by translating a linear 2-plane by a fixed point p0: {p0 + s v + t w | s, t in R}, where p0 is any point in R^n and v, w are independent.

In R^3, a plane can be described by a linear equation ax + by + cz = d. If d = 0, the plane is linear (through the origin); otherwise it is affine (offset from the origin). A normal vector n = (a, b, c) is perpendicular to the plane and can be used to write the plane equation as n · (x − p0) = 0, or equivalently n · x = d.

Key geometric properties include flatness (no curvature), two-dimensional extent, and the fact that the intersection of two non-parallel planes in R^3 is a line (parallel but distinct planes do not intersect; coincident planes coincide entirely). Planes are fundamental in geometry, computer graphics, CAD, and robotics for defining surfaces, clipping, and spatial constraints.