HalfPlane

Half-plane is a region of the two-dimensional Euclidean plane determined by a straight line. In the plane, a half-plane consists of all points on one side of a given line, together with the line itself if a non-strict inequality is used. Formally, for real numbers a, b, c with not both a and b zero, the set H = { (x, y) ∈ R^2 : a x + b y + c > 0 } is an open half-plane, while H = { (x, y) ∈ R^2 : a x + b y + c ≥ 0 } is a closed half-plane. The line a x + b y + c = 0 is the boundary of the half-plane. If the strict inequality is used, the boundary is excluded; if the non-strict inequality is used, the boundary is included. The boundary line can often be described as y = mx + d when b ≠ 0, or as x = constant when b = 0.

Half-planes are convex and unbounded; the interior of the open half-plane is connected, while the closed half-plane includes the boundary. They play a central role in linear inequalities, linear programming, and computational geometry. The intersection of half-planes yields convex regions, including polygons when finitely many are used. In higher dimensions, the analogue is a half-space, defined by a1 x1 + ... + an xn + c ≥ 0 with a boundary that is a hyperplane.