Düzlem

Düzlem is the Turkish term for a geometric plane. In Euclidean geometry it refers to a flat, two‑dimensional surface that extends without bound in all directions. A plane is one of the basic objects alongside points and lines.

In three-dimensional space, a plane is a two‑dimensional subset defined by a linear equation or by a point and a normal vector. A plane can be determined by three non‑collinear points, or by a point together with a direction perpendicular to the plane (a normal vector).

A common way to describe a plane in Cartesian coordinates is by the equation ax + by + cz + d = 0, where (a, b, c) is a normal vector to the plane. Equivalently, the point‑normal form n · (x − x0) = 0 uses a normal vector n and a point x0 on the plane. The plane can also be given parametrically as x = x0 + su + tv, where u and v are non‑parallel direction vectors in the plane.

Relations with lines are central: a line can lie in a plane, intersect it at a single point, or be parallel to it without meeting. In three‑space, the coordinate planes xOy, yOz and xOz are standard examples of planes that extend indefinitely.

Key computations include the distance from a point to a plane, given by |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2), and the projection of a point onto the plane along the normal direction.

Düzlemler are foundational in fields such as geometry, computer graphics, engineering, architecture, and navigation, where flat surfaces and their relationships to lines and other planes are routinely analyzed.