Zcoordinates

In three-dimensional space, the z-coordinate is the component of a point along the z-axis in a Cartesian coordinate system. A point is written P = (x, y, z), where x and y locate the projection on the x–y plane, and z measures the height (or depth) relative to that plane. The z-axis is perpendicular to the x–y plane.

In other common systems, z plays a similar role. In cylindrical coordinates, a point is (r, θ, z) with z equal to its Cartesian z. In spherical coordinates, a point is (ρ, θ, φ) and z can be expressed as z = ρ cos φ (depending on convention for φ as the angle from the positive z-axis).

Coordinate orientation varies by field. In mathematics, a right-handed system typically has z perpendicular to the x–y plane, with positive z pointing in the direction given by the right-hand rule. In some computer graphics APIs, a left-handed system is used, which changes how depth is interpreted.

Applications: The z-coordinate is essential for describing height in geometry, depth in computer graphics, and vertical position in geographic and physical contexts. Under rigid transformations, z changes according to rotation, translation, and scaling matrices, reflecting how points move through space.

Example: The point (3, 5, −2) lies 2 units below the x–y plane; its distance to the plane is |z|.