coplanar

Coplanar describes objects that lie in the same plane. In Euclidean geometry, a plane is a flat, two-dimensional surface extending infinitely. A set of points is coplanar if there exists a plane that contains all of them. In 3D space, this concept applies to points, lines, segments, circles, and polygons.

For points: Four points A, B, C, D are coplanar if there exists a plane containing them all. Equivalently, the volume of the tetrahedron ABCD is zero, or the scalar triple product (AB × AC) · AD equals zero; choosing A as a reference point.

For lines: Two lines L1 and L2 are coplanar if there exists a plane containing both. In 3D, this occurs if either they intersect or are parallel. A common test uses direction vectors d1, d2 and reference points P1 on L1, P2 on L2. They are coplanar when (P2 − P1) · (d1 × d2) = 0. If d1 × d2 = 0, the lines are parallel and thus coplanar.

For planes: A plane can be written as ax + by + cz + d = 0 with normal vector n = (a, b, c). A point X lies in the plane iff n · (X − P0) = 0 for some point P0 on the plane.

Polygons and polyhedra: A polygon is coplanar if all its vertices lie in a single plane. Nonplanar configurations occur in 3D modeling when vertices cannot be placed in one plane.

Applications: Coplanarity checks are used in computer graphics, CAD, geometric modeling, and spatial reasoning to determine intersections, visibility, and simplification of 3D data.