Coplanarity

In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all.

However, a set of four or more distinct points will, in general, not lie in a single plane.

Two lines in three-dimensional space are coplanar if there is a plane that includes them both.

Distance geometry provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them.

In three-dimensional space, two linearly independent vectors with the same initial point determine a plane through that point.

Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initial point will lie in the plane.

That is, the vector projections of c on a and c on b add to give the original c. Since three or fewer points are always coplanar, the problem of determining when a set of points are coplanar is generally of interest only when there are at least four points involved.

In the case that there are exactly four points, several ad hoc methods can be employed, but a general method that works for any number of points uses vector methods and the property that a plane is determined by two linearly independent vectors.

In the special case of a plane that contains the origin, the property can be simplified in the following way: A set of k points and the origin are coplanar if and only if the matrix of the coordinates of the k points is of rank 2 or less.

A polyhedron that has positive volume has vertices that are not all coplanar.