Coplanarity

In geometry, a set of points in space is coplanar if all the points lie in the same geometric plane. For example, three distinct points are always coplanar; but a fourth point or more added in space can exist in another plane, or, incoplanarly.

Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other.

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

Properties

If three vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coplanar, and $\mathbf{a}\cdot\mathbf{b} = 0$, then

$(\mathbf{c}\cdot\mathbf{\hat a})\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\mathbf{\hat b} = \mathbf{c},$

where $\mathbf{\hat a}$ denotes the unit vector in the direction of $\mathbf{a}$.

Or, the vector resolutes of $\mathbf{c}$ on $\mathbf{a}$ and $\mathbf{c}$ on $\mathbf{b}$ add to give the original $\mathbf{c}$.

Plane Formula

Another technique involves computing the formula for the planes defined by each subset of three points. First, the normal-vector for each plane is computed using some Orthogonalization technique. If the planes are parallel, then the dot-product of their normal-vectors will be 1 or -1. More specifically, the angle between the normal vectors can be computed. This is called the dihedral angle, and represents the smallest possible angle between the two planes. The formula for a plane is:

$ax+by+cz+d=0$, where $(a,b,c)$ is the normal vector of the plane.

The value $d$ can be computed by plugging in one of the points and then solving. If $d$ is the same for all subsets of three points, then the planes are the same.

One advantage of this technique is that it can work in hyper-dimensional space. For example, suppose you wanted to compute the dihedral angle between two m-dimensional hyperplanes defined by m points in n-dimensional space. If $n-m>1$, then there are an infinite number of normal vectors for each hyperplane, so the angle between two of them is not necessarily the dihedral angle. However, if you use Gram-Schmidt process using the same initial vector in both cases, then the angle between the two normal vectors will be minimal, and therefore will be the dihedral angle between the hyperplanes.
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