Coplanarity

Coplanarity is when two objects are in the same plane. In a two-dimensional space like a Euclidean plane, all objects are coplanar. With three dimensions or more, objects are coplanar only if there is a plane-shaped subset of the space that contains all of their points (an affine subspace). In general, points and objects in three or more dimensions are almost never coplanar.

The simplest way to define coplanarity mathematically is to start with vectors. A set of vectors is coplanar if and only if all vectors can be written as a linear combination of two vectors in the set. Coplanar vectors are part of a vector subspace with dimension 2.

Points are like vectors, but belong to affine spaces rather than vector spaces. A set of ${\displaystyle n}$ points can be used to construct a set of ${\displaystyle n-1}$ vectors by choosing some point ${\displaystyle p}$ and subtracting it from every other point in the set. the set of points is coplanar if the set of vectors is coplanar.

In a three-dimensional space, this can be checked using the dot product and cross product of the distances. If the four points are called ${\displaystyle p_{1},p_{2},p_{3},p_{4}}$, they are coplanar if and only if

${\displaystyle ((p_{2}-p_{1})\times (p_{3}-p_{1}))\cdot (p_{4}-p_{1})=0}$

Like how coplanar vectors define a planar vector subspace, coplanar points define a planar affine subspace.

Lines follow from the case for points. Choose two points on each line; by Euclid's postulates, there is only one line through each of these pairs of points. As any line through two points must be in the same plane as those points, the lines are coplanar only if the points are coplanar.

In Euclidean geometry, different lines are coplanar if and only if they are parallel or intersect at exactly one point.^[1] This is because of the parallel postulate.

Polygons and other two-dimensional geometric figures are coplanar if all of their points are coplanar.

1. ↑ Barron's Regents Exams and Answers. Barron's. 2016. ISBN 978-1506272016.