In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. 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. This occurs if the lines are parallel, or if they intersect each other. Two lines that are not coplanar are called skew lines.

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

Properties in three dimensions

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.¹ This leads to the following coplanarity test using a scalar triple product:

Four distinct points, x₁, x₂, x₃, x₄, are coplanar if and only if,

which is also equivalent to

If three vectors a, b, c are coplanar, then if a ⋅ b = 0 (i.e., a and b are orthogonal) then

where ⁠⁠ denotes the unit vector in the direction of a. That is, the vector projections of c on a and c on b add to give the original c.

Coplanarity of points in n dimensions whose coordinates are given

Since three or fewer points are always coplanar, the problem of determining when a set of points is 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 an n-dimensional space where n ≥ 3, a set of k points are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors is of rank 2 or less.

For example, given four points

if the matrix

is of rank 2 or less, the four points are coplanar.

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.

Geometric shapes

A skew polygon is a polygon whose vertices are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles.

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

References

1Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 647, ISBN 0-87150-341-7

External links

Weisstein, Eric W. "Coplanar". MathWorld.

Study aids for Coplanarity

Terminology

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

Terms the article defines in bold, quoted in context.

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Distance geometry
Branch of mathematics
Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points.
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Geometry
Branch of mathematics
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
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Line–line intersection
Common point(s) shared by two lines in Euclidean geometry
In Euclidean geometry, the intersection of a line and a line can be the empty set, a single point, or a line (if they coincide).
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Parallel (geometry)
Relation used in geometry
In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are infinite flat planes in the same three-dimensional space that never meet.
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Line (geometry)
Straight figure with zero width and depth
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature.
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Plane of incidence
Plane containing the surface normal and the propagation vector of the incoming radiation
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation.
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Collinearity
Property of points all lying on a single line
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear).
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Plane (mathematics)
2D surface which extends indefinitely
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.
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Skew lines
Lines not in the same plane
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.
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Skew polygon
Polygonal chain whose vertices are not all coplanar
In geometry, a skew polygon is a closed polygonal chain in Euclidean space. It is a figure similar to a polygon except its vertices are not all coplanar.
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Linked in “Properties in three dimensions”
Cross product
Mathematical operation on vectors in 3D space
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol ×.
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Matrix (mathematics)
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and columns, usually satisfying certain properties of addition and multiplication.
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Rank (linear algebra)
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A.
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Polyhedron
Flat-sided three-dimensional shape
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.
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Properties in three dimensions

Coplanarity of points in n dimensions whose coordinates are given

Geometric shapes

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Properties in three dimensions

Coplanarity of points in n dimensions whose coordinates are given

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