Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the volume of the tetrahedron defined by them is 0,

 |x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|=0.

Coplanarity is equivalent to the statement that the pair of lines determined by the four points are not skew, and can be equivalently stated in vector form as


An arbitrary number of n points x_1, ..., x_n can be tested for coplanarity by finding the point-plane distances of the points x_4, ..., x_n from the plane determined by (x_1,x_2,x_3) and checking if they are all zero. If so, the points are all coplanar.

A set of n vectors V is coplanar if the nullity of the linear mapping defined by V has dimension 1, the matrix rank of V (or equivalently, the number of its singular values) is n-1 (Abbott 2004).

Parallel lines in three-dimensional space are coplanar, but skew lines are not.

See also

Parallel Lines, Plane, Point-Plane Distance, Skew Lines

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Abbott, P. (Ed.). "In and Out: Coplanarity." Mathematica J. 9, 300-302, 2004.

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Cite this as:

Weisstein, Eric W. "Coplanar." From MathWorld--A Wolfram Web Resource.

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