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Coplanar


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

 (x_3-x_1)·[(x_2-x_1)x(x_4-x_3)]=0.

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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References

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

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Coplanar

Cite this as:

Weisstein, Eric W. "Coplanar." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Coplanar.html

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