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,
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 LinesExplore with Wolfram|Alpha
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References
Abbott, P. (Ed.). "In and Out: Coplanarity." Mathematica J. 9, 300-302, 2004.Referenced on Wolfram|Alpha
CoplanarCite this as:
Weisstein, Eric W. "Coplanar." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Coplanar.html