Coplanarity

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-dimensional space, two 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. Four distinct points, $x 1, x 2, x 3$ and $x 4$ are coplanar if and only if,

(x_3 - x_1) \cdot [(x_2 - x_1) \times (x_4 - x_3)] = 0.

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

(\mathbf{c}\cdot\mathbf{\hat a})\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\mathbf{\hat b} = \mathbf{c},

where $\mathbf{\hat a}$ 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 whose coordinates are given

In coordinate geometry, in n-dimensional space, a set of four or more distinct points are coplanar if and only if the matrix of the coordinates of these points is of rank 2 or less. For example, given four points, W = (w₁, w₂, ... , w_n), X = (x₁, x₂, ... , x_n), Y = (y₁, y₂, ... , y_n), and Z = (z₁, z₂, ... , z_n), if the matrix

\begin{bmatrix} w_1 & w_2 & \dots & w_n \\ x_1 & x_2 & \dots & x_n \\ y_1 & y_2 & \dots & y_n \\ z_1 & z_2 & \dots & z_n \end{bmatrix}

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

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

↑ Swokowski, 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.

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