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Plane-Plane Intersection


Two planes always intersect in a line as long as they are not parallel. Let the planes be specified in Hessian normal form, then the line of intersection must be perpendicular to both n_1^^ and n_2^^, which means it is parallel to

 a=n_1^^xn_2^^.
(1)

To uniquely specify the line, it is necessary to also find a particular point on it. This can be determined by finding a point that is simultaneously on both planes, i.e., a point x_0 that satisfies

n_1^^·x_0=-p_1
(2)
n_2^^·x_0=-p_2.
(3)

In general, this system is underdetermined, but a particular solution can be found by setting z_0=0 (assuming the z-component of a is not 0; or another analogous condition otherwise) and solving. The equation of the line of intersection is then

 x=x_0+ta
(4)

(Gellert et al. 1989, p. 542). A general approach avoiding the special treatment needed above is to define

m=[n_1^^ n_2^^]^(T)
(5)
b=-[p_1; p_2].
(6)

Then use a linear solving technique to find a particular solution x_0 to mx_0=b, and the direction vector will be given by the null space of m.

Let three planes be specified by a triple of points (x_(ij),y_(ij),z_(i,j)) where i,j=1, 2, 3, i denotes the plane number and j denotes the jth point of the ith plane. The point of concurrence (x,y,z) can be obtained straightforwardly (if laboriously) by simultaneously solving the three equations arising from the coplanarity of each of the planes with (x,y,z), i.e.,

 |x y z 1; x_(i1) y_(i1) z_(i1) 1; x_(i2) y_(i2) z_(i2) 1; x_(i3) y_(i3) z_(i3) 1|=0
(7)

for i=1, 2, 3 using Cramer's rule.

Plane-PlaneIntersection

If the three planes are each specified by a point x_k and a unit normal vector n_k^^, then the unique point of intersection x is given by

 x=((x_1·n_1^^)(n_2^^xn_3^^)+(x_2·n_2^^)(n_3^^xn_1^^)+(x_3·n_3^^)(n_1^^xn_2^^))/(|n_1^^ n_2^^ n_3^^|),
(8)

where |n_1^^ n_2^^ n_3^^| is the determinant of the matrix formed by writing the vectors n_i^^ side-by-side. If two of the planes are parallel, then

 |n_1^^ n_2^^ n_3^^|=0,
(9)

and there is no intersection (Gellert et al. 1989, p. 542; Goldman 1990). This condition can be checked easily for planes in Hessian normal form.

A set of planes sharing a common line is called a sheaf of planes, while a set of planes sharing a common point is called a bundle of planes.


See also

Bundle of Planes, Flat, Parallel Planes, Plane, Sheaf of Planes

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References

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 541-543, 1989.Goldman, R. "Intersection of Three Planes." In Graphics Gems I (Ed. A. S. Glassner). San Diego: Academic Press, p. 305, 1990.

Referenced on Wolfram|Alpha

Plane-Plane Intersection

Cite this as:

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

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