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 and , which means it is parallel to

(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 that satisfies

			(2)
			(3)

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

(4)

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

			(5)
			(6)

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

Let three planes be specified by a triple of points where , 2, 3, denotes the plane number and denotes the th point of the th plane. The point of concurrence can be obtained straightforwardly (if laboriously) by simultaneously solving the three equations arising from the coplanarity of each of the planes with , i.e.,

(7)

for , 2, 3 using Cramer's rule.

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

(8)

where is the determinant of the matrix formed by writing the vectors side-by-side. If two of the planes are parallel, then

(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.