Two planes always intersect in a heat as long as they space not parallel. Let the plane be stated in Hessian common form, then the line of intersection need to be perpendicular come both

and also
, which way it is parallel to

To unique specify the line, it is necessary to also find a specific point on it. This have the right to be established by recognize a point that is at the same time on both planes, i.e., a point

that satisfies

In general, this device is underdetermined, but a particular solution have the right to be discovered by setting

(assuming the
-component of
is not 0; or one more analogous problem otherwise) and also solving. The equation that the heat of intersection is then

(Gellert et al. 1989, p.542). A general technique avoiding the specialtreatment needed over is come define

 ^(T)" /> ." />

Then use a direct solving method to uncover a specific solution

come
, and the direction vector will certainly be offered by the null space of
.

You are watching: What is the intersection of two planes called

Let 3 planes be stated by a triple of point out

wherein
, 2, 3,
denotes the airplane number and also
denotes the
th allude of the
th plane. The suggest of concurrence
deserve to be derived straightforwardly (if laboriously) by all at once solving the 3 equations occurring from the coplanarity of every of the planes with
, i.e.,

for

, 2, 3 making use of Cramer"s rule.

If the three planes are each specified by a suggest

and also a unit regular vector
, then the unique allude of intersection
is offered by

where

is the determinant of the matrix created by writing the vectors
side-by-side. If 2 of the planes are parallel, then

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

A set of planes share a typical line is dubbed a sheaf of planes, if a collection of planes sharing a common suggest is called a bundle that planes.

REFERENCES:

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and also Künstner, H. (Eds.). VNR Concise Encyclopedia the Mathematics, second ed. Brand-new York: valve Nostrand Reinhold, pp.541-543, 1989.

Goldman, R. "Intersection of three Planes." In graphics Gems ns (Ed. A.S.Glassner). Mountain Diego: scholastic Press, p.305, 1990.

CITE THIS AS:

Weisstein, Eric W. "Plane-Plane Intersection."From ptcouncil.net--A ptcouncil.net net Resource. Https://ptcouncil.net/Plane-PlaneIntersection.html

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