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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