normal of plane

The perpendicularPlanetmathPlanetmath or normal line of a plane is a special case of the surface normal, but may be defined separately as follows:

A line l is a normal of a plane π, if it intersects the plane and is perpendicular to all lines passing through the intersection point in the plane.  Then the plane π is a normal planeMathworldPlanetmath of the line l.  The normal plane passing through the midpointMathworldPlanetmathPlanetmathPlanetmath ( of a line segmentMathworldPlanetmath is the center normal plane of the segment.

There is the

Theorem.  If a line (l) a plane (π) and is perpendicular to two distinct lines (m and n) passing through the cutting point (L) in the plane, then the line is a normal of the plane.

Proof.  Let a be an arbitrary line passing through the point L in the plane π.  We need to show that  al.  Set another line of the plane cutting the lines m, n and a at the points M, N and A, respectively.  Separate from l the equally segments LP and LQ.  Then


since any point of the center normal of a line segment (PQ) is equidistant from the end pointsPlanetmathPlanetmath of the segment.  Consequently,


Thus the segments PA and QA, being corresponding parts of two congruentMathworldPlanetmathPlanetmath trianglesMathworldPlanetmath, are equally long.  I.e., the point A is equidistant from the end points of the segment PQ, and it must be on the perpendicular bisector ( of PQ.  Therefore  ALPQ, i.e.  al.

PropositionPlanetmathPlanetmath 1.  All of a plane are parallelMathworldPlanetmathPlanetmath.  If a line is parallel to a normal of a plane, then it is a normal of the plane, too.

Proposition 2.  All normal planes of a line are parallel.  If a plane is parallel to a normal plane of a line, then also it is a normal plane of the line.

Title normal of plane
Canonical name NormalOfPlane
Date of creation 2013-04-19 15:03:42
Last modified on 2013-04-19 15:03:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 51M04
Synonym plane normal
Related topic AngleBetweenLineAndPlane
Related topic NormalLine
Related topic NormalVector
Related topic CongruenceMathworldPlanetmathPlanetmath
Related topic ParallelAndPerpendicularPlanes
Related topic ParallelismOfTwoPlanes
Defines normal plane
Defines center normal plane