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Plane

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A plane is a two-dimensionaldoubly ruled surface spanned by two linearly independent vectors. The generalization of the plane to higherdimensions is called ahyperplane. The angle between twointersecting planes is known as thedihedral angle.

Plane

The equation of a plane withnonzeronormal vectorn=(a,b,c) through the pointx_0=(x_0,y_0,z_0) is

n·(x-x_0)=0,
(1)

wherex=(x,y,z). Plugging in gives the general equation of a plane,

ax+by+cz+d=0,
(2)

where

d=-ax_0-by_0-cz_0.
(3)

A plane specified in this form therefore hasx-,y-, andz-intercepts at

x=-d/a
(4)
y=-d/b
(5)
z=-d/c,
(6)

and lies at adistance

D=d/(sqrt(a^2+b^2+c^2))
(7)

from theorigin.

It is especially convenient to specify planes in so-calledHessian normal form. This is obtained from (◇) by defining the components of theunit normal vectorn^^=(n_x,n_y,n_z)

n_x=a/(sqrt(a^2+b^2+c^2))
(8)
n_y=b/(sqrt(a^2+b^2+c^2))
(9)
n_z=c/(sqrt(a^2+b^2+c^2))
(10)

and the constant

p=d/(sqrt(a^2+b^2+c^2)).
(11)

Then theHessian normal form of the plane is

n^^·x=-p
(12)

(Gellertet al.1989, p. 540), the (signed) distance to a pointx_0 is

D=n^^·x_0+p,
(13)

and the distance from theorigin is simply

D=p
(14)

(Gellertet al.1989, p. 541).

PlaneIntercept

In intercept form, a plane passing through the points(a^',0,0),(0,b^',0) and(0,0,c^') is given by

x/(a^')+y/(b^')+z/(c^')=1.
(15)

The plane through(x_1,y_1,z_1) and parallel to(a_1,b_1,c_1) and(a_2,b_2,c_2) is

|x-x_1 y-y_1 z-z_1; a_1 b_1 c_1; a_2 b_2 c_2|=0.
(16)

The plane through points(x_1,y_1,z_1) and(x_2,y_2,z_2) parallel to direction(a,b,c) is

|x-x_1 y-y_1 z-z_1; x_2-x_1 y_2-y_1 z_2-z_1; a b c|=0.
(17)

The three-point form is

|x y z 1; x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1|=|x-x_1 y-y_1 z-z_1; x_2-x_1 y_2-y_1 z_2-z_1; x_3-x_1 y_3-y_1 z_3-z_1|=0.
(18)

A plane specified in three-point form can be given in terms of the general equation (◇) by

A_1x+A_2y+A_3z-A=0,
(19)

where

A=det(x_1 x_2 x_3)
(20)

andA_i is thedeterminant obtained by replacingx_i with acolumn vector of 1s. To express inHessian normal form, note that the unit normal vector can also be immediately written as

n^^=((x_3-x_1)x(x_2-x_1))/(|(x_3-x_1)x(x_2-x_1)|)
(21)

and the constantp giving the distance from the plane to the origin is

p=A/(sqrt(A_1^2+A_2^2+A_3^2)).
(22)

The (signed)point-plane distance from a point(x_0,y_0,z_0) to a plane

ax+by+cz+d=0
(23)

is

D=(ax_0+by_0+cz_0+d)/(sqrt(a^2+b^2+c^2)).
(24)

Thedihedral angle between the planes

a_1x+b_1y+c_1z+d_1=0
(25)
a_2x+b_2y+c_2z+d_2=0
(26)

which have normal vectorsn_1=(a_1,b_1,c_1) andn_2=(a_2,b_2,c_2) is simply given via thedot product of the normals,

costheta=n_1^^·n_2^^
(27)
=(a_1a_2+b_1b_2+c_1c_2)/(sqrt(a_1^2+b_1^2+c_1^2)sqrt(a_2^2+b_2^2+c_2^2)).
(28)

The dihedral angle is therefore particularly simple to compute if the planes are specified inHessian normal form (Gellertet al.1989, p. 541).

In order to specify the relative distances ofn>1 points in the plane,1+2(n-2)=2n-3 coordinates are needed, since the first can always be placed at (0, 0) and the second at(x,0), where it defines thex-axis. The remainingn-2 points need two coordinates each. However, the total number of distances is

_nC_2=(n; 2)=(n!)/(2!(n-2)!)=1/2n(n-1),
(29)

where(n; k) is abinomial coefficient, so the distances between points are subject tom relationships, where

m=1/2n(n-1)-(2n-3)=1/2(n-2)(n-3).
(30)

Forn=2 andn=3, there are no relationships. However, for aquadrilateral (withn=4), there is one (Weinberg 1972).

It is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). In four dimensions, it is possible for four planes tointersect inexactly one point. For every set ofn points in the plane, there exists a pointO in the plane having the property such thatevery straight line throughO has at least 1/3 of the points on each side of it (Honsberger 1985).

Everyrigid motion of the plane is one of the followingtypes (Singer 1995):

1.Rotation about a fixed pointP.

2.Translation in the direction of a linel.

3.Reflection across a linel.

4. Glide-reflections along a linel.

Everyrigid motion of the hyperbolic plane is oneof the previous types or a

5. Horocycle rotation.

See also

Complex Plane,Cox's Theorem,Cross Section,Dihedral Angle,Director,Doubly Ruled Surface,Elliptic Plane,Fano Plane,Hessian Normal Form,Hyperplane,Isoclinal Plane,Line-Plane Intersection,Mediator,Moufang Plane,Nirenberg's Conjecture,Normal Section,Plane-Plane Intersection,Point-Plane Distance,Projective PlaneExplore this topic in the MathWorld classroom

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References

Beyer, W. H.CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 208-209, 1987.Eisenberg, B. and Sullivan, R. "Random Trianglesn Dimensions."Amer. Math. Monthly103, 308-318, 1996.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). "Plane." InVNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, pp. 539-543, 1989.Honsberger, R.Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 189-191, 1985.Kern, W. F. and Bland, J. R. "Lines and Planes in Space." §4 inSolid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 9-12, 1948.Singer, D. A. "Isometries of the Plane."Amer. Math. Monthly102, 628-631, 1995.Weinberg, S.Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972.

Referenced on Wolfram|Alpha

Plane

Cite this as:

Weisstein, Eric W. "Plane." FromMathWorld--AWolfram Resource.https://mathworld.wolfram.com/Plane.html

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Created, developed and nurtured by Eric Weisstein at Wolfram Research
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