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Plane

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A plane is a two-dimensional doubly ruled surface spanned by two linearly independent vectors. The generalization of the plane to higher dimensions is called a hyperplane. The angle between two intersecting planes is known as the dihedral angle.

Plane

The equation of a plane with nonzero normal vector n=(a,b,c) through the point x_0=(x_0,y_0,z_0) is

n·(x-x_0)=0,
(1)

where x=(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 has x-, y-, and z-intercepts at

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

and lies at a distance

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

from the origin.

It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from (◇) by defining the components of the unit normal vector n^^=(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 the Hessian normal form of the plane is

n^^·x=-p
(12)

(Gellert et al. 1989, p. 540), the (signed) distance to a point x_0 is

D=n^^·x_0+p,
(13)

and the distance from the origin is simply

D=p
(14)

(Gellert et 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)

and A_i is the determinant obtained by replacing x_i with a column vector of 1s. To express in Hessian 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 constant p 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)

The dihedral 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 vectors n_1=(a_1,b_1,c_1) and n_2=(a_2,b_2,c_2) is simply given via the dot 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 in Hessian normal form (Gellert et al. 1989, p. 541).

In order to specify the relative distances of n>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 the x-axis. The remaining n-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 a binomial coefficient, so the distances between points are subject to m relationships, where

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

For n=2 and n=3, there are no relationships. However, for a quadrilateral (with n=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 to intersect in exactly one point. For every set of n points in the plane, there exists a point O in the plane having the property such that every straight line through O has at least 1/3 of the points on each side of it (Honsberger 1985).

Every rigid motion of the plane is one of the following types (Singer 1995):

1. Rotation about a fixed point P.

2. Translation in the direction of a line l.

3. Reflection across a line l.

4. Glide-reflections along a line l.

Every rigid motion of the hyperbolic plane is one of the previous types or a

5. Horocycle rotation.

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