Plane

A plane is a 2-D Surface spanned by two linearly independent vectors. The generalization of the plane to higher Dimensions is called a Hyperplane.

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

$${x\over a} + {y\over b} + {z\over c} = 1.$$ (1)

The equation of a plane Perpendicular to the Nonzero Vector $\hat{\bf n}=(a,b,c)$ through the point $(x_0,y_0,z_0)$ is

$$\left[{\matrix{a\cr b\cr c\cr}}\right]\cdot \left[{\matrix{x... ... y-y_0\cr z-z_0\cr}}\right] = a(x-x_0)+ b(y-y_0)+c(z-z_0) = 0,$$ (2)

so

$$ax+by+cz+d=0,$$ (3)

where

$$d\equiv -ax_0-by_0-cz_0.$$ (4)

A plane specified in this form therefore has $x$-, $y$-, and $z$-intercepts at

$$x = -{d\over a}$$ (5)

$$y = -{d\over b}$$ (6)

$$z = -{d\over c},$$ (7)

and lies at a Distance

$$h={\vert d\vert\over\sqrt{a^2+b^2+c^2}}$$ (8)

from the Origin.

The plane through $P_1$ and parallel to $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is

$$\left\vert\matrix{x-x_1 & y-y_1 & z-z_1\cr a_1 & b_1 & c_1\cr a_2 & b_2 & c_2\cr}\right\vert=0.$$ (9)

The plane through points $P_1$ and $P_2$ parallel to direction $(a,b,c)$ is

$$\left\vert\matrix{x-x_1 & y-y_1 & z-z_1\cr x_2-x_1 & y_2-y_1 & z_2-z_1\cr a & b & c\cr}\right\vert=0.$$ (10)

The three-point form is

$$\left\vert\matrix{x & y & z & 1\cr x_1 & y_1 & z_1 & 1\cr x_... ...y_1 & z_2-z_1\cr x_3-x_1 & y_3-y_1 & z_3-z_1\cr}\right\vert=0.$$ (11)

The Distance from a point $(x_1, y_1, z_1)$ to a plane

$$Ax+By+Cz+D=0$$ (12)

is

$$d={Ax_1+By_1+Cz_1+D\over\pm\sqrt{A^2+B^2+C^2}}.$$ (13)

The Dihedral Angle between the planes

$$A_1x+B_1y+C_1z+D_1 = 0$$ (14)

$$A_2x+B_2y+C_2z+D_2 = 0$$ (15)

is

$$\cos\theta={A_1A_2+B_1B_2+C_1C_2\over\sqrt{{A_1}^2+{B_1}^2+{C_1}^2}\sqrt{{A_2}^2+{B_2}^2+{C_2}^2}}.$$ (16)

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

$${}_nC_2={n\choose 2}={n!\over 2!(n-2)!}= {\textstyle{1\over 2}}n(n-1),$$ (17)

where ${n\choose k}$ is a Binomial Coefficient, so the distances between points are subject to $m$ relationships, where

$$m\equiv {\textstyle{1\over 2}}n(n-1)-(2n-3) = {\textstyle{1\over 2}}(n-2)(n-3).$$ (18)

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 4-D, 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.

See also Argand Plane, Complex Plane, Dihedral Angle, Elliptic Plane, Fano Plane, Hyperplane, Moufang Plane, Nirenberg's Conjecture, Normal Section, Point-Plane Distance, Projective Plane

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 Triangles $n$ Dimensions.'' Amer. Math. Monthly 103, 308-318, 1996.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 189-191, 1985.

Singer, D. A. ``Isometries of the Plane.'' Amer. Math. Monthly 102, 628-631, 1995.

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972.