Relative Positions of Three Planes in Space.
(and types of solution of systems of three equations in three unknowns)

 

The equation that defines a plane in 3D space is:

Ax + By + Cz + D = 0

where A, B, C y D are 4 fixed  real numbers that determine the position of the plane in space.

All sets of three real numbers x, y, z which satisfy this equation belong to the plane.  Someone could choose, for example, any couple of real values, one for the x and one for the y and compute easily the coordinate  z of the set (x, y, z) of one of the infinite points in space that belongs also to the plane.

Let's see how many different configurations can adopt 3 planes in space:

 

The systems of three equations in three unknowns have no solution (4 cases).

1) Three parallel Planes.

2) Two of the Planes are parallel.

z = -.761x - .236y + 11.272 (darker green)
z = -.761x - .236y + 5.979 (medium green)
z = -.761x - .236y - 5.184 (lighter green)

The three Planes do not share any point.
z = -2.761x - 2.236y + 1.272 (darker green)
z = -1.761x + 1.236y + 7.979 (medium green)
z =
-1.761x + 1.236y - 5.184 (lighter green)

The three Planes do not share any point.
   

3) Two Planes cut in a line, third is parallel to this line.

4) Two Planes overlap, third is parallel.

z = .473x + .805y + 8.324 (darker green)
z = -2.268x + 3.472y + 19.81 (medium green)
z = -.1316x  + 1.393y + 6.557 (lighter green)

The three Planes do not share any point.
z = -2.013x + 1.205y - 4.582 (darker green)
z = -2.013x + 1.205y - 4.582 (medium green)
z = -2.013x + 1.205y + 12.582 (lighter green)

The three Planes do not share any point.
   

The systems of three equations in three unknowns have infinite solutions (3 cases).

5) The three Planes overlap.

6) The three Planes intersect in a line.

z = -2.014x + 1.205y - 4.582 (darker green)
z = -2.014x + 1.205y - 4.582 (medium green)
z = -2.014x + 1.205y - 4.582 (lighter green)

The three Planes share all its points.
z = .473x + .805y + 8.324 (darker green)
z = -2.268x + 3.472y + 19.81 (medium green)
z = -.1316x + 1.393y + 10.86 (lighter green)

The three Planes share a line.
   

7) Two Planes overlap, the other cuts them.

 
 
z = -2.013x +1.205y - 4.582 (darker green)
z = -2.013x +1.205y - 4.582 (medium green)
z = .843x - 0.101y - 2.582 (lighter green)

The three Planes share a line.
 
   

The systems of three equations in three unknowns have one solution (1 case).

8) The three Planes intersect at a point.

 
 
z = -1.553x - 2.642y - 10.272 (darker green)
z = 1.416x - 1.92y - 10.979 (medium green)
z = -.761x - .236y - 7.184 (lighter green)

The three Planes share one point.
 
   
If we put three Planes in 3D space, we can see that they can adopt different relative positions.  The three Planes can be parallel (case 1), the three Planes can intersect in a line (case 6), the three Planes can intersect at a point (case 8)...

In this page we can see that there are eight of this different relative positions.

The equation that defines a Plane in 3D space is:

A1x + B1y + C1z + D1 = 0

you can write it also this way:

C1z = - A1x - B1y - D1

If we divide all the coefficients by C1 (the Plane does not change doing this) we obtain:

z = Ax + By + C      (where A = -A1/C1, B = -B1/C1 y C = -D1/C1)

In four of the eight different relative positions there are no points shared at once by the three Planes.  The system of three equations in three unknowns has no solution.

In three of this positions the Planes share an infinite number of points.  The system of three equations in three unknowns has infinite solutions.

In one of the positions, the three Planes share only one point (they intersect at that point).  The system has one solution.

It is very easy to find a system of three equations in three unknowns for each one of the eight different relative positions, except for one: case 6 (where the three Planes intersect in a line).

(Case 3 is almost the same as case 6 except for one thing: one of the planes is displaced paralell to itself so that it stops touching the line shared by the three planes.)

To find a Plane parallel to one given you only need to add a number to the variable C, that is, add a number to the right expression in the formula: z = Ax + By + C

Two overlapping Planes have, of course, the same formula: z = Ax + By + C

It is not very difficult that three Planes would intersect at only one point, if the coefficients Ai, Bi and Ci in the three equations are randomly chosen real numbers, the probability to obtain this (case 8) is one.

It is easy to see that, given an arbitrary Plane, the probability that other randomly chosen Plane would be parallel to it is zero.

In case 6 (three Planes that intersect in a line), you need to calculate the coefficients of the third Plane in order that it contains the cutting line of the other two, the calculations aren't that difficult.  Given, for example, the Planes:

z = A1x + B1y + C1
z = A2x + B2y + C2

if the third Plane must contain the cutting line of the other two, its coefficients A3, B3 and C3:

z = A3x + B3y + C3

must be of the form:

A3 = (k1 * A1 + k2 * A2) / (k1 + k2)
B3 = (k1 * B1 + k2 * B2) / (k1 + k2)
C3 = (k1 * C1 + k2 * C2) / (k1 + k2)

where k1 and k2 are two real numbers chosen at will.

This is the same than saying that, given two Planes (in the equivalent form we saw at the begining):

A1x + B1y + C1z + D1 = 0
A2x + B2y + C2z + D2 = 0

the third Plane must be of the form:

k1(A1x + B1y + C1z + D1) + k2(A2x + B2y + C2z + D2) = 0


The drawings have been made with the program
3D Grapher.  Download here a file generated by this program.  This file is the last example (case 8).  The one where the three Planes intersect at a point.
josechu2004@gmail.com
 

First post in: 2003-04-11
Last update
: 2006-07-31