The Intersection Of Three Planes

By:  Debra Woods
12/1993

Introduction to three dimensional space

You are already familiar with the 2-dimensional coordinate system, that is ordered pairs.  We will now add another dimension, and investigate the 3-dimensional coordinate system.  The 3-D coordinate system takes the xy-plane that you are familiar with and adds another axis that is orthogonal (perpendicular) the both the x-axis and the y-axis.  In this lesson, we will investigate the graphs of points, planes, and lines in space.  To start, let's look at the 3-D axes.

h=5;
spacer=h/10;
threedims=
	Graphics3D[{
			{Line[{{-h,0,0},{h,0,0}}]},
				Text["x",{h + spacer,0,0}],
			{Line[{{0,-h,0},{0,h,0}}]},
				Text["y",{0, h+spacer,0}],
			{Line[{{0,0,-h},{0,0,h}}]},
				Text["z",{0,0,h + spacer}]},
				Boxed->False,Axes->False,
				ViewPoint->{1.75,1.224,1.7}
				,AspectRatio->Automatic];
Show[threedims];

The points in 3-D space are ordered triples (x, y, z).  The following example shows plots of the points (2,3,1) in red, and (3,-2,-1) in green.

pts = Graphics3D[{{RGBColor[1,0,0],PointSize[0.02],
					Point[{2,3,1}]},{RGBColor[0,1,0],
					PointSize[0.02],Point[{3,-2,-1}]}}];
Show[threedims,pts];

If you would like to plot some points of your own, change the coordinates of the points in the above mathematica code.  Try changing each coordinate separately to see what happens.

Introduction to planes in space

Planes in space have equations of the form  Ax + By + Cz = D.  Let's look at a plane.  In particular, the plane with equation 6x + 4y + 3z = 12.

planeplot = Graphics3D[Polygon[{{2,0,0},{0,3,0},{0,0,4}}],
					DisplayFunction->Identity];
Show[threedims,planeplot,DisplayFunction->$DisplayFunction,
		Boxed->False,Axes->True];

The intersection of planes

Intersection of two planes

Of particular interest is the intersection of planes.  First, let's investigate the intersection of two planes.

plane1 = Graphics3D[Polygon[{{0,2,2},{0,2,-2},{0,-2,-2},
							{0,-2,2}}]];
plane2 = Graphics3D[Polygon[{{2,2,0},{2,-2,0},{-2,-2,0},
							{-2,2,0}}]];
plane3 = Graphics3D[Polygon[{{3,2,2},{3,2,-2},{3,-2,-2},
							{3,-2,2}}]];
							
Do[Show[GraphicsArray[{
		Show[plane1, plane2,Boxed->False,Axes->False,
				DisplayFunction->Identity],
		Show[plane1, plane3,Boxed->False,Axes->False,
				DisplayFunction->Identity]}],
				DisplayFunction->$DisplayFunction]];

Notice that the intersection of two planes is either a line, if the planes are not parallel, or the empty set if the planes are parallel.

Intersection of three planes

What would the intersection of three planes be?  Let's investigate several cases.

plane4 = Graphics3D[Polygon[{{2,0,2},{2,0,-2},{-2,0,-2},
							{-2,0,2}}]];
Show[plane1, plane2, plane4,Boxed->False];

The preceeding example shows three planes intersecting at a point.  Let's see if there are other possibilities.  Could three planes intersect along a line?

r = Sqrt[2];
plane5 = Graphics3D[Polygon[{{2,2,2},{2,-2,-2},{-2,-2,-2},
							{-2,2,2}}]];
Show[plane5,plane2,plane4,Boxed->False,
	ViewPoint->{1.200, 1.370, 1.070}];

Yes!!  Now, what are the possibilities of three planes not intersecting at all?  The obvious answer is if they are all parallel, but could there be other possibilities?  Let's see.

plane6 = Graphics3D[Polygon[{{2,1,2},{2,1,-2},{-2,1,-2},
							{-2,1,2}}]];
Do[Show[GraphicsArray[{
		Show[plane2,plane5,plane6,Boxed->False,
		ViewPoint->{-3.590, -1.400, 1.070},
			DisplayFunction->Identity],
		Show[plane2,plane6,plane4,Boxed->False,
		ViewPoint->{-3.590, -1.400, 1.070},
			DisplayFunction->Identity]}],
			DisplayFunction->$DisplayFunction]];

To summarize, the intersection of three planes can be a point (that is an ordered triple (a,b,c)), or a line (that is at an infinite number of points), or the null set (no point of intersection).

Recall, that the equation of a plane was Ax + By + Cz = D.  That is an equation involving three unknown variables.  We will now move on the the algebra of solving three equations in three unknowns.