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The Intersection Of Three Planes
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By: Debra Woods
12/1993
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Introduction to three dimensional space
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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.
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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.
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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.
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Introduction to planes in space
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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.
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Boxed->False,Axes->True];
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The intersection of planes
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Intersection of two planes
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Of particular interest is the intersection of planes. First, let's investigate the intersection of two planes.
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plane2 = Graphics3D[Polygon[{{2,2,0},{2,-2,0},{-2,-2,0},
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plane3 = Graphics3D[Polygon[{{3,2,2},{3,2,-2},{3,-2,-2},
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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.
:[font = subsection; inactive; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup]
Intersection of three planes
:[font = smalltext; inactive; preserveAspect; fontName = "CalcMath"]
What would the intersection of three planes be? Let's investigate several cases.
:[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup]
plane4 = Graphics3D[Polygon[{{2,0,2},{2,0,-2},{-2,0,-2},
{-2,0,2}}]];
Show[plane1, plane2, plane4,Boxed->False];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 265; pictureHeight = 286; fontName = "CalcMath"; endGroup]
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.39554 .61987 L .656 .751 .92 r
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.002 w
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.52487 .57846 L .656 .751 .92 r
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.002 w
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F P
s
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p
.52487 .57846 m .09571 .50305 L .65569 .72122 L .656 .751 .92 r
F
P
p
.002 w
.6664 .9726 m .65569 .72122 L p
.52487 .57846 L .966 .687 .452 r
F P
s
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.002 w
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.52487 .57846 L .544 .324 .532 r
F P
s
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.002 w
.27286 .65915 m .25127 .91705 L p
.52487 .57846 L .544 .324 .532 r
F P
s
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.002 w
.09571 .50305 m .27286 .65915 L p
.39554 .61987 L .656 .751 .92 r
F P
s
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.002 w
.64649 .50519 m .52487 .35349 L p
.52487 .57846 L .966 .687 .452 r
F P
s
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p
.002 w
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.66137 .53476 L .544 .324 .532 r
F P
s
P
p
.002 w
.52487 .35349 m .29129 .43897 L p
.52487 .57846 L .544 .324 .532 r
F P
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P
p
.002 w
.91647 .64728 m .80568 .48856 L p
.52487 .57846 L .656 .751 .92 r
F P
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P
p
.83276 .76234 m .52487 .35349 L .52487 .57846 L .544 .324 .532 r
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.52487 .57846 L .966 .687 .452 r
F P
s
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p
.002 w
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.52487 .57846 L .656 .751 .92 r
F P
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p
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F P
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p
.002 w
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F P
s
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p
.002 w
.36503 .40405 m .34875 .68456 L p
.44914 .49584 L .966 .687 .452 r
F P
s
P
p
.52487 .57846 m .37855 .171 L .52487 .84426 L .966 .687 .452 r
F
P
p
.002 w
.66876 .2924 m .36503 .40405 L p
.52487 .57846 L .656 .751 .92 r
F P
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P
p
.002 w
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.66137 .53476 L .544 .324 .532 r
F P
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p
.002 w
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.52487 .57846 L .656 .751 .92 r
F P
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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?
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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}];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 273; pictureHeight = 276; fontName = "CalcMath"; endGroup]
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.002 w
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.002 w
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.002 w
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.002 w
.24797 .47851 m .14048 .38408 L p
.48978 .5858 L .471 0 0 r
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p
.002 w
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.915 .607 .446 r
F P
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p
.002 w
.447 .65336 m .24797 .47851 L p
.48978 .5858 L .471 0 0 r
F P
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P
p
.002 w
.45373 .30972 m .24797 .47851 L p
.48978 .5858 L .551 .812 .994 r
F P
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P
p
.002 w
.89261 .5723 m .45373 .30972 L p
.48978 .5858 L .551 .812 .994 r
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.002 w
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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.
:[font = input; Cclosed; preserveAspect; fontName = "CalcMath"; startGroup]
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]];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 409; pictureHeight = 194; fontName = "CalcMath"; endGroup]
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F P
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F P
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F P
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.002 w
.64354 .62697 m .92907 .30368 L p
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F P
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.002 w
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.44359 .75888 L .866 .887 .849 r
F P
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.002 w
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F P
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.002 w
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.44359 .75888 L .963 .757 .577 r
F P
s
P
p
.44359 .75888 m .38066 .56747 L .30554 .6944 L .963 .757 .577 r
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p
.44359 .75888 m .30554 .6944 L .05302 .85264 L .866 .887 .849 r
F
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p
.44359 .75888 m .30554 .6944 L .21861 .84129 L .963 .757 .577 r
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.002 w
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F P
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.002 w
.22959 .47795 m .22419 .6564 L p
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F P
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.38066 .56747 m .22419 .6564 L .30554 .6944 L .963 .757 .577 r
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.002 w
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F P
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.002 w
.39725 .458 m .22419 .6564 L p
.53356 .55152 L .866 .887 .849 r
F P
s
P
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.002 w
.22419 .6564 m .05302 .85264 L p
.30554 .6944 L .866 .887 .849 r
F P
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F
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p
.002 w
.22419 .6564 m .21861 .84129 L p
.30554 .6944 L .963 .757 .577 r
F P
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p
.002 w
.64354 .62697 m .94089 .59935 L p
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.002 w
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p
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.38066 .56747 L .963 .757 .577 r
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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.
^*)