Planes

Planes

A plane is really two points, one that defines a vector(ok I’m cheating, one of the points for the vector is always at 0, 0, 0) and a point in space through which the plane passes. A plane is an infinitely perfectly flat surface, which starts to become interesting when we combine its mathematics with points and other vectors. First we must look at the mathematics of a plane.

Plane Equation

If we know the vector of a plane and a point passing through the plane, the equation of the plane is established.

Point (A) = X\tiny1 \normalsize \space; \space Y\tiny1 \normalsize \space ; \space Z\tiny1 \normalsize \\
\overrightarrow{N} = I ; J ; K \\
Plane \space Equation =  I(x−X\tiny1 \normalsize)+J(y−Y\tiny1 \normalsize)+K(z−Z\tiny1 \normalsize)=0 \\

A plane passing through the point (A) = 1 ; 3 ; 2 and has normal vector (N) = 6 ; 7 ; 5, then the equation of the plane is;

Point (A) = 1 \space; \space 3 \space ; \space 2 \\
\overrightarrow{N} = 6 \space ; \space 7 \space ; \space 5 \\
Plane \space Equation =  6(x−1 \normalsize)+7(y−3 \normalsize)+5(z−2 \normalsize)=0 \\
= 6x - 6 + 7y-21+5z-10 = 0 \\
= 6x + 7y + 5z  -6 -21 -10 = 0 \\
= 6x + 7y + 5z  -37 = 0 \\

Does Point Lay on Plane

Now we have the equation of the plane we can check to see if another point (B) = 12 ; 8 ; 1 lays on the same plane, this is achieved by substituting the new point into the planes equation;

Point (B) = 12 \space; \space 8 \space ; \space 1 \\
Plane Equation = 6x + 7y + 5z  -37 = 0 \\
= ( 6 * 12 ) + ( 7 * 8 ) + ( 5 * 1 ) - 37 = 0 \\
= 72 + 56 + 5 - 37 = 0 \\
=96 \space {=}\mathllap{/\,} \space 0 \\

Since 96 does not equal 0 Point(B) does not lay on the plane. So lets try another example where the point lay’s on the plane, Point(C) = -4.83 ; 5.461 ; 5.555;

Point (B) = -4.83 \space; \space 5.461 \space ; \space 5.555 \\
Plane Equation = 6x + 7y + 5z  -37 = 0 \\
= ( 6 * -4.83 ) + ( 7 * 5.461 ) + ( 5 * 5.555 ) - 37 = 0 \\
= -28.98 + 38.227 + 27.775 - 37 = 0 \\
=0.022 \space \approxeq \space 0 \\

If we had used values that were not trimmed to three decimal points the final result would have been 0, so in this case we can say that the result is approximately zero so the point lays on the plane.

Creating an Arbitrary Point on the Plane

With a little luck and guess work we can create an arbitrary point on the plane by guessing at two of the values for the point and then using the equation to tell us the third value. Let try with Point(D) = 5 ; 5 ; Z;

Point (D) = 5 \space; \space 5 \space ; \space Z \\
Plane Equation = 6x + 7y + 5z  -37 = 0 \\
= ( 6 * 5 ) + ( 7 * 5 ) + ( 5 * Z ) - 37 = 0 \\
= 30 + 35 +5Z -37 = 0 \\
= 5Z = -30 -35 +37 \\
= 5Z = -28 \\
= Z = -28/5 \\
= Z = -5.6 \\

We have to be careful though lets take the Point (E) = 0 ; 0 ; 0 and has normal vector (N) = 1 ; 0 ; 0, then the equation of the plane is;

Point (E) = 0 \space; \space 0 \space ; \space 0 \\
\overrightarrow{N} = 1 \space ; \space 0 \space ; \space 0 \\
Plane \space Equation =  1(x−0 \normalsize)+0(y−0 \normalsize)+0(z−0 \normalsize)=0 \\
= 1x - 6 + 0y-0+0z-0 = 0 \\
= 1x + 0y + 0z  -0 -0 -0 = 0 \\
= 1x + 0y + 0z  -0 = 0 \\

If we use the same point(D) = 5 ; 5 ; Z let’s see if we can resolve for a point on the surface;

Point (C) = 5 \space; \space 5 \space ; \space Z \\
Plane Equation = 1x + 0y + 0z  -0 = 0 \\
= ( 1 * 5 ) + ( 0 * 5 ) + ( 0 * Z ) - 0 = 0 \\
= 5 + 0 +0Z - 0 = 0 \\
= 0Z = - 5 - 0 + 0 \\
= 0Z = -5 \\
= Z = -5/0\\
= Z = NAN (Not \space A \space Number)

So since you cant divide by Zero, there is no solution for this case and there should not be since the plane is an X-Plane running through 0 ; 0 ; 0 and we gave our arbitrary point an X value of 5.

So let’s try again this time resolving for X using Point(F) X ; 5 ; 5

Point (C) = X \space; \space 5 \space ; \space 5 \\
Plane Equation = 1x + 0y + 0z  -0 = 0 \\
= ( 1 * X ) + ( 0 * 5 ) + ( 0 * 5 ) - 0 = 0 \\
= 1X + 0 + 0 - 0 = 0 \\
= 1X = - 0 - 0 + 0 \\
= 1X = 0 \\
= X = 0/1\\
= X = 0

Success we now know that we can create an arbitrary Point(F) 0 ; 5 ; 5, and it will lay on the plane. As long as the result is not divided by zero or NAN (Not A Number) then we can resolve an arbitrary point by guessing.

This is really important when projecting a point onto a plane.