# In the plane equation, what is D?

Hi all,

Just a simple question about planes in general. The explict equation is Ax + By + Cz + D = 0. I know that the normal of the plane is formed by [ A B C], but what is D?

I don’t think there’s anything particular special about D. In fact, if you “scale” the equation, D can be pretty much any constant. In other words, the plane x + y + z + 1 = 0 is the same plane as 23x + 23y + 23z + 23 = 0.

One thing possibly worth mentioning:

The x,y, and z intercepts of the plane will be -D/A, -D/B, -D/C, respectively (provided A, B, C are nonzero

It’s unitless (as compared to whatever unit of measure “X”, “Y”, and “Z” have), so I don’t think it adds anything if you’re trying to graph it on a Cartesian plane . . . It’s just a meaningless constant–which it’s been too long for me to remember what that constant does to the rest of the function (other than add to a value). I would think that you’d have to multiply it by “A”, “B”, or “C” or all three:

D(Ax + By + Cz) = 0.

Tripler
Man, it’s been awhile . . . yet it’s so simple! :smack:

I just came across http://mathworld.wolfram.com/Plane.html and it mentioned that you can find the distance of the plane from the origin. This is a ‘huh’ moment for me for I thought all planes are infintely long. What do you mean by the distance from the origin to the plane? Or is it to a point on the plane?

Thanks ago, from a maths idiot…

So, “D” is just an offset of the origin? What dimension is that it? I would think an offset of the origin needs to have some unit, be it “X”, “Y”, or “Z”, and “D” would be a value of one (or multiple) of those three units. . .

Weird. :dubious:

Tripler
I once took a class in Orbital Mechanics. This however, does not make me a “Rocket Scientist”.

I think D = d / sqrt(a^2 + b^2 + c^2) where d is the D which I asked you about (yah, I got confused too)

Notice that D isn’t the distance the plane is from the origin; the distance the plane is from the origin is D / sqrt(A[sup]2[/sup] + B[sup]2[/sup] + C[sup]2[/sup]). (Note that the denominator can’t be zero–otherwise you didn’t really have an equation for a plane in the first place).

(The units for this distance are simply the same units you’re using for your coordinate axes).

As for what that means: Pick any point at random on the plane; certainly you can measure the distance from this point to the origin.

Imagine you do that for all of the points on the plane. There’s going to be a smallest such distance–this is defined to be the distance the plane is from the origin.

Equivalently, the distance from the plane to the origin is given by: Take a line through the origin and normal to the plane. Measure the distance along this particular line to get the distance from the plane to the origin.

Bottom line is that D does have a special significance if you normalize the equation so that sqrt(A[sup]2[/sup] + B[sup]2[/sup] + C[sup]2[/sup]) = 1. In that case, D is the distance from the plane to the origin.