"Ax + By + Cz = 0 is the equation of a plane with a similar tilt but shifted such that it goes through the origin (0, 0, 0). To get the value of D, shift the equation with one of the Q, R, S points.

A(x - Qx) + B(y - Qy) + C(z - Qz) = 0

You now have a Ax + By + Cz = D equation."

Can you please enlighten me on solving/creating D? "To get the value of D, shift the equation with one of the Q,R,S points." What do you mean by shift the equation? (It would seem elementary but I need help!)

<Qx,Qy,Qz> is a point in R3. By subtracting its components from the x, y, and z terms, you've moved the plane's "origin" so the plane now contains Q. If it were a line we'd say it "passes through" Q but that phrase doesn't make as much sense here.

Doing that shift makes some constant terms fall out, collect them on the right and you should have D = AQx + BQy + CQz

So, D is a symbol that groups the constant terms in the equation once a point is substituted. Makes sense. But the timing of the D seems to be arbitrary. I've seen the plane equation below initially with and without D, but ending up with it later.

Plane eq:
n.p = 0 (sometimes n.p+D = 0 , but will use the former)
n.(p - p1) = 0
<a,b,c> . <x - x1,y - y1,z - z1 >
a(x - x1) + b(y-y1)+ c(z-z1) = 0
(d not yet, regroup)
ax -ax1 + by - by1 +cz -cz1 = 0
ax + by +cz -ax1 - by1 - cz1 = 0
ax + by + cz = ax1 + by1 + cz1
(D appears magically, to include the constants on the right)
d = ax1 + by1 + cz1
ax +by1 + cz1 - d = 0
1. how did I get my sign wrong on d
2. am I restricted in my points I choose for d

Note - writing math on a phone is pants so I'll clean up this post when I'm at a bigger keyboard.

ax + by +cz -ax1 - by1 - cz1 = 0
ax + by + cz = ax1 + by1 + cz1
(D appears magically, to include the constants on the right)
d = ax1 + by1 + cz1
ax +by1 + cz1 - d = 0
1. how did I get my sign wrong on d

It'sa matter of when you introduced the new variable, since we move terms from one side of the equation to the other by subtraction.
it is not recommended to introduce the D without warning

Since the general equation has all the terms on the same side you should arrange your work into the same form and then substitute out all the constant terms after that step but before moving them over to the RHS.
(x coefficient)*x + (y coefficient)*y + (z coefficient)*z + (whatever is left) = 0

You've quoted two similar general forms, and either works, algebraicly speaking,
(1) n*p + D = 0
(2) Ax + By + Cz = D
but they differ in the sign of D. Again, you can define things either way but keep the same definition the whole way through. I prefer (1) but you'll get points for using what your textbook started with. ; )

2. am I restricted in my points I choose for d

No, you can shift by any point in 3-space. The entire coordinate system is your Real-valued oyster. Go wild.

Forum > General Discussion > actual math/programming question "shift the equation"