Fitted curve plane equation AX + BX + C = Z to point/normal form

Question

I have fitted plane equations in a binary tree form of AX + BX + C = Z. I'm trying to figure out how to convert this to a point/normal to use my standard plane libraries. Is there a simple way to do this?

Some data examples are: Z = -1.500000X + 0.000000Y + 61.500000

Left Z = 0.057341X + 0.959100Y + -23.197208

Right Z = -1.012730X + 0.222221Y + 83.254044

Answer 1

Given,

$$z=Ax+By+C$$

We may substitute $x=0$ and $y=0$ to get $z=C$. That means a point on the plane is $(0,0,C)$. We may also write,

$$Ax+By-z+C=0$$

$$A(x-0)+B(y-0)-1(z-C)=0$$

$$\langle A,B,-1 \rangle \cdot \langle x-0,y-0,z-C \rangle=0$$

The last equation is a point-normal form equation.