calculate arbitrary points from a plane equation I understand how one can calculate a plane equation (ax+by+cz=d) from three points but how can you go in reverse?
How can you calculate arbitrary points from a plane equation?

From your comment I finally understood what you are looking for:
If you have a plane defined by $a x + b y + c z = d$ then you also have the following properties:


*

*Plane normal direction: 
$$\hat{n} = \begin{pmatrix} 
  \frac{a}{\sqrt{a^2+b^2+c^2}} \\
  \frac{b}{\sqrt{a^2+b^2+c^2}} \\
  \frac{c}{\sqrt{a^2+b^2+c^2}} \end{pmatrix}$$

*Point on plane closest to the origin (position of plane)
$$ \vec{r} = \begin{pmatrix}
  \frac{a d}{a^2+b^2+c^2} \\
  \frac{b d}{a^2+b^2+c^2} \\
  \frac{c d}{a^2+b^2+c^2} \end{pmatrix} $$

*Distance of plane from the origin
$$ r = \frac{d}{\sqrt{a^2+b^2+c^2}} $$

*Directions along the plane (not unit vectors), and perpendicular to $\hat{n}$.
$$ \begin{align}
  \hat{e}_1 & = \begin{pmatrix}    c-b \\  a-c \\ b-a \end{pmatrix} &
  \hat{e}_2 & = \begin{pmatrix}  a (b+c)-b^2-c^2 \\ b (a+c) -a^2-c^2 \\ c (a+b)-a^2 - b^2  \end{pmatrix} \end{align} $$
You can verify that $\hat{e}_1 \cdot \hat{n} =0$, $\hat{e}_2 \cdot \hat{n}=0$ and $\hat{e}_1 \cdot \hat{e}_2 =0$, where $\cdot$ is the dot (inner) product.
Confirmation via GeoGebra

NOTES: Please edit the equation to make it clear you are looking for the plane properties when given a plane in equation form.
