I have a plane represented as a 3D dataset and a normal to that plane
I'd like to represent that plane in hesse normal form in order to find the d (distance) in:
$\vec{r}.\vec{n}$ = d.
where $\vec{n}$ is the normal vector, and $\vec{r}$ is the position vector of point $\vec{P}$ that lies on the plane (I took the centroid point). Assuming that the origin is $(0,0,0)$, then the equation would be
$\vec{P}\cdot\vec{n}$ = d.
is that correct?
If the answer is yes, then I'm even more confused, since if I used another point that lies on the plane and dot product it with the normal then I'll get a different d result. Then how come that the hesse normal form is fullfilled by each point lying on the plane?
Best Answer
If $\vec r$ is a point on the plane, and $\vec n$ is the normal to the plane, then $\vec r\cdot\vec n=d$ is constant for all points on the plane. It is not possible to get a different value of $\vec r\cdot\vec n$ for other points on the plane. If you take another point $\vec s$ and get $\vec s\cdot\vec n\ne\vec r\cdot\vec n$, then $\vec s$ does not lie on the plane through $\vec r$ normal to $\vec n$, because it means that $(\vec s-\vec r)\cdot\vec n\ne0$, so the vector from $\vec r$ to $\vec s$ is not normal to $\vec n$.