I'm looking at "Fast Triangle-Triangle Intersection Test" by Tomas Moller, this is a method for computing whether or not two triangles intersect (Paper is here: http://web.stanford.edu/class/cs277/resources/papers/Moller1997b.pdf) My question in context of paper on the first page and linked with equation $(1)$
I'm not good at analytic geometry and I don't understand why in plane equation: $N \cdot X + d = 0$ we find distance $d$ from plane to origin like a dot product of $-N \cdot V $ where $N$ is a normal vector and $V$ is a radius vector of a point lying at plane. First of all, why minus? Maybe someone can explain? Any help would be greatly appreciated.
If you have three points L,M,N that is enough to determine a plane $P$ which passes through them. Simple compute two vectors for which you will use to span the plane say:
$$\vec{LM} = M-L \ \ \textrm{and} \ \ \vec{LN}= N-L$$
where the equalities come from definition. Next, you need to compute a normal vector for the plane, which can be found using cross product;
$$\vec{n} = \vec{LM} \times \vec{LN}$$
Therefore your plane equation is given by the set of points $(x,y,z) =X$ such that;
$$ (X- M) \cdot \vec{n} = 0 \iff X \cdot \vec{n}- M \cdot \vec{n} = 0$$
Here we see $d = -M \cdot \vec{n}$ and by abuse of notation we think of $X,M$ as vectors as well. Also, in the above you can choose $M,N$ or $L$, recalling that $X-M$ denotes the vector extending from $X$ to $M$.