Determine Direction of Normal vector of Convex Polyhedron in 3D

Question

How can I determine direction(point inside or outside) of normal vector drawn on one side of the polyhedron?

Known informations;

• coordinates of all corners in 3d as x,y,z

• which face of the normal vector is drawn on

For example for this points A (6,0,2) B (6,0,0) C (6,1,0) D (6,1,2) E (0,0,0) F (0,1,0) G (0,1,2) H (0,0,2) and normal vector which is drawn on ABCD face.

I need to prove which direction the normal vector is drawn to.

Example image

Answer 1

As you are already given a normal to a face of the polyhedron, you don't need to find another normal as user247327 describes. They were probably confused by your misuse of the term "polygon", like I was until I saw your picture.

Since your polyhedron is convex, for any face, all the vertices not on that face will lie on the inward side. So if $V$ lies on the face in question, and $W$ does not lie on the face, then $W - V$ will point in the general direction of the interior (though not necessarily normal to the face).

Therefore if $\vec n$ is the normal you are testing, just take the dot product:

• If $\vec n \cdot (W - V) > 0$, then $\vec n$ and $(W - V)$ are both pointing to the same side of the face, towards the inside of the polyhedron.
• If $\vec n \cdot (W - V) < 0$, then $\vec n$ and $(W - V)$ are pointing to opposite sides of the face. So $\vec n$ is pointing outward.

If your polyhedron is not convex, then greater care is needed.