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Suppose I have one convex polyhedron, I need to detect back face. And $N=(A, B, C)$ is normal vector of polygon surface and vector $V$ in viewing direction. According to books formula and internet content if $V. N>0$ then polygon is back face otherwise front face.

We know that $V. N=|V|.|N|cos\theta$ from this when we consider $\pi/2<\theta<=\pi$ then $V. N<0$ $\implies$ front face. And when we consider $0°<=\theta<=\pi/2$ then $V. N>0$ $\implies$ back face.

Case:1 Now suppose in this case for front face where $V$ and $N$ are parallel to each then they makes angle 0°.Then we get $V. N>0$ but according to books formula when $V. N<0$ then we will get front face.

:enter image description here

Case:2 Again I consider the case for back face where $V$ along with $N$ then they makes angle 180°.Then we get $V. N<0$ but according to books formula when $V. N>0$ then we will get back face. enter image description here

My question is where I mistaken to understand the detect the back face and front face for the above cases with formula $V. N >0 or <0?$

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Now suppose in this case for front face where V and N are parallel to each then they makes angle 0°.

In your example, V and N are pointing in opposite directions. The angle between them is 180 degrees, so the cosine is -1. Similarly:

Again I consider the case for back face where V along with N then they makes angle 180°.

If V and N are pointing in the same direction, then the angle between them is 0 degrees, so the cosine is 1.

You have these backwards; that's why you're getting the wrong value.

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    $\begingroup$ if I consider N for front face with the same direction of V that means if I consider N is opposite side of front face, then V and N has 0° angle, then above formula flipped for front face and back face? $\endgroup$
    – Alok Maity
    Jan 27 at 20:48
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    $\begingroup$ @Sagorika The formula is sensitive to the way you calculate N from the points of the face. If N = (A,B,C), then also -N = (C,B,A). So when checking each face, you must be careful to order the points correctly and consistently. $\endgroup$
    – luser droog ♦
    Jan 27 at 23:35
  • $\begingroup$ @luser droog if I consider N for front face with the same direction of V that means if I consider N is opposite side of front face, then V and N has 0° angle, then above formula flipped for front face and back face, that means front face formula and angles becomes back face formula and angle and vice versa? Am I correct? $\endgroup$
    – Alok Maity
    Jan 28 at 6:32
  • $\begingroup$ @Sagorika: Your comment didn't make any more sense the second time you repeated it than the first. N cannot be both the front face and the opposite side of the front face at the same time. So your question is gibberish. $\endgroup$
    – Nicol Bolas
    Jan 28 at 7:03
  • $\begingroup$ @Nicol Bolas I am saying instead of N pointing towards me, it pointing opposite side of the front face then front face formula and angle flipped with back face formula and angle? $\endgroup$
    – Alok Maity
    Jan 28 at 7:22

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