Trying to calculate the slope of a plane from a normal

Question

I have a set of 3D normal vectors for points on a 3D mesh, and I need to calculate the slope of the area below each of them. I have no idea how to do this. I don't need X or Y slope, I just need the total incline of the point in question (although to be fair, I don't know how to derive total slope from X and Y slope individually, which is part of my problem). I did see This article, but I couldn't really make heads or tails of it... The vectors are outward-facing. If anyone can explain this one to me, I'd be really grateful.

Answer 1

If you already have the normal vector, you're almost there. What you now need is the angle (look for dot product) between the normal and a vertical line (what exactly vertical means depends on your application).

If your normal vectors are actually normalized (have length 1) and the vertical is (0 0 1), the cosine of the slope angle is simply the z coordinate of the normal vector.

To demonstrate this: Take a pen and let it stand on your table. This is your table's normal vector. The angle between this vector and a vertical line is zero, as your table has no slope at all. If you tilt your table by a certain amount, the angle between the normal and a vertical line will increase by the same amount.