I have a flat object (not totally flat (let's say in range of 25µm)) which I measured two times (The measuring concept is not important here) with applying a tilt between the two times.
I have the normals in each point of the surface and I want from these normals to know the tilt that has been applied.
My approach was to calculate the average normal of each one and then calculate the angle between the normals.
Could you please suggest for me another solution or confirm mine?!
Many thanks in advance
Consider a very simple two-dimensional case where you find two normals each, and let the unit normals be vectors of the form $(x, y)$ and $(x + dx, y + dx)$ before tilting and $(u, v)$ and $(u + du, v + dv)$ after tilting, where $dx, dy, du, dv$ are errors.
Then the respective averages are $\dfrac{1}{2}(2x + dx, 2y + dy)$ and $\dfrac{1}{2}(2u + du, 2v + dv)$. Their scalar product (dot product) is:
$ \dfrac{(2x + dx)(2u + du)} 4 + \dfrac{(2y + dy)(2v + dv)} 4 \\ = xu + yv + \boxed{\dfrac{x\,du + u\,dx + y\,dv + v\,dy} 2 + \dfrac{dx\,du + dy\,dv}4} $
Another option is to find the scalar products of the respective vectors, and then take the average. This gives:
$\dfrac{(xu + yv) + [(x + dx)(u + du) + (y + dy)(v + dv)]} 2 = xu + yv + \boxed{\dfrac{x\,du + u\,dx + y\,dv + v\,dy} 2}$
(Of course, the angle is found by finding $\arccos$ of the scalar product).
The boxed quantities are the errors. Observe that in the first case (in your method), there is one extra (second-order) error term. This may indicate that the second method (that of finding the scalar product of the respective normals and then finding the average) may be better. But the usual practice is to ignore this as the first order error terms are the same. But you may want to experiment with both methods. (The same reasoning can be extended to three dimensions, and more normals).