I have a question about mathmatical notation/correctness. I am writing a report about software. This software contains a class that contains a 3D-model with points in a point cloud. After this I used a function that estimates normal unit vectors of these points. Now I am wondering how to put this correctly in my report.
Should I create two sets:
- A set with just the points $P_i$ in the point cloud. Let's call this set S. So $S = \left \lbrace P_i \right \rbrace$.
- A set with the normal vectors (containing the points $P_i$ in $S$ as position vectors) and the normal vectors $\mathbf{n_i}$ as direction vectors. Let's call this set $T$. So: $T = \left \lbrace S + \alpha \cdot n_i \right \rbrace$.
Is this the way to go, or is it possible to say that the points $p_i$ in the set $S$ contain the normal vectors?
I must admit I am not entirely sure what your normal vectors are. Does your function create some normal unit vector $n_{i}$ with respect to the position vector $P_{i}$? And then $T$ is the set containing these normal vectors and the position vectors? In that case personally I would either create a set $$N:=\{n_{i}:n_{i}\cdot P_{i}=0,\|n_{i}\|=1\}$$ containing the normal unit vectors and say $T=S\cup N$, which is the set containing both the position vectors and the normal unit vectors. Or if you want to avoid as much mathematical notation as possible you could use $$T=\{n_{i},P_{i}:i=1,...,m\}$$ where $P_{i}$ are the position vectors and $n_{i}$ is a normal unit vector with respect to $P_{i}$.
In general the notation $A+B$ means $$A+B=\{a+b:a\in A,b\in B\}$$ so this is probably not what you want. I would interpret your notation of $T$ as the set of all sets $$S+\alpha n_{i}=\{s+\alpha n_{i}:s\in S\}.$$
I hope this helps, if you give a bit more clarity on what exactly $T$ is I'd be happy to update my answer.