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Given a set of $n$ points $P$, a point $p_i\in{P}$, $1\leq i\leq n$ and a number $k<n$, I define the group $N_k(p_i)$ as the group containing $p_i$'s $k$ nearest neighbors. In addition, each point $p_i\in{P}$ has a surface normal $n_i$ associated with it, defining a surface on which this point lies.

Using $N_k(p_i)$, I want to approximate the surface. One way is to try and find circles centered on all the points in $P$. The radius for each point is defined by the following equation:

$$r_i=max_j||(p_j-p_i)-n_i^T(p_j-p_i)n_i||$$

This equation is given in a book without further explanations, and I am unable to decipher what type of vector product is being used (I have written the equation exactly as it is written in the book), and why $r_i$ is supposedly best for approximating the surface.

Any ideas would be greatly appreciated.

Edit: The question was solved by a friend of mine. The product above is a dot product. This way, $r_i$ is smaller the more the normals of $p_i$ and $p_j$ differ.

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