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Being $\bar{\mathbf{x}} \in \mathbb{R}^3$ a point and $S =\{\mathbf{x}\}_{i=1}^N \in \mathbb{R}^3$ a sample of N points. I am looking for a simple algorithm to determine the nearest point in $S$ in respect to $\bar{x}$ along a specific direction $\mathbf{\hat{u}}$

Thanks

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  • $\begingroup$ What do you mean nearest point along a specific direction? Is $S$ an unstructured point cloud? If yes, your definition is ill-posed. There might be no point at all in $\hat{\mathbf{u}}$ direction with respect to $\bar{\mathbf{x}}$. $\endgroup$
    – Mithridates the Great
    Commented Oct 23, 2019 at 21:30
  • $\begingroup$ Do you accept an approximate solution? Well, if $S$ is an unstructured point cloud, which I assume it is indeed due to your notation of labeling points by their IDs, you need to accept just an approximate solution here. Please clarify a bit more here. $\endgroup$
    – Mithridates the Great
    Commented Oct 23, 2019 at 21:33

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The question really boils down to how far two points $x,y$ are from each other in direction $u$. This is easily answered: You need to compute the component of $y-x$ onto $u$, i.e., their (signed) distance in direction $u$ is $\frac{(y-x)\cdot u}{\|u\|}$.

For your point cloud, you can then compute for each $i$ the (unsigned) distance $$ d_i = \frac{(\bar x-x_i)\cdot u}{\|u\|} $$ and then you just need to find that $i$ for which $d_i$ is smallest.

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