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I'm in the process of implementing this paper and I'm running into an issue with the process listed in portion V-A (page 5). Specifically, the paper mentions that I should store the distance from each cluster to its bounding hyperplanes.

Given that the hyperplanes are all defined for voronoi clusters and they are simply equidistant between two centroids. I assume there is an elegant mathematical solution for solving for d(x, H), where x is a point in a dataset bound on some side by hyperplane H. Assuming I already have the centroids that define that hyperplane, how would I go about calculating the distance from a point within one of those clusters to the hyperplane?

I'm not expecting a hand-fed answer, but this paper treats it as a trivial exercise, and I'm having difficulty implementing this aspect, upon which much of the paper relies. I started doing some trig style derivations, but I then was concerned that standard trig doesn't generalize to many dimensions. I am just really not sure where to start on this.

Outside of this paper, I suppose the definition of the problem would be:

Given centroids C1 and C2, and a a point x Determine the distance between point x and a hyperplane equidistant from points C1 and C2

Any help would be hugely appreciated.

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The base point to be used in the hyperplane can be $$ C = (C_1 + C_2)/ 2. $$ Next, we make a unit length vector, normal to the hyperplane, by $$ N = (C_2 - C_1)/ \parallel C_2 - C_1\parallel. $$ For some other point $X,$ the distance from $X$ to the hyperplane is $$ |(X-C) \cdot N| $$

You should draw some pictures; two dimensions is enough. We are in the $xy$ plane. Points $C_1 = (-1,0)$ and $C_2 = (1,0).$ The "hyperplane" is the $y$ axis. The fixed point is $C = (0,0).$ Finally, the normal is $N = (1,0).$ Suppose we have a new point $X = (3,5).$

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    $\begingroup$ Oh goodness. I was overcomplicating this tremendously. Thank you so much, it's been a long time since I actually did linalg. $\endgroup$ Dec 26 '14 at 3:15

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