On the [wikipedia entry for k-d trees](http://en.wikipedia.org/wiki/Kd-tree#Nearest_neighbor_search), an algorithm is presented for doing a nearest neighbor search on a k-d tree. What I don't understand is the explanation of step 3.2. How do you know there isn't a closer point just because the difference between the splitting coordinate of the search point and the current node is greater than the difference between the splitting coordinate of the search point and the current best? 

> 
> Nearest neighbor search Animation of
>  NN searching with a KD Tree in 2D
> 
> 
> The nearest neighbor (NN) algorithm
>  aims to find the point in the tree
>  which is nearest to a given input
>  point. This search can be done
>  efficiently by using the tree
>  properties to quickly eliminate large
>  portions of the search space.
>  Searching for a nearest neighbor in a
>  kd-tree proceeds as follows:
> 
> 
> 1. Starting with the root node, the algorithm moves down the tree
>  recursively, in the same way that it
>  would if the search point were being
>  inserted (i.e. it goes right or left
>  depending on whether the point is
>  greater or less than the current node
>  in the split dimension).
> 2. Once the algorithm reaches a leaf node, it saves that node point as
>  the "current best"
> 3. The algorithm unwinds the recursion of the tree, performing the
>  following steps at each node:
>  1. If the current node is closer than the current best, then it
>  becomes the current best.
>  2. The algorithm checks whether there could be any points on
>  the other side of the splitting plane
>  that are closer to the search point
>  than the current best. In concept,
>  this is done by intersecting the
>  splitting hyperplane with a
>  hypersphere around the search point
>  that has a radius equal to the current
>  nearest distance. Since the
>  hyperplanes are all axis-aligned this
>  is implemented as a simple comparison
>  to see whether the difference between
>  the splitting coordinate of the search
>  point and current node is less than
>  the distance (overall coordinates)
>  from the search point to the current
>  best.
>  1. If the hypersphere crosses the plane, there could be
>  nearer points on the other side of the
>  plane, so the algorithm must move down
>  the other branch of the tree from the
>  current node looking for closer
>  points, following the same recursive
>  process as the entire search.
>  2. If the hypersphere doesn't intersect the splitting plane,
>  then the algorithm continues walking
>  up the tree, and the entire branch on
>  the other side of that node is
>  eliminated.
> 4. When the algorithm finishes this process for the root node, then the
>  search is complete.
> 
> 
> Generally the algorithm uses squared
>  distances for comparison to avoid
>  computing square roots. Additionally,
>  it can save computation by holding the
>  squared current best distance in a
>  variable for comparison.
> 
> 
>