posts: kd-tree-revisited: add presentation

content/posts/2024-08-17-kd-tree-revisited/index.md

@@ -20,3 +20,21 @@ neighbour search (i.e: the `closest` method) for the `KdTree` I implemented in
 [my previous post]({{< relref "../2024-08-10-kd-tree/index.md" >}}).
 
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+
+## The improvement
+
+That post implemented the nearest neighbour search by keeping track of the
+tree's boundaries (through `AABB`), and each of its sub-trees (through
+`AABB.split`), and testing for the early exit condition by computing the
+distance of the search's origin to each sub-tree's boundaries.
+
+Instead of _explicitly_ keeping track of each sub-tree's boundaries, we can
+implicitly compute it when recursing down the tree.
+
+To check for the distance between the queried point and the splitting plane of
+inner nodes: we simply need to project the origin onto that plane, thus giving
+us a minimal bound on the distance of the points stored on the other side.
+
+This can be easily computed from the `axis` and `mid` values which are stored in
+the inner nodes: to project the node on the plane we simply replace its
+coordinate for this axis by `mid`.