new post about distances

posts/2016-08-29-distances-redux.md

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+---
+title: Distance Measures Redux':' Unifying and Tweaking
+tags: geometry
+description: Revisiting discrete distance measures and tweaking them
+---
+
+After [talking about distance
+calculations](/posts/2016-08-25-different-distances.html) the other
+day, I was thinking about how we could generalize the measures.
+
+Looking back on the D&D distance calculation, I think it can encompass
+civ and Manhattan distances, too.
+
+![The shortest taxi drive](/images/post_2016_08_29/taxi_path.png){width=150 height=150}
+
+First, for simplicity's sake, I'm going to refer to $x_d$ instead of
+$|x_1 - x_2|$, and likewise for $y_d$. With that convenience, I'll
+contract it to a single line rather than spreading it over
+three. Finally, I'm going to discard the floor conversion ($\lfloor
+\rfloor$) for the moment.
+
+$$d = \max(x_d, y_d) - \min(x_d, y_d) + 1.5 \times (\min(x_d, y_d))$$
+
+Note that "1.5". I think that's the trick. If we manipulate that, we
+should be able to replicate the other, simpler calculations.
+
+Remember that the 1.5 represents the cost of making a diagonal rather
+than a cardinal move. Since taxis can only move cardinally, they take
+two steps to move diagonally. Let's plug "2" into that and see what
+happens:
+
+$$d = \max(x_d, y_d) - \min(x_d, y_d) + 2 \times (\min(x_d, y_d))$$
+
+The $\min$ cancels out a bit:
+
+$$d = \max(x_d, y_d) + 1 \times (\min(x_d, y_d))$$
+
+Which further simplifies:
+
+$$d = x_d + y_d$$
+
+Which is the Manhattan formula. Success!
+
+
+![The shortest discrete drive](/images/post_2016_08_29/other_path.png){width=150 height=150}
+
+Let's try it for the Civ distance. Where taxis take 2 moves to go
+diagonally, civilizations do it in 1:
+
+$$d = \max(x_d, y_d) - \min(x_d, y_d) + 1 \times (\min(x_d, y_d))$$
+
+Those $\min$'s cancel out completely:
+
+$$d = \max(x_d, y_d)
+
+... which is, again, exactly how we calculate Civ distance. Excellent,
+well done!
+
+
+That leaves us with the general formula:
+
+$$d = \max(x_d, y_d) - \min(x_d, y_d) + k \times (\min(x_d, y_d))$$
+
+Where $k$ is the diagonal cost.
+
+### Tweaking
+
+[![A D&D distance Voronoi diagram](/images/post_2016_08_25/dandd_voronoi.png "A D&D distance Voronoi diagram"){width=150 height=150}](/images/post_2016_08_25/dandd_voronoi.png)
+
+Remember that the D&D distance ended up being closest to Euclidean. If
+$x_d$ and $y_d$ are equal -- that is, if we're going at exactly a
+45-degree angle -- the Euclidean distance is $x_d \times
+\sqrt(2)$. That $sqrt(2)$ is 1.41421....
+
+Last post, I mused that, since 1.5 was much closer to $sqrt(2)$ than
+either 1 (civ) or 2 (taxi), that is why D&D distance was such a good
+Euclidean approximation.
+
+But, could it be better? If we plug $k=1.4$ into our general distance
+equation, does it come closer?
+
+Well, so I wrote a script to test this. It's [up in the same repo as
+last time](https://github.com/jmelesky/voronoi_example).
+
+
+It generates a set of random points, then calculates the Euclidean
+distance from $(0,0)$ to each point. For each of the other measures,
+it calculates the same distance, and compares it to the Euclidean,
+keeping the difference. At the end, it prints out the total difference
+for each measure.
+
+[![A Voronoi diagram witk k=1.4](/images/post_2016_08_29/tweaked_voronoi.png "A Voronoi diagram with k=1.4"){width=150 height=150}](/images/post_2016_08_29/tweaked_voronoi.png)
+
+It looks a bit like this:
+
+~~~ { .bash }
+$ ./test_distances.py 
+taxi  diff: 12848
+civ   diff: 5146
+dandd diff: 3851
+other diff: 2071
+~~~
+
+As you can see, the "other" calculation ($k = 1.4$) has the least
+difference from the Euclidean distance, and so is a nice, quite
+accurate measure.
+
+1.4 is a tricky number. If you wanted to use it in a game of some
+sort, though, you could cost it out differently -- the thing that
+matters is the ratio. For example, it might cost 5 action points to
+move in a cardinal direction, but 7 action points to move
+diagonally. That would be quite true to Euclid, while nicely operating
+in our discrete movement world.
+
+Thanks for reading through, and hope it was interesting for you, too.