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+---
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+title: Different Measures of Distance
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+tags: geometry
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+description: A few different ways of calculating distance in noncontinuous 2d spaces
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+---
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+
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+How far is it from one place to another? How does that change if
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+you're on a flat plane? What if that plane has discrete distances,
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+like a checkerboard, or Civ map, or pixels on a screen?
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+
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+If we've got two points, *how far do you have to move to get from one
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+to the other?*
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+
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+
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+### Euclidean Distance
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+
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+{width=150 height=150}
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+
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+This is the go-to of distance calculations. If you're talking about
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+continuous, uniform planar (2d) space^[It holds for 3d and
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+higher-dimensional spaces, too.], [Euclidean
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+distance](https://en.wikipedia.org/wiki/Euclidean_distance) is the
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+shortest distance between two points.
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+
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+In the most general sense, the way you get this distance is by
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+measuring it. If you're lucky enough to be using [Cartesian
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+coordinates](https://en.wikipedia.org/wiki/Cartesian_coordinate_system),
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+then you can calculate the distance based on the distance between the
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+two coordinate points. For our purposes, I'm going to use $a$ and $b$,
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+located at $(x_1, y_1)$ and $(x_2, y_2)$.
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+
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+[Pythagoras](https://en.wikipedia.org/wiki/Pythagorean_theorem)
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+figured out how to calculate the length of the hypotenuse of a right
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+triangle. The fun part is that, with Cartesian coordinates, *any pair
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+of points* can now define a right triangle. Just take one point with
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+an x coordinate from one point and a y coordinate from the other, and
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+the triangle itself can be defined as $((x_!, y_1), (x_1, y_2), (x_2,
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+y_2))$^[Or $((x_!, y_1), (x_2, y_1), (x_2, y_2))$ if you're feeling
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+frisky].
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+
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+At that point, it becomes pretty easy. The length of the two sides is
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+just the absolute value of the difference between their x and y
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+coordinates. So, $|x_1 - x_2|$ and $|y_1 - y_2|$. Those are easy to
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+plug into the Pythagorean equation to get the distance between the
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+points ($d$):
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+
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+$$d = \sqrt{ |x_1 - x_2|^2 + |y_1 - y_2|^2 }$$
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+
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+In code, this might look something like:
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+
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+~~~ { .python }
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+def eucliddistance(x1, y1, x2, y2):
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+ xdist = abs(x1 - x2)
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+ ydist = abs(y1 - y2)
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+ return math.sqrt( (xdist ** 2) + (ydist ** 2) )
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+~~~
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+
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+This all seems a bit unnecessary, though, right? Cartesian coordinates
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+are universal. They're how we think about points. All of our screens
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+are Cartesian planes, and even our 3d worlds are Cartesian. Done and
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+done.
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+
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+Still, I think it's good to appreciate how we got here. Why? **Because
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+it doesn't work for our purposes.**
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+
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+[{width=150 height=150}](/images/post_2016_08_25/euclid_voronoi.png)
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+
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+A "point", to Euclid, is infinitesimal -- it has no size, nor
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+shape. And a Euclidean plane is continuous -- there is no "smallest
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+distance" you can move.
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+
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+Pixels, and graph squares, on the other hand, take up space. And if
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+you want to get from $(0,0)$ to $(7,14)$ on a discrete plane, there's
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+no such thing as moving 15.56247..... That's the nature of a discrete
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+space -- everything's an integer. And the other fun bit is that you
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+can't move in arbitrary directions. On a square grid, you've only got
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+8 directions to go, at most.
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+
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+It is also, frankly, a bit costly to calculate. Square roots are a bit
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+more complicated than adding and subtracting.
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+
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+Still, Euclidean distance is the definition of distance. And though we
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+can't use it to determine distance of movement, we can still use it
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+for comparative purposes.
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+
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+#### Voronoi Diagrams
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+
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+For illustration purposes, I've decided to use [Voronoi
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+diagrams](https://en.wikipedia.org/wiki/Voronoi_diagram). Voronoi
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+diagrams are a way of splitting up planes around a set of sites, such
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+that all points in the plane are associated with the closest site.
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+
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+The simplest, slowest possible way to generate such a diagram is to
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+iterate over each pixel in an image, calculate the distance between
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+that pixel and each site, sort the sites by distance, and assign that
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+site to the pixel. So I coded that up. I've put that code [up on
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+github](https://github.com/jmelesky/voronoi_example). Nothing fancy
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+going on, I assure you.
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+
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+By looking at the Voronoi diagrams for the same set of sites, using
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+different distance measures, we can get an appreciation for how those
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+measures vary, and what their characteristics are.
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+
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+
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+### Manhattan Distance
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+
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+[{width=150 height=150}](/images/post_2016_08_25/taxi_voronoi.png)
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+
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+The island of Manhattan is densly populated, and densely built. For
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+our purposes, though, it's also, notably, laid out like a grid. If you
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+want to get two blocks up and two blocks over, you're going to have to
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+move four total block lengths to get there. And that's a pretty
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+straightforward distance measure.
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+
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+It's also known as taxicab distance, rectilinear distance, and a bunch
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+of other things. And it's the simplest calculation we're going to be
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+making. Just add the lengths of each side of our triangle:
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+
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+$$d = |x_1 - x_2| + |y_1 - y_2|$$
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+
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+In code, it might look like this:
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+
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+~~~ { .python }
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+def taxidistance(x1, y1, x2, y2):
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+ return (abs(x1 - x2) + abs(y1 - y2))
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+~~~
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+
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+If you're on a checkerboard grid, and you are limited in movement to
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+only the four cardinal directions, this is how you're going to measure
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+getting from place to place. If you want to get from $(0,0)$ to
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+$(7,14)$ in a taxi, you're going to go 21 squares, which is a fair bit
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+further than 15.56.
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+
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+If you take a look at the Voronoi diagram derived from taxi distance,
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+you'll see that the shapes have changed from straight-edged polygons
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+to odd, almost art-deco-like shapes. In particular, some of the places
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+where there were very long, thin cells in the Euclidean version have
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+changed to blocky, angled, and often much distorted versions.
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+
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+### Civ Distance
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+
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+[{width=150 height=150}](/images/post_2016_08_25/civ_voronoi.png)
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+
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+If Manhattan distance is what you reckon with when you can only move
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+in the four cardinal directions, what do you do when you can move in
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+all 8 possible directions? Up until the most recent edition, the
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+[Civilization games](https://en.wikipedia.org/wiki/Civilization_%28series%29) had
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+exactly that scenario[^1]. So let's call this the Civ distance.
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+
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+Calculating this one is also easy. Instead of adding the sides
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+together, you just take the length of the longest side.
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+
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+How's that work, exactly? Well, if you're going from $(0,0)$ to
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+$(7,14)$, you go at an angle for 7 squares (moving twice as fast as
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+taxicabs). At that point, you're at $(7,7)$, and it's a straight shot
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+to $(7,14)$ only 7 squares away. Just 14 squares total.
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+
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+Mathematically, it looks like this:
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+
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+$$d = \max(|x_1 - x_2|, |y_1 - y_2|)$$
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+
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+And in code, something like this:
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+
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+~~~ {.python}
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+def civdistance(x1, y1, x2, y2):
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+ return (max(abs(x1 - x2), abs(y1 - y2)))
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+~~~
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+
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+Again, very simple. And 14 is actually shorter than 15.56. By taxi, up
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+and over is a distance of two, by Euclid that same distance is 1.41
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+and change. But by Civ distance, that same span is only one. That, of
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+course leads to some different distortions in the voronoi
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+diagram. Where the taxicab Voronoi cells stretched up, down, left, and
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+right, the Civ cells are stretching further in the diagonals.
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+
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+Overall, Civ distance is truer to Euclid than taking the taxi, but
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+it's still a very different beast.
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+
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+### D&D Distance
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+
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+Perhaps you've played some Dungeons & Dragons. Perhaps you've
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+specifically played 3.5ed^[Or its offshoot, Pathfinder; or, for that
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+matter, any of the d20 games that also derived from that source]. That
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+game was notorious for its complex, tactical, grid-based combat
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+system. Each square represented 5 feet of space, and each character
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+had a limited amount of movement per turn.
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+
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+[{width=150 height=150}](/images/post_2016_08_25/dandd_voronoi.png)
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+
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+Now, it didn't make sense to allow only movement in the four cardinal
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+directions. That would, if nothing else, be incredibly frustrating for
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+players trying to move their characters around the map.
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+
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+They also didn't want to allow willy-nilly Civ-style movement, since
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+that would offer a clear advantage to anyone who wished to use the
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+diagonals[^4ed].
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+
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+Ultimately, they decided on an iterative process which was a bit more
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+complex:
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+
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+ - The first time (and every odd time) you move a diagonal, it costs
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+ 5 feet of movement/distance, just like moving in a cardinal
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+ direction
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+
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+ - The second time (and every even time) you move a diagonal, it
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+ costs 10 feet of movement/distance, like you're taking a taxicab
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+
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+The net result is that, if you start out moving diagonally only, it's
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+5 feet, then 15, then 20, then 30, then 35, then 45, etc. Translating
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+feet to grid square units, it's 1 unit, then 3, then 4, then 6, then
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+7, then 9.
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+
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+That pattern, as it turns out, is the floor of 1.5 * the diagonal
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+distance.
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+
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+To generalize that to more than just the purely diagonal, we figure
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+out how much we can go straight, then do the rest diagonal, and add
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+them together:
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+
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+$$d_1 = \max(|x_1 - x_2|, |y_1 - y_2|) - \min(|x_1 - x_2|, |y_1 - y_2|)$$
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+$$d_2 = \lfloor 1.5 \times (\min(|x_1 - x_2|, |y_1 - y_2|))\rfloor$$
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+$$d = d_1 + d_2$$
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+
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+And in code, that looks like this[^odd]:
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+
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+~~~ {.python}
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+def dandddistance(x1, y1, x2, y2):
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+ mindist = min(abs(x1 - x2), abs(y1 - y2))
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+ maxdist = max(abs(x1 - x2), abs(y1 - y2))
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+ return ((maxdist - mindist) + (1.5 * mindist))
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+~~~
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+
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+Looking at the Voronoi diagram, this is easily the closest match to
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+Euclid. And that's not surprsing, really. When Euclid things up and
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+over is 1.41 and change, taxicabs and Civ think it's 2 and 1,
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+respectively, D&D thinks it's (about) 1.5, which is much, much closer
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+to the continuous distance. To get from $(0,0)$ to $(7,14)$ by way of
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+D&D distance is 17, which is pretty close to the 15.56 of Euclid, and
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+without the distortions of Civ.
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+
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+Of course, it's a bit more complex to calculate than Civ or taxi, but
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+it doesn't require taking any square routes, which marks it as still
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+computationally simpler than Euclid. And it's grid-native. The
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+designers of 3.5 did a good job finding a relatively simple
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+approximation for distance.
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+
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+### Conclusion
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+
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+And here we are at the end. Hopefully getting from point $a$ to point
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+$b$ wasn't too rough, and I know I learned a thing or two on the way.
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+
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+Things get more complicated when you start measuring things in three
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+dimensions. The D&D algorithm, particularly, is unlikely to scale well
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+as you end up figuring out the appropriate values for each of the 26
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+different directions you can go. Maybe I'll revisit this again.
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+
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+And let's not forget non-spatial distances! [Hamming
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+distance](https://en.wikipedia.org/wiki/Hamming_distance) is
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+particularly fun to think about. It has uses in small-scale AI tasks
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+like autocorrect, as well as in very-large-dimension binary vector
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+spaces[^duh]. I guess I'm saying that distances are fascinating
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+things, and there's so much more to talk about.
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+
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+Thanks for bearing with me.
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+
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+
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+[^1]: For an even longer time, the king piece on a chessboard had the
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+exact same abilities, which is why this is probably more commonly
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+known as "chessboard distance", or [Chebyshev
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+distance](https://en.wikipedia.org/wiki/Chebyshev_distance), after
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+Pafnuty Chebyshev, a 19th-century Russian mathematician. But I've
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+played more Civ than chess, frankly.
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+
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+[^4ed]: When it came time to publish the 4th edition of D&D, however,
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+they changed their mind on this. The complexity of the 3.5 distance
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+calculation wasn't worth the verisimilitude, so they moved to using
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+Civ distances instead. This decision also led to the infamous [square
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+fireballs](http://diceofdoom.com/blog/2009/10/powergaming-understanding-area-of-effect-in-dnd4e/),
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+but no system is perfect.
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+
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+[^odd]: You might notice that there isn't a `min` or `math.floor` call
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+in that code. There was, for a bit, but it was generating [some
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+strange artifacts](/images/post_2016_08_25/odd_dandd_voronoi.png),
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+where strange striping was occurring. I suspect that's due to an odd
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+quantizing effect, but haven't looked into it deeply. In the meantime,
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+using the un-floored numbers produce a more coherent image.
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+
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+[^duh]:
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+ *Obviously*
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+
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+ ...
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+
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+ For some machine learning tasks, you end up representing different
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+ things in very large vector spaces (one dimension for each
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+ "feature" you are keeping track of). In many cases, *binary*
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+ features are more tractable and just as useful -- for example, you
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+ might keep track of *whether* a certain word shows up in a
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+ document, rather than *how many times* it shows up. When those two
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+ situations coincide, Hamming distance becomes more palatable and
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+ tractable than Euclidean distance.]
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+
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