# Pythagorean Distance

Keywords: Theory

The geometric distance between two coordinates on the goban.

A formula for calculating the Pythagorean distance in C notation is

```   distance = sqrt( (x1 - x0) ^ 2 + (y1 - y0) ^ 2 );
```

where (x0,y0) is the numerical coordinate of one intersection and (x1,y1) is the coordinate of the other. (This is the classic "length of the hypotenuse" formula of a right triangle.) Here is a crude table with Pythagorean distances measured 'math-style' from the lower-left corner. Distances have been rounded to the nearest whole number. Hoshi are in bold.

```   18 18 18 18 18 19 19 19 20 20 21 21 22 22 23 23 24 25 25
17 17 17 17 17 18 18 18 19 19 20 20 21 21 22 23 23 24 25
16 16 16 16 16 17 17 17 18 18 19 19 20 21 21 22 23 23 24
15 15 15 15 16 16 16 17 17 17 18 19 19 20 21 21 22 23 23
14 14 14 14 15 15 15 16 16 17 17 18 18 19 20 21 21 22 23
13 13 13 13 14 14 14 15 15 16 16 17 18 18 19 20 21 21 22
12 12 12 12 13 13 13 14 14 15 16 16 17 18 18 19 20 21 22
11 11 11 11 12 12 13 13 14 14 15 16 16 17 18 19 19 20 21
10 10 10 10 11 11 12 12 13 13 14 15 16 16 17 18 19 20 21
9  9  9  9 10 10 11 11 12 13 13 14 15 16 17 17 18 19 20
8  8  8  9  9  9 10 11 11 12 13 14 14 15 16 17 18 19 20
7  7  7  8  8  9  9 10 11 11 12 13 14 15 16 17 17 18 19
6  6  6  7  7  8  8  9 10 11 12 13 13 14 15 16 17 18 19
5  5  5  6  6  7  8  9  9 10 11 12 13 14 15 16 17 18 19
4  4  4  5  6  6  7  8  9 10 11 12 13 14 15 16 16 17 18
3  3  4  4  5  6  7  8  9  9 10 11 12 13 14 15 16 17 18
2  2  3  4  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
1  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
```
Pythagorean Distances

Point a has a distance of 0 units or intersections from itself. Points b and c have a distance of 18 units from point a. Point d has a distance of approximately 25.46 units from point a.

For points a, b and c -- or for any points sharing an x-coordinate, y-coordinate or both (that is, on the same line) -- the Pythagorean Distance is identical to the Manhattan Distance.

For all other points -- points with no coordinates in common -- the Manhattan Distance is always greater than the Pythagorean Distance.

I'm not sure how useful this kind of thinking is, but it seems that neither Manhattan Distance or Pythagorean Distance can possibly describe the whole situation with respect to how related two stones are, so I went ahead and created this page. Off the top of my head, perhaps some weighted average of the two figures might give the best estimate of the strength of two stones in proximity. -- geno / 2004-01-02

Pythagorean Distance last edited by 72.66.101.13 on February 27, 2013 - 03:56