# 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 151516 16 16 17 171718 19 19 20 212122 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 9910 10 11 11 121313 14 15 16 171718 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 3 445 6 7 8 9910 11 12 13 141516 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

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