问题
I have a hexagon grid:
with template type coordinates T. How I can calculate distance between two hexagons?
For example:
dist((3,3), (5,5)) = 3
dist((1,2), (1,4)) = 2
回答1:
First apply the transform (y, x) |-> (u, v) = (x, y + floor(x / 2)).
Now the facial adjacency looks like
0 1 2 3
0*-*-*-*
|\|\|\|
1*-*-*-*
|\|\|\|
2*-*-*-*
Let the points be (u1, v1) and (u2, v2). Let du = u2 - u1 and dv = v2 - v1. The distance is
if du and dv have the same sign: max(|du|, |dv|), by using the diagonals
if du and dv have different signs: |du| + |dv|, because the diagonals are unproductive
In Python:
def dist(p1, p2):
y1, x1 = p1
y2, x2 = p2
du = x2 - x1
dv = (y2 + x2 // 2) - (y1 + x1 // 2)
return max(abs(du), abs(dv)) if ((du >= 0 and dv >= 0) or (du < 0 and dv < 0)) else abs(du) + abs(dv)
回答2:
The correct explicit formula for the distance, with your coordinate system, is given by:
d((x1,y1),(x2,y2)) = max( abs(x1 - x2),
abs((y1 + floor(x1/2)) - (y2 + floor(x2/2)))
)
回答3:
First, you need to transform your coordinates to a "mathematical" coordinate system. Every two columns you shift your coordinates by 1 unit in the y-direction. The "mathamatical" coordinates (s, t) can be calculated from your coordinates (u,v) as follows:
s = u + floor(v/2) t = v
If you call one side of your hexagons a, the basis vectors of your coordinate system are (0, -sqrt(3)a) and (3a/2, sqrt(3)a/2). To find the minimum distance between your points, you need to calculate the manhattan distance in your coordinate system, which is given by |s1-s2|+|t1-t2| where s and t are the coordinates in your system. The manhattan distance only covers walking in the direction of your basis vectors so it only covers walking like that: |/ but not walking like that: |\. You need to transform your vectors into another coordinate system with basis vectors (0, -sqrt(3)a) and (3a/2, -sqrt(3)a/2). The coordinates in this system are given by s'=s-t and t'=t so the manhattan distance in this coordinate system is given by |s1'-s2'|+|t1'-t2'|. The distance you are looking for is the minimum of the two calculated manhattan distances. Your code would look like this:
struct point
{
int u;
int v;
}
int dist(point const & p, point const & q)
{
int const ps = p.u + (p.v / 2); // integer division!
int const pt = p.v;
int const qs = q.u + (q.v / 2);
int const qt = q.v;
int const dist1 = abs(ps - qs) + abs(pt - qt);
int const dist2 = abs((ps - pt) - (qs - qt)) + abs(pt - qt);
return std::min(dist1, dist2);
}
回答4:
Here is what a did:
Taking one cell as center (it is easy to see if you choose 0,0), cells at distance dY form a big hexagon (with “radius” dY). One vertices of this hexagon is (dY2,dY). If dX<=dY2 the path is a zig-zag to the ram of the big hexagon with a distance dY. If not, then the path is the “diagonal” to the vertices, plus an vertical path from the vertices to the second cell, with add dX-dY2 cells.
Maybe better to understand: led:
dX = abs(x1 - x2);
dY = abs(y1 - y2);
dY2= floor((abs(y1 - y2) + (y1+1)%2 ) / 2);
Then:
d = d((x1,y1),(x2,y2))
= dX < dY2 ? dY : dY + dX-dY2 + y1%2 * dY%2
回答5:
Posting here after I saw a blog post of mine had gotten referral traffic from another answer here. It got voted down, rightly so, because it was incorrect; but it was a mischaracterization of the solution put forth in my post.
Your 'squiggly' axis - in terms of your x coordinate being displaced every other row - is going to cause you all sorts of headaches with trying to determine distances or doing pathfinding later on, if this is for a game of some sort. Hexagon grids lend themselves to three axes naturally, and a 'squared off' grid of hexagons will optimally have some negative coordinates, which allows for simpler math around distances.
Here's a grid with (x,y) mapped out, with x increasing to the lower right, and y increasing upwards.
By straightening things out, the third axis becomes obvious.
The neat thing about this, is that the three coordinates become interlinked - the sum of all three coordinates will always be 0.
With such a consistent coordinate system, the atomic distance between any two hexes is the largest change between the three coordinates, or:
d = max( abs(x1 - x2), abs(y1 -y2), abs( (-x1 + -y1) - (-x2 + -y2) )
Pretty straightforward. But you must fix your grid first!
回答6:
(odd-r)(without z, only x,y)
I saw some problems with realizations above. Sorry, I didn't check it all but. But maybe my solution will be helpful for someone and maybe it's a bad and not optimized solution.
The main idea to go by diagonal and then by horizontal. But for that we need to note:
1) For example, we have 0;3 (x1=0;y1=3) and to go to the y2=6 we can handle within 6 steps to each point (0-6;6) so: 0-left_border , 6-right_border
2)Calculate some offsets
#include <iostream>
#include <cmath>
int main()
{
//while(true){
int x1,y1,x2,y2;
std::cin>>x1>>y1;
std::cin>>x2>>y2;
int diff_y=y2-y1; //only up-> bottom no need abs
int left_x,right_x;
int path;
if( y1>y2 ) { // if Down->Up then swap
int temp_y=y1;
y1=y2;
y2=temp_y;
//
int temp_x=x1;
x1=x2;
x2=temp_x;
} // so now we have Up->Down
// Note that it's odd-r horizontal layout
//OF - Offset Line (y%2==1)
//NOF -Not Offset Line (y%2==0)
if( y1%2==1 && y2%2==0 ){ //OF ->NOF
left_x = x1 - ( (y2 - y1 + 1)/2 -1 ); //UP->DOWN no need abs
right_x = x1 + (y2 - y1 + 1)/2; //UP->DOWN no need abs
}
else if( y1%2==0 && y2%2==1 ){ // OF->NOF
left_x = x1 - (y2 - y1 + 1)/2; //UP->DOWN no need abs
right_x = x1 + ( (y2 - y1 + 1)/2 -1 ); //UP->DOWN no need abs
}
else{
left_x = x1 - (y2 - y1 + 1)/2; //UP->DOWN no need abs
right_x = x1 + (y2 - y1 + 1)/2; //UP->DOWN no need abs
}
/////////////////////////////////////////////////////////////
if( x2>=left_x && x2<=right_x ){
path = y2 - y1;
}
else {
int min_1 = std::abs( left_x - x2 );
int min_2 = std::abs( right_x - x2 );
path = y2 - y1 + std::min(min_1, min_2);
}
std::cout<<"Path: "<<path<<"\n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\n";
//}
return 0;
}
回答7:
I believe the answer you seek is:
d((x1,y1),(x2,y2))=max(abs(x1-x2),abs(y1-y2));
You can find a good explanation on hexagonal grid coordinate-system/distances here:
http://keekerdc.com/2011/03/hexagon-grids-coordinate-systems-and-distance-calculations/
来源：https://stackoverflow.com/questions/15919783/distance-between-2-hexagons-on-hexagon-grid