In search of the perfect digital hexagon

coreolyn has asked for the wisdom of the Perl Monks concerning the following question:

Replies are listed 'Best First'.
Re: In search of the perfect digital hexagon by japhy (Canon) on Aug 13, 2004 at 13:02 UTC
I'm not sure I understand the question... but, if you're dealing with a "perfect" hexagon (I believe you mean one with equal sides and equal angles (120 degrees)), then I think the simplest way to describe that hexagon in 2-d space would be with four values (x_c, y_c, l_s, θ): x_c is the x-coordinate of the center of the hexagon y_c is the y-coordinate of the center of the hexagon l_s is the length of a side of the hexagon θ is the angle (measured from the x-axis) of the tilt of the hexagon A tilt of 0 degrees would mean the hexagon looks like: `_ / \ \_/` [download] where a tilt of 90 degrees would mean the hexagon looks like: `/\ \|\| \/` [download] _____________________________________________________ Jeff `japhy` Pinyan, P.L., P.M., P.O.D, X.S.: Perl, regex, and `perl` hacker How can we ever be the sold short or the cheated, we who for every service have long ago been overpaid? ~~ Meister Eckhart	[reply] [d/l] [select]
Re: In search of the perfect digital hexagon by BrowserUk (Patriarch) on Aug 13, 2004 at 13:01 UTC
Perhaps you could clarify what you mean by "express perfect hexagons from data"? A quick google for "Perfect hexagons" doesn't turn up anything that immediately leaps off the page as a "well know definition". Examine what is said, not who speaks. "Efficiency is intelligent laziness." -David Dunham "Think for yourself!" - Abigail "Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon	[reply]
Re: In search of the perfect digital hexagon by pbeckingham (Parson) on Aug 14, 2004 at 01:20 UTC
I am shamelessly ripping off fletch's work here, but the following (hexagonal) grid: `-8 . . . - - -8 -7 . . . . - ^ -7 -6 . . . . - - - - -6 -5 . . - . a a - ^ ? ? - -5 -4 . . . . k o ! - ^ ? ? ? -4 -3 . - . a h j a ^ ^ ^ ^ ^ - -3 -2 . . - . a a a ^ a a - - - -2 -1 . . . a a a b + + ^ ^ - -1 0 . . . m c u a a - ^ - ^ 0 1 . - . a a . a a - - ^ 1 2 - ^ ^ a a a ^ a - - ^ 2 3 - - - - - - - . . - 3 4 ^ - - - - - ^ - - 4 5 - ^ - - ^ - - - 5` [download] Is actually the same grid as this rectangular grid: `-8 ...-- -8 -7 ....-^ -7 -6 ....---- -6 -5 ..-.aa-^??- -5 -4 ....ko!-^??? -4 -3.-.ahja^^^^^- -3 -2 ..-.aaa^aa--- -2 -1 ...aaab++^^- -1 0 ...mcuaa-^-^ 0 1 .-.aa.aa--^ 1 2 -^^aaa^a--^ 2 3 -------..- 3 4 ^-----^-- 4 5 -^--^--- 5` [download] It is just a question of How the grid is rendered. how a cell's neighboring cells are determined. Once the data structure is reduced to rectangular, the mystery of the hexagon disappears. It is this kind of thinking that reduces a checkerboard to an 8 x 4 board, for certain games. pbeckingham - typist, perishable vertebrate.	[reply] [d/l] [select]
Re: In search of the perfect digital hexagon by coreolyn (Parson) on Aug 13, 2004 at 14:34 UTC
Thanks for the responses. I'll attempt to clarify. All of the responses are relevant and have given me things to play with and research. What I keep trying to express in code is 3 dimensional space. Much like a 10 sided dice in D&D hexagons fill space perfectly. Additionally smaller hexes fill larger hexes perfectly. So what I'm looking to do is figure a coordinate system for defining what is in a space by getting code to scale space via hexagonal scaling. The crux of this is figuring the best algorithm for representing and storing a hex ( as japhy pointed out ) with 120 degree angles. Update -> After finding the most efficient hex comes translating this type of 2D thinking into 3D. :) coreolyn	[reply]
Re^2: In search of the perfect digital hexagon by VSarkiss (Monsignor) on Aug 13, 2004 at 14:55 UTC
It sounds like what you're looking for is regular solid tilings. There's actually a large body of data on the subject. Try googling for regular tiling, or take a look at some math reference sites.	[reply]
Re^3: In search of the perfect digital hexagon by coreolyn (Parson) on Aug 13, 2004 at 15:04 UTC
Ouch, but yeah.. thanks for the leads. coreolyn	[reply]
Re: In search of the perfect digital hexagon by water (Deacon) on Aug 13, 2004 at 13:06 UTC
Also not sure what you mean by "express perfect hexagons", but the reference to board games makes me think of a using a graph (vertices V_1... V_n, adjacency matrix A_ij=1 when edge between V_i and V_j) to represent the 6 neighbors of each cell.	[reply]
Re^2: In search of the perfect digital hexagon by Fletch (Bishop) on Aug 13, 2004 at 13:58 UTC
Another way is to use a two dimensional array (matrix), but only some locations are valid. Empire uses a scheme sort of like this: ---------0000000000111111111 9876543210123456789012345678 -8 . . . - - -8 -7 . . . . - ^ -7 -6 . . . . - - - - -6 -5 . . - . a a - ^ ? ? - -5 -4 . . . . k o ! - ^ ? ? ? -4 -3 . - . a h j a ^ ^ ^ ^ ^ - -3 -2 . . - . a a a ^ a a - - - -2 -1 . . . a a a b + + ^ ^ - -1 0 . . . m c u a a - ^ - ^ 0 1 . - . a a . a a - - ^ 1 2 - ^ ^ a a a ^ a - - ^ 2 3 - - - - - - - . . - 3 4 ^ - - - - - ^ - - 4 5 - ^ - - ^ - - - 5 ---------0000000000111111111 9876543210123456789012345678 (From the docs for the `map` command) To move east or west you use an increment of two to the x index (i.e. west would be `( x - 2, y )`); north / south is forbidden in this scheme since the faces are lined up W/NW/NE/E/SE/SW, although that's by convention and there's nothing saying you couldn't rule out east / west by adding to y instead (and in effect rotating the hex grid 90 degrees). To move any of the other four directions you add/subtract one to both coordinates (northeast would be `( x + 1, y - 1)`). You can easily verify if a coordinate pair's valid as both x and y have to be both odd or both even. If this is anywhere near what you were asking about you might take a gander at PEI, the Perl Empire Interface, which has code for dealing with maps of this type.	[reply]