It is recommended to keep values below a circumference of 1000. If you also need hex_round, it will be FloatHex → Hex, using helper function floatcube_to_floathex instead of cube_to_hex. Also unlike offset coordinates, we can safely add and subtract doubled coordinates, which makes them easier to work with than offset coordinates. Sometimes we don't have obvious algorithms for hex grids, but we do have algorithms for cube grids. In addition to calculating the PPI, the calculator also identifies the total number of pixels on the display and the distance between the center of the two pixels, which is also referred to as the dot pitch and is the inverse of the PPI. Octaves This number defines the intensity, or detail strength of the noise. The most common approach is to offset every other column or row. This is the oldest reference I can find for axial grids: Luczak, E. and Rosenfeld, A., There's a paper that calls cube coordinates, [1]: http://www-cs-students.stanford.edu/~amitp/gameprog.html#hex, [2]: http://www-cs-students.stanford.edu/~amitp/Articles/Hexagon2.html, [3]: http://www-cs-students.stanford.edu/~amitp/Articles/HexLOS.html, [4]: http://www-cs-students.stanford.edu/~amitp/game-programming/grids/, [5]: http://web.archive.org/web/20090205120106/http://sc.tri-bit.com/Hex_Grids, [6]: http://ondras.github.io/rot.js/manual/#hex/indexing, [7]: https://www.researchgate.net/publication/235779843_Storage_and_addressing_scheme_for_practical_hexagonal_image_processing?_sg=flKEA6rk1KmOpC4LBjQJN_-NBuiR1KJtJt-XeYRXnd0z_MNUrB2gjb2FKV3iBoKg988P2xHCpQ, [8]: https://en.wikipedia.org/wiki/Bitwise_operation#AND, [9]: https://en.wikipedia.org/wiki/Modulo_operation, [10]: https://www.researchgate.net/publication/235779843_Storage_and_addressing_scheme_for_practical_hexagonal_image_processing?_sg=flKEA6rk1KmOpC4LBjQJN_-NBuiR1KJtJt-XeYRXnd0z_MNUrB2gjb2FKV3iBoKg988P2xHCpQ, [11]: https://doi.org/10.1117/1.JEI.22.1.010502, [12]: http://3dmdesign.com/development/hexmap-coordinates-the-easy-way, [13]: http://ondras.github.io/rot.js/manual/#hex/indexing, [14]: http://ondras.github.io/rot.js/manual/#hex/indexing, [15]: https://en.wikipedia.org/wiki/Digital_differential_analyzer_(graphics_algorithm), [16]: http://zvold.blogspot.com/2010/02/line-of-sight-on-hexagonal-grid.html, [17]: http://stackoverflow.com/questions/3233522/elegant-clean-special-case-straight-line-grid-traversal-algorithm, [18]: https://doi.org/10.1111/1467-8659.1210027, [19]: http://devmag.org.za/2013/08/31/geometry-with-hex-coordinates/, [20]: https://en.wikipedia.org/wiki/Cuboid, [21]: http://devmag.org.za/2013/08/31/geometry-with-hex-coordinates/, [22]: http://gamedev.stackexchange.com/questions/15237/how-do-i-rotate-a-structure-of-hexagonal-tiles-on-a-hexagonal-grid/, [23]: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF, [24]: http://www-cs-students.stanford.edu/~amitp/Articles/HexLOS.html, [26]: https://s3.amazonaws.com/jbochi/layout.html, [27]: http://www.adammil.net/blog/v125_Roguelike_Vision_Algorithms.html, [28]: http://www.roguebasin.com/index.php?title=Pre-Computed_Visibility_Tries, [29]: http://www.roguebasin.com/index.php?title=Field_of_Vision, [30]: http://www.wolframalpha.com/input/?i=inv+%7B%7Bsqrt%283%29%2C+sqrt%283%29%2F2%7D%2C+%7B0%2C+3%2F2%7D%7D, [31]: http://www.wolframalpha.com/input/?i=inv+%7B%7B3%2F2%2C+0%7D%2C+%7Bsqrt%283%29%2F2%2C+sqrt%283%29%7D%7D, [32]: https://bl.ocks.org/patricksurry/0603b407fa0a0071b59366219c67abca, [33]: http://gamedev.stackexchange.com/a/137603/2472, [34]: https://observablehq.com/@sanderevers/hexagon-tiling-of-an-hexagonal-grid, [35]: https://observablehq.com/@sanderevers/hexmod-representation, [36]: https://www.redblobgames.com/pathfinding/a-star/introduction.html, [37]: http://www-cs-students.stanford.edu/~amitp/Articles/HexLOS.html, [38]: http://www-cs-students.stanford.edu/~amitp/Articles/Hexagon2.html, [39]: http://devmag.org.za/2013/08/31/geometry-with-hex-coordinates/, [40]: http://www.gamelogic.co.za/downloads/HexMath2.pdf, [41]: http://gamelogic.co.za/grids/documentation-contents/quick-start-tutorial/gamelogics-hex-grids-for-unity-and-amit-patels-guide-for-hex-grids/, [42]: http://www-cs-students.stanford.edu/~amitp/game-programming/grids/, [43]: http://playtechs.blogspot.com/2007/04/hex-grids.html, [44]: http://web.archive.org/web/20090205120106/http://sc.tri-bit.com/Hex_Grids, [45]: http://ondras.github.io/rot.js/manual/#hex/indexing, [46]: http://stackoverflow.com/questions/2049196/generating-triangular-hexagonal-coordinates-xyz, [47]: http://keekerdc.com/2011/03/hexagon-grids-coordinate-systems-and-distance-calculations/, [48]: http://stackoverflow.com/questions/2459402/hexagonal-grid-coordinates-to-pixel-coordinates, [49]: http://gamedev.stackexchange.com/questions/51264/get-ring-of-tiles-in-hexagon-grid, [50]: http://www.battleanalysis.com/battlefield.html, [51]: http://gamedev.stackexchange.com/questions/49718/vertical-vs-horizontal-hex-grids-pros-and-cons, [52]: http://arges-systems.com/blog/2011/01/10/hex-grid-line-of-sight-revisited/, [53]: http://hexnet.org/content/permutohedron, [54]: http://incompetech.com/graphpaper/hexagonal/, [55]: http://old.reddit.com/r/gamedev/comments/19wmvn/a_data_structure_for_a_game_board_with_hexagonal/c8s9qbe, [56]: https://www.springer.com/us/book/9781852339142, [57]: https://doi.org/10.1007/1-84628-203-9, [58]: http://hexgridutilities.codeplex.com/documentation, [59]: https://doi.org/10.1109/TC.1976.1674642, [61]: https://doi.org/10.1109/PROC.1979.11356, [64]: http://old.reddit.com/r/gamedev/comments/1dz1tr/, [65]: https://news.ycombinator.com/item?id=5809724, [66]: http://www.metafilter.com/128649/Hexagonal-Grids, [67]: https://trello.com/card/hexagonal-grids-2-0/4f1dbfdc0fc2508c1b238d7d/52. Another approach is to convert the offset coordinates into cube/axial coordinates, then use the cube/axial to pixel conversion. This comes up in line drawing and pixel to hex. In the pointy orientation, a hexagon has width w = sqrt(3) * size and height h = 2 * size. This allows the in-between values for the hexes that are halfway in between: I haven't found much information about this system — tri-bit.com called it interlaced[5], rot.js calls it double width[6], and this paper[7] calls it rectangular. Since we have a constraint x + y + z = 0, there's some redundancy, and we don't need to store all three coordinates. There are many other ways to convert pixel to hex; see this page for the ones I know of. Very simple to use. palette: [ However, in most statically typed languages, you'll need a separate class/struct type for float coordinates, and cube_round will have type FloatCube → Cube. If you're using graph-based pathfinding such as A* or Dijkstra's algorithm or Floyd-Warshall, pathfinding on hex grids isn't different from pathfinding on square grids. scale: 30, A 180° rotation flips the signs but the coordinates have rotated back to where they originally were. To draw a hexagon outline, use those vertices, and then draw a line back to hex_corner(…, 0).
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