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> The cube coordinates are a reasonable choice for a hex grid coordinate system. The constraint is that q + r + s = 0 so the algorithms must preserve that. The

> The cube coordinates are a reasonable choice for a hex grid coordinate system. The constraint is that q + r + s = 0 so the algorithms must preserve that. The constraint also ensures that there's a canonical coordinate for each hex. If the dimensions aren't independent and/or orthogonal, can this even be called a coordinate system? For example, you can have (1, 4, -5) but you can't have (1, 4, -4).

itishappy 3y ago
I don't think coordinate systems are required to be independent or orthogonal. https://en.wikipedia.org/wiki/Coordinate_system https://en.wikipedia.org/wiki/Coordinate_system https://en.wikipedia.org/wiki/Curvilinear_coordinates https://en.wikipedia.org/wiki/Curvilinear_coordinates https://en.wikipedia.org/wiki/Skew_coordinates https://en.wikipedia.org/wiki/Skew_coordinates https://en.wikipedia.org/wiki/Generalized_coordinates https://en.wikipedia.org/wiki/Generalized_coordinates
- SeanLuke 3y ago
  Actually, I think all the examples you provided were independent dimensions. I'll concede orthogonality. More specifically: can you think of coordinate systems with explicit binary or ternary constraints on the parameters in the coordinates? I can't immediately think of any. Holonomic constraints etc. are constraints on the validity of regions of the space, not constraints on the ability of coordinates to exist.
  - itishappy 3y ago
    Good point! Hmm... If I understand correctly, homogeneous coordinates have additional independent dimensions, but are many-to-one instead of constraining their parameters. Maybe normalized homogeneous coordinates? That's not a particularly interesting system. I'm now second guessing my definition of coordinate system...
    - SeanLuke 3y ago
      Would points in a simplex qualify as a coordinate system? Their dimensions would have to sum to 1. That is, so-called "normalized" barycentric coordinates maybe, as opposed to standard barycentric coordinates? https://en.wikipedia.org/wiki/Simplex https://en.wikipedia.org/wiki/Simplex https://en.wikipedia.org/wiki/Barycentric_coordinate_system https://en.wikipedia.org/wiki/Barycentric_coordinate_system
unflxw 3y ago
Note that any combination of two of the three "dimensions" does constitute an independent system. That is, given that q + r + s = 0, then q + r = -s, so you could use only (q, r) as your coordinates (and indeed that's likely how you'd canonically refer to them) The "third coordinate" is only for convenience when working with the three main axes of an hexagonal grid. Without it, the three directions of your grid are {+q, +r, -q-r}, which introduces an ugly asymmetry for one of the three axes.

itishappy 3y ago

I don't think coordinate systems are required to be independent or orthogonal. https://en.wikipedia.org/wiki/Coordinate_system https://en.wikipedia.org/wiki/Coordinate_system https://en.wikipedia.org/wiki/Curvilinear_coordinates https://en.wikipedia.org/wiki/Curvilinear_coordinates https://en.wikipedia.org/wiki/Skew_coordinates https://en.wikipedia.org/wiki/Skew_coordinates https://en.wikipedia.org/wiki/Generalized_coordinates https://en.wikipedia.org/wiki/Generalized_coordinates

SeanLuke 3y ago
Actually, I think all the examples you provided were independent dimensions. I'll concede orthogonality. More specifically: can you think of coordinate systems with explicit binary or ternary constraints on the parameters in the coordinates? I can't immediately think of any. Holonomic constraints etc. are constraints on the validity of regions of the space, not constraints on the ability of coordinates to exist.
- itishappy 3y ago
  Good point! Hmm... If I understand correctly, homogeneous coordinates have additional independent dimensions, but are many-to-one instead of constraining their parameters. Maybe normalized homogeneous coordinates? That's not a particularly interesting system. I'm now second guessing my definition of coordinate system...
  - SeanLuke 3y ago
    Would points in a simplex qualify as a coordinate system? Their dimensions would have to sum to 1. That is, so-called "normalized" barycentric coordinates maybe, as opposed to standard barycentric coordinates? https://en.wikipedia.org/wiki/Simplex https://en.wikipedia.org/wiki/Simplex https://en.wikipedia.org/wiki/Barycentric_coordinate_system https://en.wikipedia.org/wiki/Barycentric_coordinate_system

unflxw 3y ago

Note that any combination of two of the three "dimensions" does constitute an independent system. That is, given that q + r + s = 0, then q + r = -s, so you could use only (q, r) as your coordinates (and indeed that's likely how you'd canonically refer to them) The "third coordinate" is only for convenience when working with the three main axes of an hexagonal grid. Without it, the three directions of your grid are {+q, +r, -q-r}, which introduces an ugly asymmetry for one of the three axes.