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correcting vertex number (20 is the number of faces, not vertices)

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edited Feb 7, 2019 at 20:05

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Would you be amenable to using a triangular grid as opposed to a square one? If so, I think the worst-case distortion can be reduced, relative to using a square grid.

If your planet is "really" an icosahedron before your rounding effect is applied, then each face can be covered with a grid of equilateral triangles, as densely as you like. The icosahedron approximates a sphere more closely than any other platonic solid. The only distortion occurs at the 2012 vertices of the icosahedron, where you'd have 5 triangles joining instead of 6. (A shortfall of 60 degrees, a third less than you'd get with squares).

If you can control where the continents end up, you can ensure these 2012 problem points are located in the middle of oceans, so they don't impact typical gameplay. (See, for instance, the points of the Dymaxion projection)

Your unit of construction would then be a triangular prism, rather than a cube. These are a bit trickier to work with, but you may be able to make that a natural part of your game's aesthetic.

Would you be amenable to using a triangular grid as opposed to a square one? If so, I think the worst-case distortion can be reduced, relative to using a square grid.

If your planet is "really" an icosahedron before your rounding effect is applied, then each face can be covered with a grid of equilateral triangles, as densely as you like. The icosahedron approximates a sphere more closely than any other platonic solid. The only distortion occurs at the 20 vertices of the icosahedron, where you'd have 5 triangles joining instead of 6. (A shortfall of 60 degrees, a third less than you'd get with squares).

If you can control where the continents end up, you can ensure these 20 problem points are located in the middle of oceans, so they don't impact typical gameplay. (See, for instance, the points of the Dymaxion projection)

Your unit of construction would then be a triangular prism, rather than a cube. These are a bit trickier to work with, but you may be able to make that a natural part of your game's aesthetic.

Would you be amenable to using a triangular grid as opposed to a square one? If so, I think the worst-case distortion can be reduced, relative to using a square grid.

If your planet is "really" an icosahedron before your rounding effect is applied, then each face can be covered with a grid of equilateral triangles, as densely as you like. The icosahedron approximates a sphere more closely than any other platonic solid. The only distortion occurs at the 12 vertices of the icosahedron, where you'd have 5 triangles joining instead of 6. (A shortfall of 60 degrees, a third less than you'd get with squares).

If you can control where the continents end up, you can ensure these 12 problem points are located in the middle of oceans, so they don't impact typical gameplay. (See, for instance, the points of the Dymaxion projection)

Your unit of construction would then be a triangular prism, rather than a cube. These are a bit trickier to work with, but you may be able to make that a natural part of your game's aesthetic.

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answered Dec 1, 2013 at 23:42

DMGregory ♦

140.9k
23
258
401

Would you be amenable to using a triangular grid as opposed to a square one? If so, I think the worst-case distortion can be reduced, relative to using a square grid.

If your planet is "really" an icosahedron before your rounding effect is applied, then each face can be covered with a grid of equilateral triangles, as densely as you like. The icosahedron approximates a sphere more closely than any other platonic solid. The only distortion occurs at the 20 vertices of the icosahedron, where you'd have 5 triangles joining instead of 6. (A shortfall of 60 degrees, a third less than you'd get with squares).

Your unit of construction would then be a triangular prism, rather than a cube. These are a bit trickier to work with, but you may be able to make that a natural part of your game's aesthetic.