Then when the player moves away we’ll adjust them back, out of sight. The idea is to adjust triangular regions near the poles to be adjacent to the ones the player is in. We need to translate both positions and directions to the new triangle. This doesn’t work near the triple points (where the pentagons are) but if you can’t get close to the pentagons anyway it shouldn’t be a problem. If x goes off the left/right side of a triangle region into a red area, you are teleported into a nearby valid green area region. If you try to exceed the y bound, you stop at the edge. Use a single x,y coordinate for the entire world, but limit it to the blue area shape. While global calculations may be a little better, distances and angles are distorted locally. The triangles are distorted though, especially near the poles. If you try to exceed the x bound, you wrap around. Use a single x,y coordinate for the entire world, but limit it to this shape. Since region crossing is complicated, it’d be best to avoid combat or even land masses crossing regions. There are no distance or angle distortions for local calculations. When crossing a boundary into another region, calculate the new x,y.ĭistances and vectors that cross a region boundary becomes hard. Keep track of which region you are in, and then keep a local x,y coordinate inside of that triangle. The third problem is: what should happen when you pass a pole? I’m going to rotate the camera and make it only let you see the local area. This would only work in games where you’re walking/driving and not games where you can zoom out or freely control the camera. I’m going to try hiding the pentagons by just not letting you walk near them. The second problem is: what should I do with those leftover pentagons? They won’t work with a flat tile map. (Note: if you want to use square tiles instead of hexagon tiles, unwrap a cube.) That icosahedron, when unfolded, can look like this: The first problem is: what coordinate system should I use to represent the player’s position and direction, in a way that makes walking around the pole shorter than walking around the equator? A common way of tesselating a sphere is to subdivide an icosahedron. (Note: the /x/ in the url is where I put my quick, unpolished projects like this one.) So it looks like it may not be practical to flatten a sphere onto a flat grid, but I wanted to explore it anyway. It either needs to rotate freely or it needs to flip upside down when you pass the pole. You can’t have a camera that’s always facing north.
The need to flip direction when passing a pole makes camera orientation tricky.The need for polar regions to be narrower means it has a more complicated coordinate system that doesn’t allow rectangular grid maps.In the image on the right it’s easy to spot them, but even in a map with millions of tiles, you will still have those twelve pentagons hiding among millions of hexagons. If you try to tile with hexagons (by subdividing an icosahedron), you end up with twelve pentagons left over. You can’t tile a sphere with uniform regular square or hexagon tiles ( Euler said so! ).Wouldn’t that be cool? Why don’t tile map games use this? Walking west/east to wrap around the map takes less time (shorter distance) near the poles than near the equator.You can walk past the north/south poles and stay at that pole but are facing the south/north instead of north/south.You can walk north/south but you do not wrap around to the other side.You can walk west/east and wrap around the map.Most wraparound tile map games don’t use a sphere like an actual planet. Some games also let you go off the top side and warp to the bottom, and vice versa.
You can’t go off the top or bottom of the map. Civilization for example lets you go off the left side of the map and you warp to the right side. These are the formulas that allow us to convert from spherical to cylindrical coordinates.Some games have tile maps that wrap around. Using a bit of geometry, we also know that φ is the angle between ρ and the vertical side of the triangle. We know that the angle between the z axis and ρ is equal to φ. To facilitate the derivation of the transformation formulas, we can start by transforming from spherical to cylindrical coordinates and then, we can transform from cylindrical to Cartesian coordinates. To transform from spherical to Cartesian coordinates, we have to use right triangles and trigonometry. On the other hand, three-dimensional Cartesian coordinates have the form (x, y, z). In this form, ρ represents the distance from the origin to the point, θ represents the angle in the xy plane with respect to the x axis and φ represents the angle with respect to the z axis. Spherical coordinates have the form (ρ, θ, φ).
Spherical to cartesian coordinates how to#
How to transform from spherical coordinates to Cartesian coordinates?
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