Sub-projects in this repository:
- Experiments with texture mappings (source, web app)
- A presentation on map/texture transformations (source, web app)
The most common way to render a sphere is based on the latitude/longitude grid as it is used for the earth: Select two opposite points on the sphere as the "poles", a bunch of equidistant "meridians", and a bunch of equidistant "parallels". Introduce a vertex at each intersection of a meridian and a parallel and connect neighboring vertices. This divides the sphere into a set of quadrangles. Dividing each quadrangle along one of its diagonals gives a triangulation of the sphere.
The natural texture for such a "lat/long sphere" is a map with a equirectangular projection. This is easy to implement and good enough for many applications. But while the distance between meridians gets smaller and smaller near the poles on the sphere, the meridians have a constant distance on the map. This leads to some drawbacks:
- Equirectangular maps are heavily distorted.
- Texture memory is wasted.
- Half of the triangles around the poles are degenerated on the sphere and thus the corresponding parts of the texture/map are not rendered.
To avoid these drawbacks this project investigates texture mappings based on octaspheres and icospheres.
See the README files of the subprojects for more details.
Projections from a sphere to a plane are not only needed for textures in computer graphics, but also for traditional paper maps of the Earth. There are several desirable properties of such a projection such as the accurate representation of distances, areas, angles and compass directions. It is not possible to achieve all these goals in a single projection. Thus over the centuries many different projections have been developed, each based on some choice or compromise between the goals according to some use case. (And actually the simplicity of a mapping is another aspect that might be considered in the trade-off.)
From the vast amount of resources discussing sphere-to-plane projections here are just a few:
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A nice overview video on map projections with some math background.
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Gene Keyes' discussion of map projections is interesting here for two reasons:
- It discusses projections based on an octahedron and an icosahedron. (See appendix 2 of part 2 for an icosahedral map similar to ours.)
- It deals with paper maps that are actually folded to 3D objects.
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This video (mentioned by Keyes) shows a map unfolding somewhat similar to ours.
Notice the following differences between mappings to paper and mappings to a texture:
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In contrast to paper maps, textures should be easily applicable to a triangulated sphere.
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For projections between a sphere and a texture it is a bit less important to minimize various kinds of distortions if the distortions introduced by the projection will be reversed when the texture is applied to a sphere. But while the user will not see a distorted surface, mapping distortions will cause the available level of detail to vary across the sphere. And typically we want to limit this variation.
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Maps try to be contiguous (at least on the continents) so that nearby points on the earth will also be nearby on the map. For textures this is not important, provided we have a good "sewing" mechanism. This allows us to use more cuts in the mapping to improve other aspects such as reducing distortion and arranging the map in a memory-efficient way.
Thus the trade-offs differ between paper maps and textures and may lead to different solutions. For example in our octahedral and icosahedral projections we save memory by moving triangles from the south-pole region to the gaps between triangles in the north-pole region. Such a representation would be confusing on a map directly viewed by a user.