Spherical Textures
courtesy of vterrain.org



There are many ways to apply a texture map to a sphere.  It is an important process because of applications such as texture-mapping the entire surface
of the earth.

• naive approach: directly map latitude-longitude onto sphere
  • source: a single rectangular texture with a proportion of 2x1
  • uv mapping: u goes from 0 to 1 around the equator, v goes from 0 to 1 from pole to pole
  • pro: simple, most data sources are already in this format
  • con: wastes the majority of the texture resolution at the poles, where
    it is usually least needed
  • BlueMarbleViewer uses this approach
• cubic mapping
• source: 6 textures corresponding to sides of a cube surrounding the sphere
• uv mapping is projected (e.g. with Gnomonic Projection) from the surface of the sphere to the sides of the cube
• pro: more evenly distribution of texels than the naive approach
• con:
  • the projection is a non-linear warp, making the computation difficult
  • still has wide variation in ratio of texels/surface area
• Omnitect mapping
  • like cubic mapping, uses four equatorial textures and two polar
  • however, the equatorial textures use a direct lat-lon mapping, and the poles are treated specially by joining four spherical right triangles of the top and bottom latitudes
  • pro: takes advantage of simple direct lat-lon mapping for most of the non-polar surface
  • con
    • complicated special cases for the tile boundaries
    • two types of mapping with different requirements for subdivision and projection
  • detail: see slides on global tiling system
• icosahedral mapping
  • like cubic mapping, but using an icosahedron
  • source: pairs of the icosahedron's faces can be joined into quads, for 10 square source textures
  • uv mapping: uv is simply (0,0), (1,0), (0,1) at the corners of each triangular face
  • pro:
    • less distortion than other approaches, evenly distributed texels/surface area
    • elegant system of identical regular triangles
  • con:
    • rendering must explicitly store and draw all 20 faces of the icosahedron
    • more tile boundaries to deal with
  • when applied to the earth with boundaries arranged to avoid breaking land masses, this is known as Buckminster Fuller's Dymaxion Projection
    • overview: Fuller Projection at the BFI site
    • great resource: Robert W. Gray's Notes to Buckminster Fuller's World Maps includes comparison of related projections, and source code!
    • also available is a Perl implementation Geo::Dymaxion by Schuyler Erle
  • this is now implemented in the VTP software
    • a pre-processing utility projects a conventional earth texture map onto 20 faces (10 square texture maps)
    • at runtime, spherical icosahedron is created, with one mesh per face, in Enviro
• octahedral
  • see Exoflight on the spherical LOD page
• Aasgaard's map projection
  • developed for the SINTEF Virtual Globe, Rune Aasgaard's map projection page has a great PowerPoint presentation illustrating the approach, example source code of the transformation, and other supporting documents
  • basically maps a quadtree onto a Mercator-like projection, with the depth
    of the tree's branches decreasing towards the poles
  • benefit is that the texels stay close to square
  • disadvantage is that the poles behave badly (distortion, wasted texture memory) which is often not a problem since they are "small and usually insignificant"
  • published academically as "Projecting a Regular Grid onto a Sphere or Ellipsoid", Rune Aasgaard, in: "Advances in Spatial Data Handling", Dianne Richardson and Peter van Oosterom (eds.), Springer-Verlag 2002, pp 339-350

Resources

• Map Projections and Spatial Referencing for Global Data Sets
• NGDC/NOAA Icosahedron Globe
• International Conference on Discrete Global Grids (March 2000)
• Discrete
  Global Grids for Digital Earth (pdf), Michael Goodchild, UCSB
• should locate:
  • John P. Snyder, "An Equal-Area Map Projection For Polyhedral Globes", Cartographica, Vol. 29, No. 1, 1992, pp. 10-21
  • Dutton, G. (1998) A Hierarchical Coordinate System for Geoprocessing and Cartography. Lecture Notes in Earth Science 79.