the distance between two points isn't the straight-line distance but the great circle distance. The problem is that the flat space might have to have more dimensions than the original space to allow it to take up the curvature.įor example, if you live on the surface of a 2D sphere then you have to measure distance using the appropriate metric, e.g. This is great because it means that you can do all the work in the simple flat space. a Riemann manifold, in a perfectly flat, i.e. John Forbes Nash, the same mathematician responsible for the Nash Equilibrium and who featured in the movie "A Beautiful Mind" worked out that if you could always embed a curved space, i.e. This is where the Nash embedding theorem comes in. You have to work with a curvature tensor and minimize the error within the curved space. The problem can be thought of as trying to best fit the mesh to a curved surface but it is the curvature that makes the problem difficult. It can be shown that the anisotropy of the triangles should depend on the ratio of the curvature in two directions. For such applications we would need to fit a mesh according to other physical conditions not just curvature. The same idea can make accurate simulations such as fluid dynamics faster. An anisotropic mesh can represent a surface more accurately with fewer triangles making rendering faster but creating such a mesh is difficult. The next level of improvement comes from dividing up the surface using different shapes and sizes of triangle - a so called Anisotropic mesh. The resulting mesh is generally referred to as Non-uniform Isotropic because it uses the same shape of triangle. The idea is that where the curvature of the surface is large you need more triangles and where it is approaches flat you need far fewer. In many cases the triangles are all the same size but you can get a better representation by varying the size of the triangle. Most 3D models make use of a mesh of simple triangles. The result is faster and more accurate meshes. Researchers at UT Dallas have found a really clever way of solving the anisotropic mesh problem - do it in higher dimensions and then map it back to 3D. Better 3D Meshes Using The Nash Embedding Theorem
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |