![]() ![]() ![]() In 2D, the symplectic form is equivalent to the determinant: note that this is an oriented or signed area rather than an absolute-value one. Symplectic geometry is built on a skew-symmetric bilinear map (the symplectic form) so that w(u,v)=-w(v,u). the inner product is a symmetric bilinear map. On the other hand: length (even if curved) in that direction is the same as length in this direction, i.e. You learn a lot about this (not in the "curve" differential-geometric sense, but most of the intuition carries over) in college linear algebra: you learn to think about entire vector subspaces having orthogonal complements as defined by the inner product. ![]() Put it this way: Riemannian (and as a special case, "ordinary") geometry is based around the inner product, which gives you both angles and projection (and 2D shadows from 3D objects, for example) and lengths. My dissertation is about simulating conservative physical systems on a computer but it is based in symplectic geometry, which exists only in even dimensions. Now imagine being lifted out the 3D plane of existence so that you could behold the entirety of the 3D world at once. Now, like seeing a 1D amount of information about a 2D maze while live inside it, we see a 2D amount of information about a 3D world around us (a picture demonstrate's this 2D amount of information - it's planar). Before, your vision was blocked by the walls, now you see the walls and what's on the other side of the walls simultaneously in a way that's entirely distinct from simply seeing through a transparent object. Now you can see the entire 2D extent of the maze at once. Now contrast that to if you were plucked up vertically 'above' the game's level to look down upon it. You can infer depths to objects if you have two eyes. You're really seeing a 1D amount of information about a 2D world (in fact, this is how the calculation is for Wolf3D and other games of its era). Imagine living in a world where everything is constant vertically, like those old 3D maze screen savers or Wolfenstein 3D. The comparison here is, as usual, to go down a dimension. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |