metric tensor

I don''t know, what do you think?

The Metric Tensor is a matrix used to define the geometry of space, so you could probably use it for some cool distortion effects in games.

For flat space, the tensor is:
[ i 0 0 ]
[ 0 j 0 ]
[ 0 0 k ]

When you multiply that by a vector (x,y,z) you get (i*x,j*y,k*z), so by changing the values of i,j,k to something other than 1.0 you can stretch space. By fiddling with some of the other values in the tensor you get a "non-flat" space. Try it and see what you get

Disclaimer: some or all of this may be wrong. It's been a long time since I did anything to do with tensors.

[teamonkey]

[edited by - teamonkey on December 29, 2003 4:32:53 AM]

The metric tensor is used to generalize concepts like distance, volume, and the dot product to arbitrary curvilinear coordinate systems. For example, the 3D dot product is generalized to

U dot V = sum(g[i,j] * Ui * Vj),

where g is the 3x3 metric tensor and the sum ranges over all values of i and j between 1 and 3. In flat Cartesian coordinates, g is the identity matrix and the above equation reduces to the conventional dot product.

This is probably major overkill for games, unless you're planning to simulate general relativistic effects. Nevertheless, I do include a short dicussion about the metric tensor in Appendix C of my book, Mathematics for 3D Game Programming and Computer Graphics:

http://www.terathon.com/books/mathgames2.html


[edited by - elengyel on December 30, 2003 4:14:37 PM]