volume of a k-simplex...

[Han-Wen Nienhuys]

aupetit at dase.bruyeres.cea.fr writes:
>            | det (W*W^t) | ^(1/2)
> Volume_k = ----------------------
>                     k!
> 
> on the site:
> http://www.math.washington.edu/~hillman/PUB/volume
> 
> with W the matrix with k rows and n columns
> where W=(v_1-v_0)
>         (v_2-v_0)
>         (  ...  )
>         (v_k-v_0)
> 
> with row vectors v_i the k+1 vertices of the
> k-simplex in R^n.
> 
> W^t denotes the transpose of W.
> 
> Are both formulae equivalent when n=k?

Almost. When W has rank n, then det (W*W^T) = det(W)^2.  The above
formula always yields a positive answer. For an n-dimensional simplex,
the determinant of the coordinate vectors can be negative depending on
the orientation of the simplex.

The formulation which uses a (k+1)x(k+1) determinant is more symmetric
than the one that selects an origin vector (v_0 above) and uses the k
x k determinant. I'm not sure if this has consequences for numerical
precision and/or stability when using inexact computation.

-- 
Han-Wen Nienhuys   |   hanwen at cs.uu.nl    | http://www.cs.uu.nl/~hanwen/


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