volume of a k-simplex...

[aupetit]

William Flis wrote:

> > I need to compute the volume of a k-simplex
> > knowing the coordinates of its vertices.
>
> For a triangle:
>
> Area = Abs(| 1  x1  y1 |)
>            | 1  x2  y2 |
>            | 1  x3  y3 |
>            -------------
>                  2
>
> For a tetrahedron:
>
> Volume = Abs(| 1  x1  y1  z1 |)
>              | 1  x2  y2  z2 |
>              | 1  x3  y3  z3 |
>              | 1  x4  y4  z4 |
>              -----------------
>                      6
>
> I believe this generalizes to any dimension k, with the denominator equal to
> (k!).
>
> William J. Flis   Director of Research
> DE Technologies, Inc.
> 3620 Horizon Drive
> King of Prussia, PA 19406
> Voice: 610-270-9700 x130
> Fax: 610-270-9733
> mailto:flis at detk.com

I forgot to mention that in my case
the vertices of a k-simplex are given in R^n
where n may be greater than k.

I got this formula

           | 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?

Thanks

Michael
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