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Topic: Obtaining the circumsphere of a simplex from the Cayley Menger matrix
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Gerard Westendorp

Posts: 59
Registered: 12/13/04
Obtaining the circumsphere of a simplex from the Cayley Menger matrix
Posted: May 20, 2013 5:02 PM
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It is well known how to obtain the circumsphere of a simplex, given the
coordinates of the vertices. But there is a rather nice expression that
uses only the squared distances. One can read of the radius and
barycentric coordinates directly from the Cayley-Menger matrix.

In 3D, if the inverse Cayley-Menger matrix is:

  ( -2r^2  a     b    c     d  )
  (   a    0   C_12  C_13  C_14)
  (   b   C_21   0   C_23  C_24)
  (   c   C_31  C_32   0   C_34)
  (   d   C_41  C_42 C_43   0  )

Then the barycentric coordinates of the circumsphere are (a,b,c,d) and
r is the radius

I put the proof with on this web site:
http://westy31.home.xs4all.nl/Circumsphere/ncircumsphere.htm


Gerard



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