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Obtaining the circumsphere of a simplex from the Cayley Menger matrix
Posted:
May 20, 2013 5:02 PM


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 CayleyMenger matrix.
In 3D, if the inverse CayleyMenger 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



