Obtaining the circumsphere of a simplex from the Cayley Menger matrix

[Gerard Westendorp]

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