# How do i find the circumsphere of a tetrahedron?

I'm looking for the most minimized equation to find the center coordinates and the radius of a tetrahedron circumsphere given four 3D points.

What I found on the internet mainly deal with the circum sphere of a flat 3D triangle, or some rough mathematical definitions, or some very single case such as regular tetrahedrons. Anyway I managed to find the equation below but I missed something :

    ->  ->      ->
let d1, d2, and d3 three vectors of any face of the triangle :

| d1x  d1y  d1z |   | x |   | d1^2 |
2 * | d2x  d2y  d2z | * | y | = | d2^2 |
| d3x  d3y  d3z |   | z |   | d3^2 |


My knowledge in this field has its limits but I think I can handle matrices and vector operations. But is the right part of the equation the square of the norm of each vectors ? (which are into a vector). Is the equation valid ? Is it just the writer who lazely forgot to write |d1|^2 ? Or Is it a common way to define some mathematical property.

PS : It's for a Delaunay Triangulation implementation. The equation (number 9) is in the following link : https://www2.mps.mpg.de/homes/daly/CSDS/t4h/tetra.htm

• Try maths stackexchange. – Majte Oct 24 '15 at 2:10
• Thanks I've found a way to compute the circumsphere there ! – herme5 Oct 24 '15 at 3:07
• @JamesAMD the link is www2.mps.mpg.de/homes/daly/CSDS/t4h/tetra.htm. – herme5 Oct 24 '15 at 11:02
• @herme5, feel free to post your own answer here on how you are calculating the answer. Many people may come here in the future hoping to find the answer, and you sharing it will be valuable to them. It's totally acceptable to post your own answer and even accept it. – Tim Holt Feb 4 '18 at 23:57
• Thanks for the notice @TimHolt. I’ll do it ! Nevertheless I am not sure anymore how I did it, it was more than 2 years ago ! just let me find and take a look at my old implementation – herme5 Feb 5 '18 at 1:53