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Re: Re: Re: Sphere through 4 points


Daniel Lichtblau wrote:
> danl at wolfram.com wrote:
> 
>>>	Given 4 points in 3-space not lying in a plane, I want the
>>>center and radius of the incident sphere. I know about the elegant
>>>determinant solution (4 or 5 determinants, each 4x4, plus a few
>>>arithmetic operations) and I'm using that, but maybe someone has coded
>>>a faster version.
>>>	Thanks for any information.
>>>
>>>Steve Gray
>>> [...]
> 
> If raw speed is the issue, the variant below is faster.
> 
> centerAndRadius2[data_] :=
>   Module[{x, y, z, x0, y0, z0, rsqr, coeffs, rules, polys, lpolys, cen},
>    rules = (Thread[{x, y, z} -> #1] &) /@ data;
>    polys = x^2 - 2*x*x0 + y^2 - 2*y*y0 + z^2 - 2*z*z0 /. rules;
>    lpolys = Rest[polys] - First[polys];
>    coeffs = Normal[CoefficientArrays[lpolys, {x0, y0, z0}]];
>    cen = LinearSolve[Last[coeffs], -First[coeffs]];
>    {cen, Norm[cen - First[data]]}]
> [...]

Last time on this, I promise (well, I hope, anyway). Significantly 
faster still, as well as working in arbitrary dimension, is:

centerAndRadius3[data_] :=
   Module[{coeffs, rhs, soln, cen},
    rhs = Map[#.#&,data];
    coeffs = Map[Append[#,1]&,data];
    soln = LinearSolve[coeffs,rhs];
    cen = Most[soln];
    {cen/2,Sqrt[Last[soln]+cen.cen/4]}
    ]

Daniel Lichtblau
Wolfram Research


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