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Re: Sphere formula

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109524] Re: Sphere formula
  • From: Ray Koopman <koopman at sfu.ca>
  • Date: Mon, 3 May 2010 06:12:25 -0400 (EDT)
  • References: <hr65rj$jq6$1@smc.vnet.net> <hrbach$hq9$1@smc.vnet.net>

On May 2, 2:35 am, Ray Koopman <koop... at sfu.ca> wrote:
> On May 1, 3:51 am, "Alexander Elkins" <alexander_elk... at hotmail.com> wrote:
>> "S. B. Gray" <stev... at ROADRUNNER.COM> wrote in messagenews:hrbach$hq9$1 at smc.vnet.net...
>>> On 4/27/2010 1:05 AM, S. B. Gray wrote:
>>>> 1. The center of a sphere through 4 points has a very nice determinant
>>>> form. (http://mathworld.wolfram.com/Sphere.html) What I want is a nice
>>>> formula for the center of a sphere through 3 points, where the center is
>>>> in the plane of the three points. I have a formula but it's a horrible
>>>> mess of hundreds of lines, even after FullSimplify.
>>>>
>>>> 2. (Unlikely) Is there a way to get Mathematica to put a long formula
>>>> into a matrix/determinant form if there is a nice one?
>>>>
>>>> Any tips will be appreciated.
>>>>
>>>> Steve Gray
>>>
>>> Thanks to everyone who answered my question, but there is a simpler
>>> answer. I forgot the simple fact that any linear combination of two
>>> vectors lies in the plane of the two vectors.
>>>
>>> Let the three points be p1,p2,p3. Consider the linear function
>>> p=b(p2-p1)+c(p3-p1) where b,c are to be determined and p is the desired
>>> center. Now do
>>>
>>> Solve[{Norm(p-p1)==Norm(p-p2),Norm(p-p1)==Norm(p-p3)},{b,c}].
>>>
>>> This gives b,c and therefore p, which will be equidistant from p1,p2,
>>> and p3 and lie in their plane. Very simple. (I used (p-p1).(p-p1) etc.
>>> instead of Norm.)
>>>
>>> Steve Gray
>>
>> Unless I misinterpreted something in this posting, the following does not
>> give the expected center point {1, 2, 3} using Ray Koopman' s posted values
>> for {p1, p2, p3} as the result:
>>
>> In[1]:=With[{p1={0.018473,-1.1359,-0.768653},
>>              p2={2.51514,5.25315,6.48158},
>>              p3={0.818313,-0.881007,-1.0825}},
>>              With[{p=b(p2-p1)+c(p3-p1)},
>>                   p/.NSolve[{(p-p1).(p-p1)==(p-p2).(p-p2),
>>                              (p-p1).(p-p1)==(p-p3).(p-p3)},{b,c}]]]
>>
>> Out[1]={{0.797494,2.34596,2.76487}}
>> [...]
>
> The fix is simple: just change the definition of p.
>
> Block[{p1 = {0.018473,-1.1359,-0.768653},
>        p2 = {2.51514,5.25315,6.48158},
>        p3 = {0.818313,-0.881007,-1.0825}, b,c,p},
>       p = (1-b-c)p1 + b*p2 + c*p3;
>       p /. Flatten@Solve[{(p-p1).(p-p1)==(p-p2).(p-p2),
>                           (p-p1).(p-p1)==(p-p3).(p-p3)},{b,c}]]
>
> {1.,2.,3.}

Or you could shift to put one point at the origin:

Block[{p1 = {0.018473,-1.1359,-0.768653},
       p2 = {2.51514,5.25315,6.48158},
       p3 = {0.818313,-0.881007,-1.0825}, b,c,q,q2,q3},
      q2 = p2-p1; q3 = p3-p1; q = b*q2 + c*q3;
      p1 + q/.Flatten at Solve[{q.q==(q-q2).(q-q2),
                             q.q==(q-q3).(q-q3)},{b,c}]]

{1.,2.,3.}


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