Re: Sphere formula

*To*: mathgroup at smc.vnet.net*Subject*: [mg109538] Re: Sphere formula*From*: Ray Koopman <koopman at sfu.ca>*Date*: Tue, 4 May 2010 06:29:42 -0400 (EDT)*References*: <hr65rj$jq6$1@smc.vnet.net> <hrbach$hq9$1@smc.vnet.net>

On May 3, 3:10 am, "Alexander Elkins" <alexander_elk... at hotmail.com> wrote: > "Ray Koopman" <koop... at sfu.ca> wrote in message [...] >> 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.} > > As shown, Solve complains and gives no result: > > In[1]:=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}]] > > During evaluation of In[1]:= Solve::tdep: The equations appear to \ > involve the variables to be solved for in an essentially \ > non-algebraic way. >> > > During evaluation of In[1]:= Solve::tdep: The equations appear to \ > involve the variables to be solved for in an essentially \ > non-algebraic way. >> > > During evaluation of In[1]:= ReplaceAll::reps: {Solve[<<1>>,{b,c}]} \ > is neither a list of replacement rules nor a valid dispatch table, \ > and so cannot be used for replacing. >> > > During evaluation of In[1]:= Solve::tdep: The equations appear to \ > involve the variables to be solved for in an essentially \ > non-algebraic way. >> > > During evaluation of In[1]:= General::stop: Further output of \ > Solve::tdep will be suppressed during this calculation. >> > > During evaluation of In[1]:= ReplaceAll::reps: {Solve[<<1>>,{b,c}]} \ > is neither a list of replacement rules nor a valid dispatch table, \ > and so cannot be used for replacing. >> > > Out[1]= {2.51514 b + 0.018473 (1 - b - c) + 0.818313 c, > 5.25315 b - 1.1359 (1 - b - c) - 0.881007 c, > 6.48158 b - 0.768653 (1 - b - c) - > 1.0825 c} /.Solve[{(0.768653+ 6.48158 b - 0.768653 (1 - b - c) - > 1.0825 c)^2 + (1.1359+ 5.25315 b - 1.1359 (1 - b - c) - > 0.881007 c)^2 + (-0.018473 + 2.51514 b + > 0.018473 (1 - b - c) + 0.818313 c)^2 == (-6.48158 + > 6.48158 b - 0.768653 (1 - b - c) - 1.0825 c)^2 + (-5.25315= + > 5.25315 b - 1.1359 (1 - b - c) - 0.881007 c)^2 + (-2.51514= + > 2.51514 b + 0.018473 (1 - b - c) + 0.818313 c)^2, (0.76865= 3+ > 6.48158 b - 0.768653 (1 - b - c) - 1.0825 c)^2 + (1.1359+ > 5.25315 b - 1.1359 (1 - b - c) - 0.881007 c)^2 + (-0.01847= 3 + > 2.51514 b + 0.018473 (1 - b - c) + 0.818313 c)^2 == (1= .0825+ > 6.48158 b - 0.768653 (1 - b - c) - 1.0825 c)^2 + (0.881007= + > 5.25315 b - 1.1359 (1 - b - c) - 0.881007 c)^2 + (-0.81831= 3 + > 2.51514 b + 0.018473 (1 - b - c) + 0.818313 c)^2}, {b, c}] > > If Solve is changed to NSolve the correct result is returned: > > In[2]:=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@NSolve[{(p-p1).(p-p1)==(p-p2).(p-p2), > (p-p1).(p-p1)==(p-p3).(p-p3)},{b,c}]] > > Out[2]={1.,2.,3.} The code I posted works nicely in v5.2, which is what I usually use. It will also give a symbolic solution if the points are defined as p1 = {x1,y1,z1}, etc., but the result is quite long.