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MathGroup Archive 2006

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Re: Sovling non-linear eq-sys (beginners question)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71232] Re: [mg71207] Sovling non-linear eq-sys (beginners question)
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 11 Nov 2006 03:38:17 -0500 (EST)
  • Reply-to: hanlonr at cox.net

A numeric solution is more practical

Needs["Graphics`"];

ellipseParameters[pts_List/;Length[pts]==4,
      {ox_Symbol,oy_Symbol,s_Symbol,b_Symbol}]:=
    Module[{x,y,eqn,sub,param},
      eqn=(x-ox)^2/s^2+(y-oy)^2/b^2==1;
      sub=Thread[{x,y}->#]&/@pts;
      param=Select[Solve[eqn/.sub,{ox,oy,s,b}],
            FreeQ[#,Complex]&&(b>0 && s>0/.#)&]//Flatten];

While[pts=Table[
        {Random[Real,{1/4,3/4}],Random[Real,{1/4,3/4}]},{4}];
    Length[param=ellipseParameters[pts,{ox,oy,s,b}]]<1,];

eqn=(x-ox)^2/s^2+(y-oy)^2/b^2==1/.param

ImplicitPlot[eqn, {x,-0.01,1.5},
    Epilog->{Red,AbsolutePointSize[6],Point/@pts},
    ImageSize->288,Frame->True,Axes->False,
    PlotRange->{{0,1},{0,1}}];


Bob Hanlon

---- 17538 at student.hhs.se wrote: 
> I wonder how I can use Mathematica to solve a system of non-linear
> equations.
> 
> If I want to find the parameters of an ellipse, given coordinates of
> some points on it, then I have a system of four non-linear equations.
> Where should I go from here in order to make Mathematica help me with
> the substitutions necessary to express each parameter as a function of
> the coordinates of the points?
> 
> Here follows an example:
> 
> The formula of an ellipse (with an axis parallell to the x-axis of the
> coordinate system) is:
> 
> (x-ox)^2 / s^2 + (y-oy)^2 / b^2 = 1
> 
> where
> s: half length of small axis.
> b: half length of big axis.
> ox: x-coordinate of the center of the ellipse.
> oy: y-coordinate of the center of the ellipse.
> x,y: coordinates of a point on perimeter of the ellipse.
> 
> I solve for each of the 4 parameters s, b, ox, oy and insert unique
> {x,y}-values in each equation:
> s = f(x1,y1,b,ox,oy)
> b = f(x2,y2,s,ox,oy)
> ox = f(x3,y3,b,s,oy)
> oy = f(x4,y4,b,s,ox)
> 
> [Of course, there are two functions for each parameter, one describing
> the upper half and the other describing the lower half, so to speak.]
> 
> Now I should be able to substitute my way to a solution like this:
> 
> Inserting b = f(x2,y2,s,ox,oy) in s = f(x1,y1,b,ox,oy) gives:
> s = f(x1,y1,f(x2,y2,s,ox,oy),ox,oy)
> 
> Solving for s gives:
> s = f(x1,y1,x2,y2,ox,oy)
> 
> And then continue with inserting oy = f(...) in s' = f(...) above and
> again solving for s, then the same with ox so that I arrive at s =
> f(x1,y1,x2,y2,x3,y3,x4,y4). Then the procedure is repeated with each of
> the other parameters b, ox, oy.
> 
> The thing is that it gets quite tedious even for a pretty simple
> problem like this. And what about doing it for an ellipse with a tilted
> axis (not parallell with the x-axis of the coordinate system)? That
> would introduce a new parameter, a new equation in the system and
> significantly longer equations too (maybe it is pointless to work with
> algebraic solutions in that case anyway).
> 
> So, what tricks can Mathematica perform in order to do the work for me?
> 
> I'm grateful for any comments or reference that could help me!
> 


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