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Topic: A challenging 3 equations and 3 unknowns
Replies: 20   Last Post: Apr 4, 2009 10:50 PM

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 Axel Vogt Posts: 993 Registered: 5/5/07
Re: A challenging 3 equations and 3 unknowns
Posted: Apr 1, 2009 4:07 PM

pnachtwey wrote:
> The is post on this forum about finding the location and size of a
> circle using three ultrasonic sensors. The thread is here:
>
> My wxMaxima doesn't seem to be getting the job done.
>
> eq1: (x1-x0)^2+(y1-y0)^2-(r0+r1)^2;
> eq2: (x2-x0)^2+(y2-y0)^2-(r0+r2)^2;
> eq3: (x3-x0)^2+(y3-y0)^2-(r0+r3)^2;
> eq4: solve([eq1,eq2,eq3],[x0,y0,r0]);
> grind(%);
>
> where the center of the inner circle is x0,y0 and the radius is r0.
> x1,y1 and x2,y2 and x3,y3 are the locations of the ultrasonic sensors.
> r1, r2, r3 are the distances between the sensors and the inner circle.
> It looks easy but my wxMaxima has been working on the solution for
> about a half hour and so far there is no solution.
>
> Perhaps there is a better way to approach this but I can't see how.
> Thanks
>
> Peter Nachtwey

What do you get for x0=y0 = 0? May be I got trapped, but I was
thinking of quadratic normal forms (though I do not have code
for simultaneous situations):

Each is equivalent to x[1]^2+x[2]^2-x[3]^2 over the Reals, so one
may look at x[1]^2+x[2]^2-(x[3]-r[i])^2, r[1]=0 and the other are
free.

May be I got it wrong ... but what for that special case?

Date Subject Author
3/31/09 Peter Nachtwey
3/31/09 Roman Pearce
3/31/09 Daniel Lichtblau
3/31/09 Peter Nachtwey
4/1/09 Robert H. Lewis
4/1/09 Robert H. Lewis
4/1/09 Peter Nachtwey
4/1/09 Robert H. Lewis
4/2/09 Peter Nachtwey
4/2/09 Robert H. Lewis
4/2/09 Dave Rusin
4/4/09 Peter Nachtwey
4/1/09 Daniel Lichtblau
4/1/09 Axel Vogt
4/1/09 Peter Nachtwey
4/2/09 Robert H. Lewis
4/2/09 Dave Rusin
4/4/09 Robert H. Lewis
4/4/09 Roman Pearce
4/4/09 Peter Nachtwey
4/4/09 Robert H. Lewis