Re: A challenging 3 equations and 3 unknowns

The is post on this forum about finding the
location
and size of a
circle using three ultrasonic sensors.  The thread is
here:


http://www.plctalk.net/qanda/showthread.php?p=316629#post316629

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]);

This sort of thing is very easy to do
symbolically
with resultants. I only have a minute now, but
you
might look at


http://fordham.academia.edu/RobertLewis/Papers/82784/
Apollonius-Probl...

Robert H. Lewis
Fordham University

  I computed the 3 resultants. Each takes about
0.019 seconds.  This probably duplicates Dan
Lichtblau's result, but here is the resultant for x0
in nested form: .......
+ ((-2y1^2)x2^2 - y1^4 + (-2x1^2)y1^2 - x1^4)y2^2 +
((2y1^3 + (2x1^2)y1)x2^2)y2 + (-y1^2)
x2^4

It's just quadratic, so this ought to be quite
useful.


Someone should write a universal graphical interface
to Fermat using
wxWidgets or Java Swing or similar.

Please do!


If you read the first post I posted a link to the
problem on www.plcs.net. This for for someone else and I doubt they need a 3d
version. I have been e-mail the complete solution. At least I think
it is. It is so long it is hard to make heads or tails out of it. In
any case the OP on the plc forum said this must run on a PLC so the
symbolic solution is way too complex. I agree with Daniel Lichtblau
that a practical solution will require iteration.

Peter Nachtwey

I just read the post you quote from the other forum. What is a PLC? I gather this is some kind
of very small processor?

If, as the original post indicates, the three points are on an equilateral triangle, that could probably simplify the formulas.

Robert H. Lewis
Fordham University
.