Help solving equations

[quicksilver]

I have these equations I was working on that although seemed simple, turned out to be deceptively hard.


r^2/s1^4 = (x-x1)^2 + (y-y1)^2
r^2/s2^4 = (x-x2)^2 + (y-y2)^2
r^2/s3^4 = (x-x3)^2 + (y-y3)^2

I need to solve for r, x, and y in terms of s1, s2, s3, x1, x2, x3, y1, y2, y3.

I tried solving by hand and was able to get it down to just one equation with one unknown but the equation was so complicated I could not figure out how to solve it. Perhaps my approach was not correct, could anyone give me some pointers?


Below is about how far I got, but the equation got to be far to complicated to work with.

We can easily remove the r from the first equation and simplify it to two equations.


s1^4*((x-x1)^2+(y-y1)^2) = s2^4*((x-x2)^2+(y-y2)^2)
s1^4*((x-x1)^2+(y-y1)^2) = s3^4*((x-x3)^2+(y-y3)^2)

We can take the first equation above and solve for Y:
s1^4*(x^2-2x*x1+x1^2+y^2-2y*y1+y1^2) = s2^4*(x^2-2x*x2+x2^2+y^2-2y*y2+y2^2)

s1^4*x^2-s1^4*2x*x1+s1^4*x1^2+s1^4*y^2-s1^4*2y*y1+s1^4*y1^2 = s2^4*x^2-s2^4*2x*x2+s2^4*x2^2+s2^4*y^2-s2^4*2y*y2+s2^4*y2^2
s1^4*y^2-s1^4*2y*y1-s2^4*y^2+s2^4*2y*y2 = s2^4*x^2-s2^4*2x*x2+s2^4*x2^2+s2^4*y2^2-s1^4*x^2+s1^4*2x*x1-s1^4*x1^2+s1^4*2y*y1-s1^4*y1^2
(s1^4-s2^4)*y^2+(2*s2^4*y1-2*s1^4*y2)*y = s2^4*(x^2-2x*x2+x2^2+y2^2) - s1^4(x^2-2x*x1+x1^2+y1^2)
(s1^4-s2^4)*y^2+(2*s2^4*y1-2*s1^4*y2)*y + (s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)) = 0
y = (2*s1^4*y2-2*s2^4*y1 \pm{\sqrt{(2*s2^4*y1-2*s1^4*y2)^2-4*(s1^4-s2^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)}})/(2s1^4-2s2^4)

We can solve for Y of the second equation the same way giving us:

y = (2*s1^4*y3-2*s3^4*y1 \pm{\sqrt{(2*s3^4*y1-2*s1^4*y3)^2-4*(s1^4-s3^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s3^4*(x^2-2x*x3+x3^2+y3^2)}})/(2s1^4-2s3^4)

By combining these two equations we get:

(2*s1^4*y2-2*s2^4*y1 \pm{\sqrt{(2*s2^4*y1-2*s1^4*y2)^2-4*(s1^4-s2^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)}})/(2s1^4-2s2^4) = (2*s1^4*y3-2*s3^4*y1 \pm{\sqrt{(2*s3^4*y1-2*s1^4*y3)^2-4*(s1^4-s3^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s3^4*(x^2-2x*x3+x3^2+y3^2)}})/(2s1^4-2s3^4)

This is one equation with only one unknown (x). However solving this for x eludes me. There must be a simplier way.

 

[tkhunny]

How sure are you that there IS a solution?

You've three circles with different centers and different radii. One would be wise to ponder existence of a solution (given x, y, & s) before wading through death-defying algebra.

Have you considered Polar Coordinates?

 

[quicksilver]

There is definitly a solution. If I use a ti-89 and I put the known variables in first I can use the solve function and get the answer I was looking for, however I need to find a general solution since I will not be able to use the solve feature each time for every set of input variables.

I have not tried polar coordinates since I am pretty unfamiliar with how those work.

 

[Deleted member 4993]

You will not have a solution in every case.

You have three intersecting circles. There could be one point where all the circles go through. It is similar - but harder - to find solution for three straight lines.

You can envision that there three points (centers) - now you need to choose radii such that the circles intersect at a point.

Even if there is a solution - there is no guarantee of closed-form solution.

Try to reduce confusion by renaming a group of constants - by a single constant.

for example:

Let

A = 2*s1^4*y2-2*s2^4*y1

and so on...

 

[tkhunny]

There is definitly a solution..

Only in the Complex Plane. If you must stay Real, you must consider existence.

If your centers are 10 apart and your radii are all 2, there is no Real solution. Develop existence criteria. They are rather simple. When a case fails, don't even bother to run the solution program.