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Re: Computation does not stop when solving system quadratic eqn

  • To: mathgroup at smc.vnet.net
  • Subject: [mg82563] Re: Computation does not stop when solving system quadratic eqn
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 25 Oct 2007 05:59:53 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <ffmvn3$5dl$1@smc.vnet.net>

sapsi wrote:

> I have two quadratic equations in two unknowns l1 , l2. I try to solve
> it using Solve (i could do it by hand but it's quite messy and wanted
> to Mathematica to take the trouble).
> 
> Solve[{32 k l2 r^2 - 16 l2^2 r^2 +
>     l1^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
>        16 r^2) +
>     k^2 (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
>        16 r^2) +
>     l1 (-32 l2 r^2 +
>        2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
>           nyb p0y + p0y^2 - 16 r^2)) == 0,
>   -d^2 l1^2 + l1 (-2 d^2 k - 2 d^2 l2) +
>     k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 - 2 ay nyb + nyb^2) +
>     l2^2 (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2) +
>     2 k l2 (ax^2 + ay^2 - d^2 + nxb p0x - ax (nxb + p0x) + nyb p0y -
>        ay (nyb + p0y)) == 0}, {l1, l2}]
> 
> However, the calculation does not stop and seems to run for ever.
> Could some one please suggest any modifications i need to do?

Rewrite temporarily the coefficients of l1 and l2 in a simpler form.

In[1]:= rules = {c[1] -> 32 k l2 r^2 - 16 l2^2 r^2,
    c[2] -> (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
       16 r^2),
    c[3] ->
     k^2 (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
        16 r^2),
    c[4] -> (-32 l2 r^2 +
       2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
          nyb p0y + p0y^2 - 16 r^2)),
    c[5] -> -d^2,
    c[6] -> (-2 d^2 k - 2 d^2 l2),
    c[7] ->
     k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 - 2 ay nyb + nyb^2),
    c[8] -> (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2),
    c[9] -> 2 k ,
    c[10] -> (ax^2 + ay^2 - d^2 + nxb p0x - ax (nxb + p0x) + nyb p0y -
       ay (nyb + p0y))};

In[2]:= sols =
  Solve[{c[1] + l1^2 c[2] + c[3] + l1 c[4] == 0,
    c[5] l1^2 + l1 c[6] + c[7] + l2^2 c[8] + c[9] l2 c[10] == 0}, {l1,
    l2}]

Out[2]= {{l2 -> (1/(
    4 c[2]^2 c[
      8]))(-2 c[2]^2 c[9] c[
        10] - \[Sqrt](-8 c[
           2]^2 (-2 c[1] c[2] c[5] - 2 c[2] c[3] c[5] + c[4]^2 c[5] +
            c[4] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[5] -
            c[2] c[4] c[6] -
            c[2] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[6] +
            2 c[2]^2 c[7]) c[8] + 4 c[2]^4 c[9]^2 c[10]^2)),
   l1 -> (-c[4] - Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2])/(
    2 c[2])}, {l2 -> (1/(
    4 c[2]^2 c[
      8]))(-2 c[2]^2 c[9] c[
        10] + \[Sqrt](-8 c[
           2]^2 (-2 c[1] c[2] c[5] - 2 c[2] c[3] c[5] + c[4]^2 c[5] +
            c[4] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[5] -
            c[2] c[4] c[6] -
            c[2] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[6] +
            2 c[2]^2 c[7]) c[8] + 4 c[2]^4 c[9]^2 c[10]^2)),
   l1 -> (-c[4] - Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2])/(
    2 c[2])}, {l2 -> (1/(
    4 c[2]^2 c[
      8]))(-2 c[2]^2 c[9] c[
        10] - \[Sqrt](-8 c[
           2]^2 (-2 c[1] c[2] c[5] - 2 c[2] c[3] c[5] + c[4]^2 c[5] -
            c[4] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[5] -
            c[2] c[4] c[6] +
            c[2] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[6] +
            2 c[2]^2 c[7]) c[8] + 4 c[2]^4 c[9]^2 c[10]^2)),
   l1 -> (-c[4] + Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2])/(
    2 c[2])}, {l2 -> (1/(
    4 c[2]^2 c[
      8]))(-2 c[2]^2 c[9] c[
        10] + \[Sqrt](-8 c[
           2]^2 (-2 c[1] c[2] c[5] - 2 c[2] c[3] c[5] + c[4]^2 c[5] -
            c[4] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[5] -
            c[2] c[4] c[6] +
            c[2] Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2] c[6] +
            2 c[2]^2 c[7]) c[8] + 4 c[2]^4 c[9]^2 c[10]^2)),
   l1 -> (-c[4] + Sqrt[-4 c[1] c[2] - 4 c[2] c[3] + c[4]^2])/(
    2 c[2])}}

In[3]:= sols /. rules

Out[3]= {{l2 -> (-4 k (ax^2 + ay^2 - d^2 + nxb p0x - ax (nxb + p0x) +
          nyb p0y - ay (nyb + p0y)) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
          2 ay p0y + p0y^2 -
          16 r^2)^2 - \[Sqrt](16 k^2 (ax^2 + ay^2 - d^2 + nxb p0x -
             ax (nxb + p0x) + nyb p0y - ay (nyb + p0y))^2 (ax^2 +
             ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 - 16 r^2)^4 -
          8 (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2 -
             16 r^2)^2 (2 k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 -
                2 ay nyb + nyb^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
                2 ay p0y + p0y^2 - 16 r^2)^2 +
             2 d^2 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 -
                2 nyb p0y + p0y^2 - 16 r^2) +
             2 d^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (32 k l2 r^2 -
                16 l2^2 r^2) - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2)) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2))^2 - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2))^2) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2)) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 -
                     16 r^2))^2))))/(4 (ax^2 + ay^2 - d^2 - 2 ax p0x +
         p0x^2 - 2 ay p0y + p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
         2 ay p0y + p0y^2 - 16 r^2)^2),
   l1 -> (32 l2 r^2 -
       2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
          nyb p0y + p0y^2 -
          16 r^2) - \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
             2 ay p0y + p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x +
             p0x^2 - 2 nyb p0y + p0y^2 - 16 r^2) -
          4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
             16 r^2) (32 k l2 r^2 - 16 l2^2 r^2) + (-32 l2 r^2 +
            2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
               nyb p0y + p0y^2 - 16 r^2))^2))/(2 (ax^2 + ay^2 -
         2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
         16 r^2))}, {l2 -> (-4 k (ax^2 + ay^2 - d^2 + nxb p0x -
          ax (nxb + p0x) + nyb p0y - ay (nyb + p0y)) (ax^2 + ay^2 -
          2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
          16 r^2)^2 + \[Sqrt](16 k^2 (ax^2 + ay^2 - d^2 + nxb p0x -
             ax (nxb + p0x) + nyb p0y - ay (nyb + p0y))^2 (ax^2 +
             ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 - 16 r^2)^4 -
          8 (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2 -
             16 r^2)^2 (2 k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 -
                2 ay nyb + nyb^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
                2 ay p0y + p0y^2 - 16 r^2)^2 +
             2 d^2 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 -
                2 nyb p0y + p0y^2 - 16 r^2) +

             2 d^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (32 k l2 r^2 -
                16 l2^2 r^2) - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2)) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2))^2 - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2))^2) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -

                   16 r^2)) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 -
                     16 r^2))^2))))/(4 (ax^2 + ay^2 - d^2 - 2 ax p0x +
         p0x^2 - 2 ay p0y + p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
         2 ay p0y + p0y^2 - 16 r^2)^2),
   l1 -> (32 l2 r^2 -
       2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
          nyb p0y + p0y^2 -
          16 r^2) - \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
             2 ay p0y + p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x +
             p0x^2 - 2 nyb p0y + p0y^2 - 16 r^2) -
          4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
             16 r^2) (32 k l2 r^2 - 16 l2^2 r^2) + (-32 l2 r^2 +
            2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
               nyb p0y + p0y^2 - 16 r^2))^2))/(2 (ax^2 + ay^2 -
         2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
         16 r^2))}, {l2 -> (-4 k (ax^2 + ay^2 - d^2 + nxb p0x -
          ax (nxb + p0x) + nyb p0y - ay (nyb + p0y)) (ax^2 + ay^2 -
          2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
          16 r^2)^2 - \[Sqrt](16 k^2 (ax^2 + ay^2 - d^2 + nxb p0x -
             ax (nxb + p0x) + nyb p0y - ay (nyb + p0y))^2 (ax^2 +
             ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 - 16 r^2)^4 -
          8 (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2 -
             16 r^2)^2 (2 k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 -
                2 ay nyb + nyb^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
                2 ay p0y + p0y^2 - 16 r^2)^2 +
             2 d^2 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 -
                2 nyb p0y + p0y^2 - 16 r^2) +
             2 d^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (32 k l2 r^2 -
                16 l2^2 r^2) - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2)) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2))^2 + (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2))^2) +
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2)) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 -
                     16 r^2))^2))))/(4 (ax^2 + ay^2 - d^2 - 2 ax p0x +
         p0x^2 - 2 ay p0y + p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
         2 ay p0y + p0y^2 - 16 r^2)^2),
   l1 -> (32 l2 r^2 -
       2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
          nyb p0y + p0y^2 -
          16 r^2) + \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
             2 ay p0y + p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x +
             p0x^2 - 2 nyb p0y + p0y^2 - 16 r^2) -
          4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
             16 r^2) (32 k l2 r^2 - 16 l2^2 r^2) + (-32 l2 r^2 +
            2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
               nyb p0y + p0y^2 - 16 r^2))^2))/(2 (ax^2 + ay^2 -
         2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
         16 r^2))}, {l2 -> (-4 k (ax^2 + ay^2 - d^2 + nxb p0x -
          ax (nxb + p0x) + nyb p0y - ay (nyb + p0y)) (ax^2 + ay^2 -
          2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
          16 r^2)^2 + \[Sqrt](16 k^2 (ax^2 + ay^2 - d^2 + nxb p0x -
             ax (nxb + p0x) + nyb p0y - ay (nyb + p0y))^2 (ax^2 +
             ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 - 16 r^2)^4 -
          8 (ax^2 + ay^2 - d^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
             p0y^2 -
             16 r^2)^2 (2 k^2 (ax^2 + ay^2 - d^2 - 2 ax nxb + nxb^2 -
                2 ay nyb + nyb^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
                2 ay p0y + p0y^2 - 16 r^2)^2 +
             2 d^2 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x + p0x^2 -
                2 nyb p0y + p0y^2 - 16 r^2) +
             2 d^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                p0y^2 - 16 r^2) (32 k l2 r^2 -
                16 l2^2 r^2) - (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2)) -
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2))^2 + (-2 d^2 k - 2 d^2 l2) (ax^2 + ay^2 -
                2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
                16 r^2) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 - 16 r^2))^2) +
             d^2 (-32 l2 r^2 +
                2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                   ay (nyb - p0y) - nyb p0y + p0y^2 -
                   16 r^2)) \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x +
                    p0x^2 - 2 ay p0y + p0y^2 - 16 r^2) (nxb^2 +
                    nyb^2 - 2 nxb p0x + p0x^2 - 2 nyb p0y + p0y^2 -
                    16 r^2) -
                 4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y +
                    p0y^2 - 16 r^2) (32 k l2 r^2 -
                    16 l2^2 r^2) + (-32 l2 r^2 +
                   2 k (ax (nxb - p0x) - nxb p0x + p0x^2 +
                     ay (nyb - p0y) - nyb p0y + p0y^2 -
                     16 r^2))^2))))/(4 (ax^2 + ay^2 - d^2 - 2 ax p0x +
         p0x^2 - 2 ay p0y + p0y^2) (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
         2 ay p0y + p0y^2 - 16 r^2)^2),
   l1 -> (32 l2 r^2 -
       2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
          nyb p0y + p0y^2 -
          16 r^2) + \[Sqrt](-4 k^2 (ax^2 + ay^2 - 2 ax p0x + p0x^2 -
             2 ay p0y + p0y^2 - 16 r^2) (nxb^2 + nyb^2 - 2 nxb p0x +
             p0x^2 - 2 nyb p0y + p0y^2 - 16 r^2) -
          4 (ax^2 + ay^2 - 2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 -
             16 r^2) (32 k l2 r^2 - 16 l2^2 r^2) + (-32 l2 r^2 +
            2 k (ax (nxb - p0x) - nxb p0x + p0x^2 + ay (nyb - p0y) -
               nyb p0y + p0y^2 - 16 r^2))^2))/(2 (ax^2 + ay^2 -
         2 ax p0x + p0x^2 - 2 ay p0y + p0y^2 - 16 r^2))}}

Regards,
-- 
Jean-Marc


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