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Topic: Designing a baseball (Read 5027 times) |
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SMQ
wu::riddles Moderator
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Re: Designing a baseball
« Reply #25 on: Sep 4th, 2005, 5:49am » |
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on Sep 3rd, 2005, 6:38am, JocK wrote:| Yes, that proves that the optimal planar shape leads to a seam that can not be located on a spheroid. |
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Hmm, the differences are so small I don't feel confident they're not an unintended artifact of my hand-coded numeric solver... or even just a floating-point artifact. I 'suppose I could run the integrals through Mathematica or Maple to be more sure.
EDIT: OK, after crashing my newly-downloaded-and-registered trial Mathematica kernel several times, I finally obtained a convincing result:
In[1] := y[x_, a_, d_, i_, j_] := (1 - Abs[x/a]^i)^(1/j)
In[2] := z[x_, a_, d_, i_, j_] := (1 - Abs[-d + x/a]^i)^(1/j)
In[3] := V[a_, d_, i_, j_] := 4Integrate[y[x, a, d, i, j]z[x, a, d, i, j], {x, a(-1 + d), a}]
In[4] := L[a_, d_, i_, j_] := 4Integrate[Sqrt[1 + D[y[x, a, d, i, j], x]^2 + D[z[x, a, d, i, j], x]^2], {x, a(-1 + d), a}]
In[5] := W[a_, d_, i_, j_] := V[a, d, i, j]/L[a, d, i, j]^3
In[6] := W[a, d, i, j]
Out[6] := Integrate[((1 - Abs[x/a]^i)^(1/j))((1 - Abs[-d + x/a]^i)^(1/j)), {x, a(-1 + d), a}] /
(16Integrate[Sqrt[1 + ((i^2)(Abs[x/a]^(-2 + 2i))((1 - Abs[x/a]^i)^(-2 + 2/j))(Abs'[x/a]^2))/((a^2)(j^2)) + ((i^2)(Abs[-d + x/a]^(-2 + 2i))((1 - Abs[-d + x/a]^i)^(-2 + 2/j))(Abs'[-d + x/a]^2))/((a^2)(j^2))], {x, a(-1 + d), a}]^3)
In[7] := WN[a_?NumberQ, d_?NumberQ, i_?NumberQ, j_?NumberQ] :=
Re[NIntegrate[Re[((1 - Abs[x/a]^i)^(1/j))((1 - Abs[-d + x/a]^i)^(1/j))], {x, a(-1 + d), a}] /
(16NIntegrate[Re[Sqrt[1 + ((i^2)(Abs[x/a]^(-2 + 2i))Re[(1 - Abs[x/a]^i)^(-2 + 2/j)])/((a^2)(j^2)) + ((i^2)(Abs[-d + x/a]^(-2 + 2i))Re[(1 - Abs[-d + x/a]^i)^(-2 + 2/j)])/((a^2)(j^2))]], {x, a(-1 + d), a}]^3)]
In[8] := NMaximize[{WN[a, d, 2, 2], .5 <= a <= 2, 0 <= d < 2}, {a, d}]
Out[8] := {0.00379605, {a -> 1.12984, d -> 0.966366}}
In[9] := NMaximize[{WN[a, d, i, j], .5 <= a <= 2, 0 <= d < 2, 1 < i <= 3, 1 < j <= 3}, {a, d, i, j}]
NIntegrate::pwncvb: NIntegrate failed to converge to prescribed accuracy after 7 recursive bisections.
Out[9] := {0.00379652, {a -> 1.18269, d -> 1.01795, i -> 2.12355, j -> 2.02463}}
So I guess that convinces me that no, the optimum solution does not lie on the surface of an ellipsoid.
--SMQ
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| « Last Edit: Sep 6th, 2005, 12:13pm by SMQ » |
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--SMQ
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