sampling_using_integral_transforms.nb

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Cell["\<\
Random variate generation/Importance sampling using integral transforms\
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 StyleBox["Wellington, New Zealand\nApril 5, 2020     (updated May 10, 2020)",
  
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We discuss various analytic and exact methods for deriving \
importance-sampling procedures for 1D distributions f(r) on the half line r \
\[Element] [0,\[Infinity]).  Specifically:\
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Sampling as a superposition of exponentials using the inverse Laplace \
transform\
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 "Simulating the linear transport of light or particles often involves \
drawing random variates from some given distribution on the half line, such \
as free-path length sampling in homogeneous classical media where we draw \
samples from the collision-rate density ",
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There are times where we wish to sample other distributions f(r) on the half \
line where the CDF inversion method is not possible analytically.  For \
example, in non-classical transport where free-path lengths between collision \
are not exponentially-distributed, we need sampling procedures for a large \
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that is normalized on the half line ",
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 "If f(r) can be expressed as a convolution of more than 1 distribution, each \
of which has known sampling procedures, then f(r) can be sampled as a whole \
by sampling each of the distributions in its convolution and summing the \
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We find the exponential, which we know how to sample.  Therefore, Erlang-2 \
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       RoundingRadius->0], RectangleBox[{0.14, 0}, {0.16, 1.36},
       RoundingRadius->0], RectangleBox[{0.16, 0}, {0.18, 1.23},
       RoundingRadius->0], RectangleBox[{0.18, 0}, {0.2, 1.072},
       RoundingRadius->0], RectangleBox[{0.2, 0}, {0.22, 1.14},
       RoundingRadius->0], RectangleBox[{0.22, 0}, {0.24, 1.076},
       RoundingRadius->0], RectangleBox[{0.24, 0}, {0.26, 1.045},
       RoundingRadius->0], RectangleBox[{0.26, 0}, {0.28, 0.945},
       RoundingRadius->0], RectangleBox[{0.28, 0}, {0.3, 0.884},
       RoundingRadius->0], RectangleBox[{0.3, 0}, {0.32, 0.843},
       RoundingRadius->0], RectangleBox[{0.32, 0}, {0.34, 0.796},
       RoundingRadius->0], RectangleBox[{0.34, 0}, {0.36, 0.828},
       RoundingRadius->0], RectangleBox[{0.36, 0}, {0.38, 0.758},
       RoundingRadius->0], RectangleBox[{0.38, 0}, {0.4, 0.718},
       RoundingRadius->0], RectangleBox[{0.4, 0}, {0.42, 0.712},
       RoundingRadius->0], RectangleBox[{0.42, 0}, {0.44, 0.666},
       RoundingRadius->0], RectangleBox[{0.44, 0}, {0.46, 0.652},
       RoundingRadius->0], RectangleBox[{0.46, 0}, {0.48, 0.693},
       RoundingRadius->0], RectangleBox[{0.48, 0}, {0.5, 0.607},
       RoundingRadius->0], RectangleBox[{0.5, 0}, {0.52, 0.554},
       RoundingRadius->0], RectangleBox[{0.52, 0}, {0.54, 0.59},
       RoundingRadius->0], RectangleBox[{0.54, 0}, {0.56, 0.524},
       RoundingRadius->0], RectangleBox[{0.56, 0}, {0.58, 0.497},
       RoundingRadius->0], RectangleBox[{0.58, 0}, {0.6, 0.508},
       RoundingRadius->0], RectangleBox[{0.6, 0}, {0.62, 0.474},
       RoundingRadius->0], RectangleBox[{0.62, 0}, {0.64, 0.5},
       RoundingRadius->0], RectangleBox[{0.64, 0}, {0.66, 0.445},
       RoundingRadius->0], RectangleBox[{0.66, 0}, {0.68, 0.462},
       RoundingRadius->0], RectangleBox[{0.68, 0}, {0.7, 0.44},
       RoundingRadius->0], RectangleBox[{0.7, 0}, {0.72, 0.419},
       RoundingRadius->0], RectangleBox[{0.72, 0}, {0.74, 0.394},
       RoundingRadius->0], RectangleBox[{0.74, 0}, {0.76, 0.41},
       RoundingRadius->0], RectangleBox[{0.76, 0}, {0.78, 0.379},
       RoundingRadius->0], RectangleBox[{0.78, 0}, {0.8, 0.372},
       RoundingRadius->0], RectangleBox[{0.8, 0}, {0.82, 0.364},
       RoundingRadius->0], RectangleBox[{0.82, 0}, {0.84, 0.308},
       RoundingRadius->0], RectangleBox[{0.84, 0}, {0.86, 0.333},
       RoundingRadius->0], RectangleBox[{0.86, 0}, {0.88, 0.337},
       RoundingRadius->0], RectangleBox[{0.88, 0}, {0.9, 0.303},
       RoundingRadius->0], RectangleBox[{0.9, 0}, {0.92, 0.3},
       RoundingRadius->0], RectangleBox[{0.92, 0}, {0.94, 0.304},
       RoundingRadius->0], RectangleBox[{0.94, 0}, {0.96, 0.292},
       RoundingRadius->0], RectangleBox[{0.96, 0}, {0.98, 0.276},
       RoundingRadius->0], RectangleBox[{0.98, 0}, {1., 0.281},
       RoundingRadius->0], RectangleBox[{1., 0}, {1.02, 0.261},
       RoundingRadius->0], RectangleBox[{1.02, 0}, {1.04, 0.256},
       RoundingRadius->0], RectangleBox[{1.04, 0}, {1.06, 0.253},
       RoundingRadius->0], RectangleBox[{1.06, 0}, {1.08, 0.234},
       RoundingRadius->0], RectangleBox[{1.08, 0}, {1.1, 0.241},
       RoundingRadius->0], RectangleBox[{1.1, 0}, {1.12, 0.222},
       RoundingRadius->0], RectangleBox[{1.12, 0}, {1.14, 0.218},
       RoundingRadius->0], RectangleBox[{1.14, 0}, {1.16, 0.222},
       RoundingRadius->0], RectangleBox[{1.16, 0}, {1.18, 0.213},
       RoundingRadius->0], RectangleBox[{1.18, 0}, {1.2, 0.208},
       RoundingRadius->0], RectangleBox[{1.2, 0}, {1.22, 0.204},
       RoundingRadius->0], RectangleBox[{1.22, 0}, {1.24, 0.186},
       RoundingRadius->0], RectangleBox[{1.24, 0}, {1.26, 0.19},
       RoundingRadius->0], RectangleBox[{1.26, 0}, {1.28, 0.22},
       RoundingRadius->0], RectangleBox[{1.28, 0}, {1.3, 0.173},
       RoundingRadius->0], RectangleBox[{1.3, 0}, {1.32, 0.152},
       RoundingRadius->0], RectangleBox[{1.32, 0}, {1.34, 0.183},
       RoundingRadius->0], RectangleBox[{1.34, 0}, {1.36, 0.152},
       RoundingRadius->0], RectangleBox[{1.36, 0}, {1.38, 0.145},
       RoundingRadius->0], RectangleBox[{1.38, 0}, {1.4, 0.14},
       RoundingRadius->0], RectangleBox[{1.4, 0}, {1.42, 0.151},
       RoundingRadius->0], RectangleBox[{1.42, 0}, {1.44, 0.147},
       RoundingRadius->0], RectangleBox[{1.44, 0}, {1.46, 0.147},
       RoundingRadius->0], RectangleBox[{1.46, 0}, {1.48, 0.152},
       RoundingRadius->0], RectangleBox[{1.48, 0}, {1.5, 0.136},
       RoundingRadius->0], RectangleBox[{1.5, 0}, {1.52, 0.118},
       RoundingRadius->0], RectangleBox[{1.52, 0}, {1.54, 0.134},
       RoundingRadius->0], RectangleBox[{1.54, 0}, {1.56, 0.125},
       RoundingRadius->0], RectangleBox[{1.56, 0}, {1.58, 0.127},
       RoundingRadius->0], RectangleBox[{1.58, 0}, {1.6, 0.14},
       RoundingRadius->0], RectangleBox[{1.6, 0}, {1.62, 0.097},
       RoundingRadius->0], RectangleBox[{1.62, 0}, {1.64, 0.095},
       RoundingRadius->0], RectangleBox[{1.64, 0}, {1.66, 0.105},
       RoundingRadius->0], RectangleBox[{1.66, 0}, {1.68, 0.111},
       RoundingRadius->0], RectangleBox[{1.68, 0}, {1.7, 0.114},
       RoundingRadius->0], RectangleBox[{1.7, 0}, {1.72, 0.104},
       RoundingRadius->0], RectangleBox[{1.72, 0}, {1.74, 0.088},
       RoundingRadius->0], RectangleBox[{1.74, 0}, {1.76, 0.097},
       RoundingRadius->0], RectangleBox[{1.76, 0}, {1.78, 0.087},
       RoundingRadius->0], RectangleBox[{1.78, 0}, {1.8, 0.085},
       RoundingRadius->0], RectangleBox[{1.8, 0}, {1.82, 0.082},
       RoundingRadius->0], RectangleBox[{1.82, 0}, {1.84, 0.088},
       RoundingRadius->0], RectangleBox[{1.84, 0}, {1.86, 0.088},
       RoundingRadius->0], RectangleBox[{1.86, 0}, {1.88, 0.087},
       RoundingRadius->0], RectangleBox[{1.88, 0}, {1.9, 0.077},
       RoundingRadius->0], RectangleBox[{1.9, 0}, {1.92, 0.07},
       RoundingRadius->0], RectangleBox[{1.92, 0}, {1.94, 0.077},
       RoundingRadius->0], RectangleBox[{1.94, 0}, {1.96, 0.078},
       RoundingRadius->0], RectangleBox[{1.96, 0}, {1.98, 0.075},
       RoundingRadius->0], RectangleBox[{1.98, 0}, {2., 0.071},
       RoundingRadius->0], RectangleBox[{2., 0}, {2.02, 0.067},
       RoundingRadius->0], RectangleBox[{2.02, 0}, {2.04, 0.071},
       RoundingRadius->0], RectangleBox[{2.04, 0}, {2.06, 0.063},
       RoundingRadius->0], RectangleBox[{2.06, 0}, {2.08, 0.049},
       RoundingRadius->0], RectangleBox[{2.08, 0}, {2.1, 0.056},
       RoundingRadius->0], RectangleBox[{2.1, 0}, {2.12, 0.072},
       RoundingRadius->0], RectangleBox[{2.12, 0}, {2.14, 0.062},
       RoundingRadius->0], RectangleBox[{2.14, 0}, {2.16, 0.069},
       RoundingRadius->0], RectangleBox[{2.16, 0}, {2.18, 0.056},
       RoundingRadius->0], RectangleBox[{2.18, 0}, {2.2, 0.065},
       RoundingRadius->0], RectangleBox[{2.2, 0}, {2.22, 0.046},
       RoundingRadius->0], RectangleBox[{2.22, 0}, {2.24, 0.054},
       RoundingRadius->0], RectangleBox[{2.24, 0}, {2.26, 0.059},
       RoundingRadius->0], RectangleBox[{2.26, 0}, {2.28, 0.053},
       RoundingRadius->0], RectangleBox[{2.28, 0}, {2.3, 0.044},
       RoundingRadius->0], RectangleBox[{2.3, 0}, {2.32, 0.046},
       RoundingRadius->0], RectangleBox[{2.32, 0}, {2.34, 0.04},
       RoundingRadius->0], RectangleBox[{2.34, 0}, {2.36, 0.039},
       RoundingRadius->0], RectangleBox[{2.36, 0}, {2.38, 0.041},
       RoundingRadius->0], RectangleBox[{2.38, 0}, {2.4, 0.044},
       RoundingRadius->0], RectangleBox[{2.4, 0}, {2.42, 0.057},
       RoundingRadius->0], RectangleBox[{2.42, 0}, {2.44, 0.037},
       RoundingRadius->0], RectangleBox[{2.44, 0}, {2.46, 0.038},
       RoundingRadius->0], RectangleBox[{2.46, 0}, {2.48, 0.04},
       RoundingRadius->0], RectangleBox[{2.48, 0}, {2.5, 0.039},
       RoundingRadius->0], RectangleBox[{2.5, 0}, {2.52, 0.04},
       RoundingRadius->0], RectangleBox[{2.52, 0}, {2.54, 0.039},
       RoundingRadius->0], RectangleBox[{2.54, 0}, {2.56, 0.043},
       RoundingRadius->0], RectangleBox[{2.56, 0}, {2.58, 0.029},
       RoundingRadius->0], RectangleBox[{2.58, 0}, {2.6, 0.043},
       RoundingRadius->0], RectangleBox[{2.6, 0}, {2.62, 0.036},
       RoundingRadius->0], RectangleBox[{2.62, 0}, {2.64, 0.035},
       RoundingRadius->0], RectangleBox[{2.64, 0}, {2.66, 0.029},
       RoundingRadius->0], RectangleBox[{2.66, 0}, {2.68, 0.031},
       RoundingRadius->0], RectangleBox[{2.68, 0}, {2.7, 0.035},
       RoundingRadius->0], RectangleBox[{2.7, 0}, {2.72, 0.036},
       RoundingRadius->0], RectangleBox[{2.72, 0}, {2.74, 0.031},
       RoundingRadius->0], RectangleBox[{2.74, 0}, {2.76, 0.033},
       RoundingRadius->0], RectangleBox[{2.76, 0}, {2.78, 0.03},
       RoundingRadius->0], RectangleBox[{2.78, 0}, {2.8, 0.03},
       RoundingRadius->0], RectangleBox[{2.8, 0}, {2.82, 0.033},
       RoundingRadius->0], RectangleBox[{2.82, 0}, {2.84, 0.032},
       RoundingRadius->0], RectangleBox[{2.84, 0}, {2.86, 0.024},
       RoundingRadius->0], RectangleBox[{2.86, 0}, {2.88, 0.021},
       RoundingRadius->0], RectangleBox[{2.88, 0}, {2.9, 0.024},
       RoundingRadius->0], RectangleBox[{2.9, 0}, {2.92, 0.028},
       RoundingRadius->0], RectangleBox[{2.92, 0}, {2.94, 0.026},
       RoundingRadius->0], RectangleBox[{2.94, 0}, {2.96, 0.022},
       RoundingRadius->0], RectangleBox[{2.96, 0}, {2.98, 0.012},
       RoundingRadius->0], RectangleBox[{2.98, 0}, {3., 0.024},
       RoundingRadius->0], RectangleBox[{3., 0}, {3.02, 0.027},
       RoundingRadius->0], RectangleBox[{3.02, 0}, {3.04, 0.02},
       RoundingRadius->0], RectangleBox[{3.04, 0}, {3.06, 0.01},
       RoundingRadius->0], RectangleBox[{3.06, 0}, {3.08, 0.017},
       RoundingRadius->0], RectangleBox[{3.08, 0}, {3.1, 0.024},
       RoundingRadius->0], RectangleBox[{3.1, 0}, {3.12, 0.017},
       RoundingRadius->0], RectangleBox[{3.12, 0}, {3.14, 0.013},
       RoundingRadius->0], RectangleBox[{3.14, 0}, {3.16, 0.022},
       RoundingRadius->0], RectangleBox[{3.16, 0}, {3.18, 0.013},
       RoundingRadius->0], RectangleBox[{3.18, 0}, {3.2, 0.016},
       RoundingRadius->0], RectangleBox[{3.2, 0}, {3.22, 0.018},
       RoundingRadius->0], RectangleBox[{3.22, 0}, {3.24, 0.012},
       RoundingRadius->0], RectangleBox[{3.24, 0}, {3.26, 0.029},
       RoundingRadius->0], RectangleBox[{3.26, 0}, {3.28, 0.008},
       RoundingRadius->0], RectangleBox[{3.28, 0}, {3.3, 0.019},
       RoundingRadius->0], RectangleBox[{3.3, 0}, {3.32, 0.019},
       RoundingRadius->0], RectangleBox[{3.32, 0}, {3.34, 0.019},
       RoundingRadius->0], RectangleBox[{3.34, 0}, {3.36, 0.017},
       RoundingRadius->0], RectangleBox[{3.36, 0}, {3.38, 0.015},
       RoundingRadius->0], RectangleBox[{3.38, 0}, {3.4, 0.014},
       RoundingRadius->0], RectangleBox[{3.4, 0}, {3.42, 0.009},
       RoundingRadius->0], RectangleBox[{3.42, 0}, {3.44, 0.006},
       RoundingRadius->0], RectangleBox[{3.44, 0}, {3.46, 0.009},
       RoundingRadius->0], RectangleBox[{3.46, 0}, {3.48, 0.011},
       RoundingRadius->0], RectangleBox[{3.48, 0}, {3.5, 0.021},
       RoundingRadius->0], RectangleBox[{3.5, 0}, {3.52, 0.004},
       RoundingRadius->0], RectangleBox[{3.52, 0}, {3.54, 0.008},
       RoundingRadius->0], RectangleBox[{3.54, 0}, {3.56, 0.014},
       RoundingRadius->0], RectangleBox[{3.56, 0}, {3.58, 0.014},
       RoundingRadius->0], RectangleBox[{3.58, 0}, {3.6, 0.013},
       RoundingRadius->0], RectangleBox[{3.6, 0}, {3.62, 0.006},
       RoundingRadius->0], RectangleBox[{3.62, 0}, {3.64, 0.013},
       RoundingRadius->0], RectangleBox[{3.64, 0}, {3.66, 0.01},
       RoundingRadius->0], RectangleBox[{3.66, 0}, {3.68, 0.009},
       RoundingRadius->0], RectangleBox[{3.68, 0}, {3.7, 0.007},
       RoundingRadius->0], RectangleBox[{3.7, 0}, {3.72, 0.01},
       RoundingRadius->0], RectangleBox[{3.72, 0}, {3.74, 0.006},
       RoundingRadius->0], RectangleBox[{3.74, 0}, {3.76, 0.012},
       RoundingRadius->0], RectangleBox[{3.76, 0}, {3.78, 0.009},
       RoundingRadius->0], RectangleBox[{3.78, 0}, {3.8, 0.01},
       RoundingRadius->0], RectangleBox[{3.8, 0}, {3.82, 0.01},
       RoundingRadius->0], RectangleBox[{3.82, 0}, {3.84, 0.006},
       RoundingRadius->0], RectangleBox[{3.84, 0}, {3.86, 0.006},
       RoundingRadius->0], RectangleBox[{3.86, 0}, {3.88, 0.008},
       RoundingRadius->0], RectangleBox[{3.88, 0}, {3.9, 0.007},
       RoundingRadius->0], RectangleBox[{3.9, 0}, {3.92, 0.003},
       RoundingRadius->0], RectangleBox[{3.92, 0}, {3.94, 0.008},
       RoundingRadius->0], RectangleBox[{3.94, 0}, {3.96, 0.008},
       RoundingRadius->0], RectangleBox[{3.96, 0}, {3.98, 0.006},
       RoundingRadius->0], RectangleBox[{3.98, 0}, {4., 0.006},
       RoundingRadius->0], RectangleBox[{4., 0}, {4.02, 0.011},
       RoundingRadius->0], RectangleBox[{4.02, 0}, {4.04, 0.008},
       RoundingRadius->0], RectangleBox[{4.04, 0}, {4.06, 0.006},
       RoundingRadius->0], RectangleBox[{4.06, 0}, {4.08, 0.004},
       RoundingRadius->0], RectangleBox[{4.08, 0}, {4.1, 0.007},
       RoundingRadius->0], RectangleBox[{4.1, 0}, {4.12, 0.008},
       RoundingRadius->0], RectangleBox[{4.12, 0}, {4.14, 0.006},
       RoundingRadius->0], RectangleBox[{4.14, 0}, {4.16, 0.009},
       RoundingRadius->0], RectangleBox[{4.16, 0}, {4.18, 0.008},
       RoundingRadius->0], RectangleBox[{4.18, 0}, {4.2, 0.005},
       RoundingRadius->0], RectangleBox[{4.2, 0}, {4.22, 0.011},
       RoundingRadius->0], RectangleBox[{4.22, 0}, {4.24, 0.007},
       RoundingRadius->0], RectangleBox[{4.24, 0}, {4.26, 0.005},
       RoundingRadius->0], RectangleBox[{4.26, 0}, {4.28, 0.004},
       RoundingRadius->0], RectangleBox[{4.28, 0}, {4.3, 0.005},
       RoundingRadius->0], RectangleBox[{4.3, 0}, {4.32, 0.003},
       RoundingRadius->0], RectangleBox[{4.32, 0}, {4.34, 0.006},
       RoundingRadius->0], RectangleBox[{4.34, 0}, {4.36, 0.002},
       RoundingRadius->0], RectangleBox[{4.36, 0}, {4.38, 0.004},
       RoundingRadius->0], RectangleBox[{4.38, 0}, {4.4, 0.001},
       RoundingRadius->0], RectangleBox[{4.4, 0}, {4.42, 0.002},
       RoundingRadius->0], RectangleBox[{4.42, 0}, {4.44, 0.006},
       RoundingRadius->0], RectangleBox[{4.44, 0}, {4.46, 0.001},
       RoundingRadius->0], RectangleBox[{4.46, 0}, {4.48, 0.005},
       RoundingRadius->0], RectangleBox[{4.48, 0}, {4.5, 0.003},
       RoundingRadius->0], RectangleBox[{4.5, 0}, {4.52, 0.003},
       RoundingRadius->0], RectangleBox[{4.52, 0}, {4.54, 0.003},
       RoundingRadius->0], RectangleBox[{4.54, 0}, {4.56, 0.004},
       RoundingRadius->0], RectangleBox[{4.56, 0}, {4.58, 0.004},
       RoundingRadius->0], RectangleBox[{4.58, 0}, {4.6, 0.005},
       RoundingRadius->0], RectangleBox[{4.6, 0}, {4.62, 0.003},
       RoundingRadius->0], RectangleBox[{4.62, 0}, {4.64, 0.003},
       RoundingRadius->0], RectangleBox[{4.64, 0}, {4.66, 0.003},
       RoundingRadius->0], RectangleBox[{4.66, 0}, {4.68, 0.003},
       RoundingRadius->0], RectangleBox[{4.68, 0}, {4.7, 0.004},
       RoundingRadius->0], RectangleBox[{4.7, 0}, {4.72, 0.003},
       RoundingRadius->0], RectangleBox[{4.72, 0}, {4.74, 0.001},
       RoundingRadius->0], RectangleBox[{4.74, 0}, {4.76, 0.004},
       RoundingRadius->0], RectangleBox[{4.76, 0}, {4.78, 0.002},
       RoundingRadius->0], RectangleBox[{4.78, 0}, {4.8, 0.005},
       RoundingRadius->0], RectangleBox[{4.8, 0}, {4.82, 0.005},
       RoundingRadius->0], RectangleBox[{4.82, 0}, {4.84, 0.002},
       RoundingRadius->0], RectangleBox[{4.84, 0}, {4.86, 0.001},
       RoundingRadius->0], RectangleBox[{4.86, 0}, {4.88, 0.002},
       RoundingRadius->0], RectangleBox[{4.88, 0}, {4.9, 0.002},
       RoundingRadius->0], RectangleBox[{4.9, 0}, {4.92, 0.003},
       RoundingRadius->0], RectangleBox[{4.92, 0}, {4.94, 0.001},
       RoundingRadius->0], RectangleBox[{4.94, 0}, {4.96, 0.004},
       RoundingRadius->0], RectangleBox[{4.96, 0}, {4.98, 0.002},
       RoundingRadius->0], RectangleBox[{4.98, 0}, {5., 0.007},
       RoundingRadius->0], RectangleBox[{5., 0}, {5.02, 0.002},
       RoundingRadius->0], RectangleBox[{5.02, 0}, {5.04, 0.001},
       RoundingRadius->0], RectangleBox[{5.04, 0}, {5.06, 0.002},
       RoundingRadius->0], RectangleBox[{5.06, 0}, {5.08, 0.001},
       RoundingRadius->0], RectangleBox[{5.08, 0}, {5.1, 0.001},
       RoundingRadius->0], RectangleBox[{5.1, 0}, {5.12, 0.001},
       RoundingRadius->0], RectangleBox[{5.14, 0}, {5.16, 0.001},
       RoundingRadius->0], RectangleBox[{5.18, 0}, {5.2, 0.003},
       RoundingRadius->0], RectangleBox[{5.2, 0}, {5.22, 0.004},
       RoundingRadius->0], RectangleBox[{5.22, 0}, {5.24, 0.002},
       RoundingRadius->0], RectangleBox[{5.26, 0}, {5.28, 0.003},
       RoundingRadius->0], RectangleBox[{5.28, 0}, {5.3, 0.002},
       RoundingRadius->0], RectangleBox[{5.32, 0}, {5.34, 0.002},
       RoundingRadius->0], RectangleBox[{5.34, 0}, {5.36, 0.001},
       RoundingRadius->0], RectangleBox[{5.36, 0}, {5.38, 0.004},
       RoundingRadius->0], RectangleBox[{5.38, 0}, {5.4, 0.002},
       RoundingRadius->0], RectangleBox[{5.42, 0}, {5.44, 0.003},
       RoundingRadius->0], RectangleBox[{5.44, 0}, {5.46, 0.001},
       RoundingRadius->0], RectangleBox[{5.48, 0}, {5.5, 0.002},
       RoundingRadius->0], RectangleBox[{5.5, 0}, {5.52, 0.001},
       RoundingRadius->0], RectangleBox[{5.54, 0}, {5.56, 0.002},
       RoundingRadius->0], RectangleBox[{5.56, 0}, {5.58, 0.002},
       RoundingRadius->0], RectangleBox[{5.58, 0}, {5.6, 0.001},
       RoundingRadius->0], RectangleBox[{5.6, 0}, {5.62, 0.003},
       RoundingRadius->0], RectangleBox[{5.62, 0}, {5.64, 0.001},
       RoundingRadius->0], RectangleBox[{5.64, 0}, {5.66, 0.003},
       RoundingRadius->0], RectangleBox[{5.66, 0}, {5.68, 0.001},
       RoundingRadius->0], RectangleBox[{5.74, 0}, {5.76, 0.001},
       RoundingRadius->0], RectangleBox[{5.76, 0}, {5.78, 0.002},
       RoundingRadius->0], RectangleBox[{5.78, 0}, {5.8, 0.001},
       RoundingRadius->0], RectangleBox[{5.8, 0}, {5.82, 0.002},
       RoundingRadius->0], RectangleBox[{5.82, 0}, {5.84, 0.001},
       RoundingRadius->0], RectangleBox[{5.88, 0}, {5.9, 0.002},
       RoundingRadius->0], RectangleBox[{5.9, 0}, {5.92, 0.002},
       RoundingRadius->0], RectangleBox[{5.92, 0}, {5.94, 0.001},
       RoundingRadius->0], RectangleBox[{5.96, 0}, {5.98, 0.001},
       RoundingRadius->0], RectangleBox[{6.04, 0}, {6.06, 0.001},
       RoundingRadius->0], RectangleBox[{6.06, 0}, {6.08, 0.003},
       RoundingRadius->0], RectangleBox[{6.1, 0}, {6.12, 0.001},
       RoundingRadius->0], RectangleBox[{6.2, 0}, {6.22, 0.001},
       RoundingRadius->0], RectangleBox[{6.28, 0}, {6.3, 0.001},
       RoundingRadius->0], RectangleBox[{6.3, 0}, {6.32, 0.001},
       RoundingRadius->0], RectangleBox[{6.32, 0}, {6.34, 0.001},
       RoundingRadius->0], RectangleBox[{6.46, 0}, {6.48, 0.001},
       RoundingRadius->0], RectangleBox[{6.54, 0}, {6.56, 0.001},
       RoundingRadius->0], RectangleBox[{6.56, 0}, {6.58, 0.001},
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The 1D Gaussian/Normal distribution on the half line with variance s is:\
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This one-sided Gaussian/Normal distribution can be sampled using inverseErf[] \
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We now show how to solve for isotropic random flights in d dimension that \
project onto a single axis to give some desired symmetric distribution.  This \
is a generalization of polar methods [Devroye 2006 - Section 1.2].

We need the surface area of the sphere of radius r in dD:\
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Given a desired distribution f(r) on the half line we wish to sample, we \
first form the Fourier transform\
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and then search for some dimension d such that the inverse Hankel transform\
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gives a distribution p(r) that we can sample analytically.  If so, then \
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Suppose we wish to sample a random variate from a one-sided Gaussian/Normal \
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Following the 2 steps above, and knowing that the Box-Muller transform works \
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Cell["Example 7.2 - Sampling GGX distribution of visible normals", \
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In sampling the distribution of visible normals for the GGX normal \
distribution the following distribution requires sampling [Heitz and d\
\[CloseCurlyQuote]Eon 2014] in order to sample slope q conditioned on having \
sampled slope p:\
\>", "Text",
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This was sampled using an approximate inverse in [Heitz and d\
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We consider here an example that combines several of the above techniques to \
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procedures for the GGX vNDF sampling [Heitz 2018].  However, the procedure \
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GGX is simply a superposition of Beckmanns:\
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we find that the GGX vNDF can be sampled using an exact vNDF procedure for \
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       RoundingRadius->0], RectangleBox[{1.3, 0}, {1.31, 0.173},
       RoundingRadius->0], RectangleBox[{1.31, 0}, {1.32, 0.174},
       RoundingRadius->0], RectangleBox[{1.32, 0}, {1.33, 0.149},
       RoundingRadius->0], RectangleBox[{1.33, 0}, {1.34, 0.192},
       RoundingRadius->0], RectangleBox[{1.34, 0}, {1.35, 0.174},
       RoundingRadius->0], RectangleBox[{1.35, 0}, {1.36, 0.135},
       RoundingRadius->0], RectangleBox[{1.36, 0}, {1.37, 0.137},
       RoundingRadius->0], RectangleBox[{1.37, 0}, {1.38, 0.153},
       RoundingRadius->0], RectangleBox[{1.38, 0}, {1.39, 0.154},
       RoundingRadius->0], RectangleBox[{1.39, 0}, {1.4, 0.168},
       RoundingRadius->0], RectangleBox[{1.4, 0}, {1.41, 0.162},
       RoundingRadius->0], RectangleBox[{1.41, 0}, {1.42, 0.151},
       RoundingRadius->0], RectangleBox[{1.42, 0}, {1.43, 0.149},
       RoundingRadius->0], RectangleBox[{1.43, 0}, {1.44, 0.152},
       RoundingRadius->0], RectangleBox[{1.44, 0}, {1.45, 0.125},
       RoundingRadius->0], RectangleBox[{1.45, 0}, {1.46, 0.176},
       RoundingRadius->0], RectangleBox[{1.46, 0}, {1.47, 0.164},
       RoundingRadius->0], RectangleBox[{1.47, 0}, {1.48, 0.154},
       RoundingRadius->0], RectangleBox[{1.48, 0}, {1.49, 0.148},
       RoundingRadius->0], RectangleBox[{1.49, 0}, {1.5, 0.141},
       RoundingRadius->0], RectangleBox[{1.5, 0}, {1.51, 0.157},
       RoundingRadius->0], RectangleBox[{1.51, 0}, {1.52, 0.164},
       RoundingRadius->0], RectangleBox[{1.52, 0}, {1.53, 0.142},
       RoundingRadius->0], RectangleBox[{1.53, 0}, {1.54, 0.119},
       RoundingRadius->0], RectangleBox[{1.54, 0}, {1.55, 0.128},
       RoundingRadius->0], RectangleBox[{1.55, 0}, {1.56, 0.149},
       RoundingRadius->0], RectangleBox[{1.56, 0}, {1.57, 0.13},
       RoundingRadius->0], RectangleBox[{1.57, 0}, {1.58, 0.113},
       RoundingRadius->0], RectangleBox[{1.58, 0}, {1.59, 0.126},
       RoundingRadius->0], RectangleBox[{1.59, 0}, {1.6, 0.124},
       RoundingRadius->0], RectangleBox[{1.6, 0}, {1.61, 0.119},
       RoundingRadius->0], RectangleBox[{1.61, 0}, {1.62, 0.111},
       RoundingRadius->0], RectangleBox[{1.62, 0}, {1.63, 0.12},
       RoundingRadius->0], RectangleBox[{1.63, 0}, {1.64, 0.12},
       RoundingRadius->0], RectangleBox[{1.64, 0}, {1.65, 0.124},
       RoundingRadius->0], RectangleBox[{1.65, 0}, {1.66, 0.102},
       RoundingRadius->0], RectangleBox[{1.66, 0}, {1.67, 0.107},
       RoundingRadius->0], RectangleBox[{1.67, 0}, {1.68, 0.105},
       RoundingRadius->0], RectangleBox[{1.68, 0}, {1.69, 0.11},
       RoundingRadius->0], RectangleBox[{1.69, 0}, {1.7, 0.105},
       RoundingRadius->0], RectangleBox[{1.7, 0}, {1.71, 0.099},
       RoundingRadius->0], RectangleBox[{1.71, 0}, {1.72, 0.124},
       RoundingRadius->0], RectangleBox[{1.72, 0}, {1.73, 0.113},
       RoundingRadius->0], RectangleBox[{1.73, 0}, {1.74, 0.093},
       RoundingRadius->0], RectangleBox[{1.74, 0}, {1.75, 0.126},
       RoundingRadius->0], RectangleBox[{1.75, 0}, {1.76, 0.099},
       RoundingRadius->0], RectangleBox[{1.76, 0}, {1.77, 0.117},
       RoundingRadius->0], RectangleBox[{1.77, 0}, {1.78, 0.084},
       RoundingRadius->0], RectangleBox[{1.78, 0}, {1.79, 0.112},
       RoundingRadius->0], RectangleBox[{1.79, 0}, {1.8, 0.101},
       RoundingRadius->0], RectangleBox[{1.8, 0}, {1.81, 0.114},
       RoundingRadius->0], RectangleBox[{1.81, 0}, {1.82, 0.12},
       RoundingRadius->0], RectangleBox[{1.82, 0}, {1.83, 0.106},
       RoundingRadius->0], RectangleBox[{1.83, 0}, {1.84, 0.09},
       RoundingRadius->0], RectangleBox[{1.84, 0}, {1.85, 0.1},
       RoundingRadius->0], RectangleBox[{1.85, 0}, {1.86, 0.092},
       RoundingRadius->0], RectangleBox[{1.86, 0}, {1.87, 0.097},
       RoundingRadius->0], RectangleBox[{1.87, 0}, {1.88, 0.063},
       RoundingRadius->0], RectangleBox[{1.88, 0}, {1.89, 0.087},
       RoundingRadius->0], RectangleBox[{1.89, 0}, {1.9, 0.078},
       RoundingRadius->0], RectangleBox[{1.9, 0}, {1.91, 0.095},
       RoundingRadius->0], RectangleBox[{1.91, 0}, {1.92, 0.088},
       RoundingRadius->0], RectangleBox[{1.92, 0}, {1.93, 0.076},
       RoundingRadius->0], RectangleBox[{1.93, 0}, {1.94, 0.069},
       RoundingRadius->0], RectangleBox[{1.94, 0}, {1.95, 0.082},
       RoundingRadius->0], RectangleBox[{1.95, 0}, {1.96, 0.099},
       RoundingRadius->0], RectangleBox[{1.96, 0}, {1.97, 0.087},
       RoundingRadius->0], RectangleBox[{1.97, 0}, {1.98, 0.08},
       RoundingRadius->0], RectangleBox[{1.98, 0}, {1.99, 0.1},
       RoundingRadius->0], RectangleBox[{1.99, 0}, {2., 0.088},
       RoundingRadius->0], RectangleBox[{2., 0}, {2.01, 0.073},
       RoundingRadius->0], RectangleBox[{2.01, 0}, {2.02, 0.075},
       RoundingRadius->0], RectangleBox[{2.02, 0}, {2.03, 0.071},
       RoundingRadius->0], RectangleBox[{2.03, 0}, {2.04, 0.07},
       RoundingRadius->0], RectangleBox[{2.04, 0}, {2.05, 0.077},
       RoundingRadius->0], RectangleBox[{2.05, 0}, {2.06, 0.083},
       RoundingRadius->0], RectangleBox[{2.06, 0}, {2.07, 0.079},
       RoundingRadius->0], RectangleBox[{2.07, 0}, {2.08, 0.07},
       RoundingRadius->0], RectangleBox[{2.08, 0}, {2.09, 0.072},
       RoundingRadius->0], RectangleBox[{2.09, 0}, {2.1, 0.077},
       RoundingRadius->0], RectangleBox[{2.1, 0}, {2.11, 0.072},
       RoundingRadius->0], RectangleBox[{2.11, 0}, {2.12, 0.075},
       RoundingRadius->0], RectangleBox[{2.12, 0}, {2.13, 0.071},
       RoundingRadius->0], RectangleBox[{2.13, 0}, {2.14, 0.068},
       RoundingRadius->0], RectangleBox[{2.14, 0}, {2.15, 0.075},
       RoundingRadius->0], RectangleBox[{2.15, 0}, {2.16, 0.082},
       RoundingRadius->0], RectangleBox[{2.16, 0}, {2.17, 0.067},
       RoundingRadius->0], RectangleBox[{2.17, 0}, {2.18, 0.076},
       RoundingRadius->0], RectangleBox[{2.18, 0}, {2.19, 0.068},
       RoundingRadius->0], RectangleBox[{2.19, 0}, {2.2, 0.067},
       RoundingRadius->0], RectangleBox[{2.2, 0}, {2.21, 0.078},
       RoundingRadius->0], RectangleBox[{2.21, 0}, {2.22, 0.059},
       RoundingRadius->0], RectangleBox[{2.22, 0}, {2.23, 0.047},
       RoundingRadius->0], RectangleBox[{2.23, 0}, {2.24, 0.052},
       RoundingRadius->0], RectangleBox[{2.24, 0}, {2.25, 0.056},
       RoundingRadius->0], RectangleBox[{2.25, 0}, {2.26, 0.07},
       RoundingRadius->0], RectangleBox[{2.26, 0}, {2.27, 0.068},
       RoundingRadius->0], RectangleBox[{2.27, 0}, {2.28, 0.05},
       RoundingRadius->0], RectangleBox[{2.28, 0}, {2.29, 0.055},
       RoundingRadius->0], RectangleBox[{2.29, 0}, {2.3, 0.047},
       RoundingRadius->0], RectangleBox[{2.3, 0}, {2.31, 0.052},
       RoundingRadius->0], RectangleBox[{2.31, 0}, {2.32, 0.052},
       RoundingRadius->0], RectangleBox[{2.32, 0}, {2.33, 0.047},
       RoundingRadius->0], RectangleBox[{2.33, 0}, {2.34, 0.044},
       RoundingRadius->0], RectangleBox[{2.34, 0}, {2.35, 0.046},
       RoundingRadius->0], RectangleBox[{2.35, 0}, {2.36, 0.065},
       RoundingRadius->0], RectangleBox[{2.36, 0}, {2.37, 0.043},
       RoundingRadius->0], RectangleBox[{2.37, 0}, {2.38, 0.044},
       RoundingRadius->0], RectangleBox[{2.38, 0}, {2.39, 0.061},
       RoundingRadius->0], RectangleBox[{2.39, 0}, {2.4, 0.04},
       RoundingRadius->0], RectangleBox[{2.4, 0}, {2.41, 0.05},
       RoundingRadius->0], RectangleBox[{2.41, 0}, {2.42, 0.046},
       RoundingRadius->0], RectangleBox[{2.42, 0}, {2.43, 0.054},
       RoundingRadius->0], RectangleBox[{2.43, 0}, {2.44, 0.052},
       RoundingRadius->0], RectangleBox[{2.44, 0}, {2.45, 0.046},
       RoundingRadius->0], RectangleBox[{2.45, 0}, {2.46, 0.043},
       RoundingRadius->0], RectangleBox[{2.46, 0}, {2.47, 0.06},
       RoundingRadius->0], RectangleBox[{2.47, 0}, {2.48, 0.058},
       RoundingRadius->0], RectangleBox[{2.48, 0}, {2.49, 0.046},
       RoundingRadius->0], RectangleBox[{2.49, 0}, {2.5, 0.036},
       RoundingRadius->0], RectangleBox[{2.5, 0}, {2.51, 0.049},
       RoundingRadius->0], RectangleBox[{2.51, 0}, {2.52, 0.057},
       RoundingRadius->0], RectangleBox[{2.52, 0}, {2.53, 0.04},
       RoundingRadius->0], RectangleBox[{2.53, 0}, {2.54, 0.045},
       RoundingRadius->0], RectangleBox[{2.54, 0}, {2.55, 0.044},
       RoundingRadius->0], RectangleBox[{2.55, 0}, {2.56, 0.053},
       RoundingRadius->0], RectangleBox[{2.56, 0}, {2.57, 0.038},
       RoundingRadius->0], RectangleBox[{2.57, 0}, {2.58, 0.036},
       RoundingRadius->0], RectangleBox[{2.58, 0}, {2.59, 0.038},
       RoundingRadius->0], RectangleBox[{2.59, 0}, {2.6, 0.041},
       RoundingRadius->0], RectangleBox[{2.6, 0}, {2.61, 0.041},
       RoundingRadius->0], RectangleBox[{2.61, 0}, {2.62, 0.046},
       RoundingRadius->0], RectangleBox[{2.62, 0}, {2.63, 0.048},
       RoundingRadius->0], RectangleBox[{2.63, 0}, {2.64, 0.041},
       RoundingRadius->0], RectangleBox[{2.64, 0}, {2.65, 0.026},
       RoundingRadius->0], RectangleBox[{2.65, 0}, {2.66, 0.045},
       RoundingRadius->0], RectangleBox[{2.66, 0}, {2.67, 0.037},
       RoundingRadius->0], RectangleBox[{2.67, 0}, {2.68, 0.045},
       RoundingRadius->0], RectangleBox[{2.68, 0}, {2.69, 0.043},
       RoundingRadius->0], RectangleBox[{2.69, 0}, {2.7, 0.044},
       RoundingRadius->0], RectangleBox[{2.7, 0}, {2.71, 0.036},
       RoundingRadius->0], RectangleBox[{2.71, 0}, {2.72, 0.029},
       RoundingRadius->0], RectangleBox[{2.72, 0}, {2.73, 0.039},
       RoundingRadius->0], RectangleBox[{2.73, 0}, {2.74, 0.031},
       RoundingRadius->0], RectangleBox[{2.74, 0}, {2.75, 0.037},
       RoundingRadius->0], RectangleBox[{2.75, 0}, {2.76, 0.026},
       RoundingRadius->0], RectangleBox[{2.76, 0}, {2.77, 0.033},
       RoundingRadius->0], RectangleBox[{2.77, 0}, {2.78, 0.033},
       RoundingRadius->0], RectangleBox[{2.78, 0}, {2.79, 0.031},
       RoundingRadius->0], RectangleBox[{2.79, 0}, {2.8, 0.034},
       RoundingRadius->0], RectangleBox[{2.8, 0}, {2.81, 0.043},
       RoundingRadius->0], RectangleBox[{2.81, 0}, {2.82, 0.033},
       RoundingRadius->0], RectangleBox[{2.82, 0}, {2.83, 0.038},
       RoundingRadius->0], RectangleBox[{2.83, 0}, {2.84, 0.028},
       RoundingRadius->0], RectangleBox[{2.84, 0}, {2.85, 0.033},
       RoundingRadius->0], RectangleBox[{2.85, 0}, {2.86, 0.04},
       RoundingRadius->0], RectangleBox[{2.86, 0}, {2.87, 0.032},
       RoundingRadius->0], RectangleBox[{2.87, 0}, {2.88, 0.038},
       RoundingRadius->0], RectangleBox[{2.88, 0}, {2.89, 0.03},
       RoundingRadius->0], RectangleBox[{2.89, 0}, {2.9, 0.032},
       RoundingRadius->0], RectangleBox[{2.9, 0}, {2.91, 0.029},
       RoundingRadius->0], RectangleBox[{2.91, 0}, {2.92, 0.03},
       RoundingRadius->0], RectangleBox[{2.92, 0}, {2.93, 0.023},
       RoundingRadius->0], RectangleBox[{2.93, 0}, {2.94, 0.026},
       RoundingRadius->0], RectangleBox[{2.94, 0}, {2.95, 0.019},
       RoundingRadius->0], RectangleBox[{2.95, 0}, {2.96, 0.017},
       RoundingRadius->0], RectangleBox[{2.96, 0}, {2.97, 0.015},
       RoundingRadius->0], RectangleBox[{2.97, 0}, {2.98, 0.028},
       RoundingRadius->0], RectangleBox[{2.98, 0}, {2.99, 0.021},
       RoundingRadius->0], RectangleBox[{2.99, 0}, {3., 0.03},
       RoundingRadius->0], RectangleBox[{3., 0}, {3.01, 0.028},
       RoundingRadius->0], RectangleBox[{3.01, 0}, {3.02, 0.031},
       RoundingRadius->0], RectangleBox[{3.02, 0}, {3.03, 0.029},
       RoundingRadius->0], RectangleBox[{3.03, 0}, {3.04, 0.019},
       RoundingRadius->0], RectangleBox[{3.04, 0}, {3.05, 0.023},
       RoundingRadius->0], RectangleBox[{3.05, 0}, {3.06, 0.026},
       RoundingRadius->0], RectangleBox[{3.06, 0}, {3.07, 0.026},
       RoundingRadius->0], RectangleBox[{3.07, 0}, {3.08, 0.023},
       RoundingRadius->0], RectangleBox[{3.08, 0}, {3.09, 0.018},
       RoundingRadius->0], RectangleBox[{3.09, 0}, {3.1, 0.029},
       RoundingRadius->0], RectangleBox[{3.1, 0}, {3.11, 0.022},
       RoundingRadius->0], RectangleBox[{3.11, 0}, {3.12, 0.026},
       RoundingRadius->0], RectangleBox[{3.12, 0}, {3.13, 0.024},
       RoundingRadius->0], RectangleBox[{3.13, 0}, {3.14, 0.021},
       RoundingRadius->0], RectangleBox[{3.14, 0}, {3.15, 0.027},
       RoundingRadius->0], RectangleBox[{3.15, 0}, {3.16, 0.014},
       RoundingRadius->0], RectangleBox[{3.16, 0}, {3.17, 0.02},
       RoundingRadius->0], RectangleBox[{3.17, 0}, {3.18, 0.02},
       RoundingRadius->0], RectangleBox[{3.18, 0}, {3.19, 0.023},
       RoundingRadius->0], RectangleBox[{3.19, 0}, {3.2, 0.025},
       RoundingRadius->0], RectangleBox[{3.2, 0}, {3.21, 0.028},
       RoundingRadius->0], RectangleBox[{3.21, 0}, {3.22, 0.019},
       RoundingRadius->0], RectangleBox[{3.22, 0}, {3.23, 0.022},
       RoundingRadius->0], RectangleBox[{3.23, 0}, {3.24, 0.016},
       RoundingRadius->0], RectangleBox[{3.24, 0}, {3.25, 0.019},
       RoundingRadius->0], RectangleBox[{3.25, 0}, {3.26, 0.012},
       RoundingRadius->0], RectangleBox[{3.26, 0}, {3.27, 0.016},
       RoundingRadius->0], RectangleBox[{3.27, 0}, {3.28, 0.015},
       RoundingRadius->0], RectangleBox[{3.28, 0}, {3.29, 0.016},
       RoundingRadius->0], RectangleBox[{3.29, 0}, {3.3, 0.016},
       RoundingRadius->0], RectangleBox[{3.3, 0}, {3.31, 0.021},
       RoundingRadius->0], RectangleBox[{3.31, 0}, {3.32, 0.027},
       RoundingRadius->0], RectangleBox[{3.32, 0}, {3.33, 0.02},
       RoundingRadius->0], RectangleBox[{3.33, 0}, {3.34, 0.016},
       RoundingRadius->0], RectangleBox[{3.34, 0}, {3.35, 0.022},
       RoundingRadius->0], RectangleBox[{3.35, 0}, {3.36, 0.021},
       RoundingRadius->0], RectangleBox[{3.36, 0}, {3.37, 0.023},
       RoundingRadius->0], RectangleBox[{3.37, 0}, {3.38, 0.021},
       RoundingRadius->0], RectangleBox[{3.38, 0}, {3.39, 0.017},
       RoundingRadius->0], RectangleBox[{3.39, 0}, {3.4, 0.014},
       RoundingRadius->0], RectangleBox[{3.4, 0}, {3.41, 0.016},
       RoundingRadius->0], RectangleBox[{3.41, 0}, {3.42, 0.02},
       RoundingRadius->0], RectangleBox[{3.42, 0}, {3.43, 0.012},
       RoundingRadius->0], RectangleBox[{3.43, 0}, {3.44, 0.018},
       RoundingRadius->0], RectangleBox[{3.44, 0}, {3.45, 0.015},
       RoundingRadius->0], RectangleBox[{3.45, 0}, {3.46, 0.023},
       RoundingRadius->0], RectangleBox[{3.46, 0}, {3.47, 0.015},
       RoundingRadius->0], RectangleBox[{3.47, 0}, {3.48, 0.017},
       RoundingRadius->0], RectangleBox[{3.48, 0}, {3.49, 0.016},
       RoundingRadius->0], RectangleBox[{3.49, 0}, {3.5, 0.019},
       RoundingRadius->0], RectangleBox[{3.5, 0}, {3.51, 0.018},
       RoundingRadius->0], RectangleBox[{3.51, 0}, {3.52, 0.015},
       RoundingRadius->0], RectangleBox[{3.52, 0}, {3.53, 0.017},
       RoundingRadius->0], RectangleBox[{3.53, 0}, {3.54, 0.01},
       RoundingRadius->0], RectangleBox[{3.54, 0}, {3.55, 0.016},
       RoundingRadius->0], RectangleBox[{3.55, 0}, {3.56, 0.016},
       RoundingRadius->0], RectangleBox[{3.56, 0}, {3.57, 0.015},
       RoundingRadius->0], RectangleBox[{3.57, 0}, {3.58, 0.011},
       RoundingRadius->0], RectangleBox[{3.58, 0}, {3.59, 0.011},
       RoundingRadius->0], RectangleBox[{3.59, 0}, {3.6, 0.02},
       RoundingRadius->0], RectangleBox[{3.6, 0}, {3.61, 0.014},
       RoundingRadius->0], RectangleBox[{3.61, 0}, {3.62, 0.013},
       RoundingRadius->0], RectangleBox[{3.62, 0}, {3.63, 0.015},
       RoundingRadius->0], RectangleBox[{3.63, 0}, {3.64, 0.016},
       RoundingRadius->0], RectangleBox[{3.64, 0}, {3.65, 0.012},
       RoundingRadius->0], RectangleBox[{3.65, 0}, {3.66, 0.012},
       RoundingRadius->0], RectangleBox[{3.66, 0}, {3.67, 0.009},
       RoundingRadius->0], RectangleBox[{3.67, 0}, {3.68, 0.014},
       RoundingRadius->0], RectangleBox[{3.68, 0}, {3.69, 0.015},
       RoundingRadius->0], RectangleBox[{3.69, 0}, {3.7, 0.014},
       RoundingRadius->0], RectangleBox[{3.7, 0}, {3.71, 0.012},
       RoundingRadius->0], RectangleBox[{3.71, 0}, {3.72, 0.008},
       RoundingRadius->0], RectangleBox[{3.72, 0}, {3.73, 0.011},
       RoundingRadius->0], RectangleBox[{3.73, 0}, {3.74, 0.009},
       RoundingRadius->0], RectangleBox[{3.74, 0}, {3.75, 0.009},
       RoundingRadius->0], RectangleBox[{3.75, 0}, {3.76, 0.009},
       RoundingRadius->0], RectangleBox[{3.76, 0}, {3.77, 0.016},
       RoundingRadius->0], RectangleBox[{3.77, 0}, {3.78, 0.009},
       RoundingRadius->0], RectangleBox[{3.78, 0}, {3.79, 0.01},
       RoundingRadius->0], RectangleBox[{3.79, 0}, {3.8, 0.015},
       RoundingRadius->0], RectangleBox[{3.8, 0}, {3.81, 0.012},
       RoundingRadius->0], RectangleBox[{3.81, 0}, {3.82, 0.01},
       RoundingRadius->0], RectangleBox[{3.82, 0}, {3.83, 0.013},
       RoundingRadius->0], RectangleBox[{3.83, 0}, {3.84, 0.006},
       RoundingRadius->0], RectangleBox[{3.84, 0}, {3.85, 0.017},
       RoundingRadius->0], RectangleBox[{3.85, 0}, {3.86, 0.012},
       RoundingRadius->0], RectangleBox[{3.86, 0}, {3.87, 0.012},
       RoundingRadius->0], RectangleBox[{3.87, 0}, {3.88, 0.009},
       RoundingRadius->0], RectangleBox[{3.88, 0}, {3.89, 0.005},
       RoundingRadius->0], RectangleBox[{3.89, 0}, {3.9, 0.008},
       RoundingRadius->0], RectangleBox[{3.9, 0}, {3.91, 0.009},
       RoundingRadius->0], RectangleBox[{3.91, 0}, {3.92, 0.008},
       RoundingRadius->0], RectangleBox[{3.92, 0}, {3.93, 0.012},
       RoundingRadius->0], RectangleBox[{3.93, 0}, {3.94, 0.011},
       RoundingRadius->0], RectangleBox[{3.94, 0}, {3.95, 0.01},
       RoundingRadius->0], RectangleBox[{3.95, 0}, {3.96, 0.005},
       RoundingRadius->0], RectangleBox[{3.96, 0}, {3.97, 0.01},
       RoundingRadius->0], RectangleBox[{3.97, 0}, {3.98, 0.009},
       RoundingRadius->0], RectangleBox[{3.98, 0}, {3.99, 0.012},
       RoundingRadius->0], RectangleBox[{3.99, 0}, {4., 0.006},
       RoundingRadius->0], RectangleBox[{4., 0}, {4.01, 0.003},
       RoundingRadius->0], RectangleBox[{4.01, 0}, {4.02, 0.006},
       RoundingRadius->0], RectangleBox[{4.02, 0}, {4.03, 0.007},
       RoundingRadius->0], RectangleBox[{4.03, 0}, {4.04, 0.012},
       RoundingRadius->0], RectangleBox[{4.04, 0}, {4.05, 0.012},
       RoundingRadius->0], RectangleBox[{4.05, 0}, {4.06, 0.008},
       RoundingRadius->0], RectangleBox[{4.06, 0}, {4.07, 0.007},
       RoundingRadius->0], RectangleBox[{4.07, 0}, {4.08, 0.008},
       RoundingRadius->0], RectangleBox[{4.08, 0}, {4.09, 0.006},
       RoundingRadius->0], RectangleBox[{4.09, 0}, {4.1, 0.004},
       RoundingRadius->0], RectangleBox[{4.1, 0}, {4.11, 0.01},
       RoundingRadius->0], RectangleBox[{4.11, 0}, {4.12, 0.003},
       RoundingRadius->0], RectangleBox[{4.12, 0}, {4.13, 0.004},
       RoundingRadius->0], RectangleBox[{4.13, 0}, {4.14, 0.008},
       RoundingRadius->0], RectangleBox[{4.14, 0}, {4.15, 0.006},
       RoundingRadius->0], RectangleBox[{4.15, 0}, {4.16, 0.007},
       RoundingRadius->0], RectangleBox[{4.16, 0}, {4.17, 0.011},
       RoundingRadius->0], RectangleBox[{4.17, 0}, {4.18, 0.009},
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