// Boost.Geometry
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// Copyright (c) 2015-2018 Oracle and/or its affiliates.
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// Contributed and/or modified by Vissarion Fysikopoulos, on behalf of Oracle
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// Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
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// Use, modification and distribution is subject to the Boost Software License,
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// Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
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// http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
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#define BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
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#include <boost/geometry/core/radian_access.hpp>
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#include <boost/geometry/formulas/flattening.hpp>
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#include <boost/geometry/util/math.hpp>
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#include <boost/math/special_functions/hypot.hpp>
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namespace boost { namespace geometry { namespace formula
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{
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/*!
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\brief Formulas for computing spherical and ellipsoidal polygon area.
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The current class computes the area of the trapezoid defined by a segment
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the two meridians passing by the endpoints and the equator.
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\author See
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- Danielsen JS, The area under the geodesic. Surv Rev 30(232):
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61–66, 1989
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- Charles F.F Karney, Algorithms for geodesics, 2011
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https://arxiv.org/pdf/1109.4448.pdf
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*/
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template <
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typename CT,
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std::size_t SeriesOrder = 2,
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bool ExpandEpsN = true
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>
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class area_formulas
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{
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public:
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//TODO: move the following to a more general space to be used by other
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// classes as well
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/*
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Evaluate the polynomial in x using Horner's method.
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*/
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template <typename NT, typename IteratorType>
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static inline NT horner_evaluate(NT const& x,
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IteratorType begin,
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IteratorType end)
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{
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NT result(0);
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IteratorType it = end;
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do
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{
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result = result * x + *--it;
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}
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while (it != begin);
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return result;
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}
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/*
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Clenshaw algorithm for summing trigonometric series
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https://en.wikipedia.org/wiki/Clenshaw_algorithm
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*/
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template <typename NT, typename IteratorType>
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static inline NT clenshaw_sum(NT const& cosx,
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IteratorType begin,
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IteratorType end)
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{
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IteratorType it = end;
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bool odd = true;
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CT b_k, b_k1(0), b_k2(0);
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do
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{
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CT c_k = odd ? *--it : NT(0);
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b_k = c_k + NT(2) * cosx * b_k1 - b_k2;
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b_k2 = b_k1;
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b_k1 = b_k;
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odd = !odd;
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}
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while (it != begin);
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return *begin + b_k1 * cosx - b_k2;
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}
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template<typename T>
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static inline void normalize(T& x, T& y)
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{
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T h = boost::math::hypot(x, y);
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x /= h;
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y /= h;
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}
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/*
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Generate and evaluate the series expansion of the following integral
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I4 = -integrate( (t(ep2) - t(k2*sin(sigma1)^2)) / (ep2 - k2*sin(sigma1)^2)
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* sin(sigma1)/2, sigma1, pi/2, sigma )
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where
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t(x) = sqrt(1+1/x)*asinh(sqrt(x)) + x
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valid for ep2 and k2 small. We substitute k2 = 4 * eps / (1 - eps)^2
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and ep2 = 4 * n / (1 - n)^2 and expand in eps and n.
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The resulting sum of the series is of the form
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sum(C4[l] * cos((2*l+1)*sigma), l, 0, maxpow-1) )
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The above expansion is performed in Computer Algebra System Maxima.
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The C++ code (that yields the function evaluate_coeffs_n below) is generated
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by the following Maxima script and is based on script:
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http://geographiclib.sourceforge.net/html/geod.mac
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// Maxima script begin
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taylordepth:5$
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ataylor(expr,var,ord):=expand(ratdisrep(taylor(expr,var,0,ord)))$
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jtaylor(expr,var1,var2,ord):=block([zz],expand(subst([zz=1],
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ratdisrep(taylor(subst([var1=zz*var1,var2=zz*var2],expr),zz,0,ord)))))$
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compute(maxpow):=block([int,t,intexp,area, x,ep2,k2],
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maxpow:maxpow-1,
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t : sqrt(1+1/x) * asinh(sqrt(x)) + x,
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int:-(tf(ep2) - tf(k2*sin(sigma)^2)) / (ep2 - k2*sin(sigma)^2)
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* sin(sigma)/2,
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int:subst([tf(ep2)=subst([x=ep2],t),
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tf(k2*sin(sigma)^2)=subst([x=k2*sin(sigma)^2],t)],
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int),
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int:subst([abs(sin(sigma))=sin(sigma)],int),
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int:subst([k2=4*eps/(1-eps)^2,ep2=4*n/(1-n)^2],int),
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intexp:jtaylor(int,n,eps,maxpow),
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area:trigreduce(integrate(intexp,sigma)),
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area:expand(area-subst(sigma=%pi/2,area)),
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for i:0 thru maxpow do C4[i]:coeff(area,cos((2*i+1)*sigma)),
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if expand(area-sum(C4[i]*cos((2*i+1)*sigma),i,0,maxpow)) # 0
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then error("left over terms in I4"),
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'done)$
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printcode(maxpow):=
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block([tab2:" ",tab3:" "],
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print(" switch (SeriesOrder) {"),
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for nn:1 thru maxpow do block([c],
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print(concat(tab2,"case ",string(nn-1),":")),
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c:0,
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for m:0 thru nn-1 do block(
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[q:jtaylor(subst([n=n],C4[m]),n,eps,nn-1),
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linel:1200],
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for j:m thru nn-1 do (
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print(concat(tab3,"coeffs_n[",c,"] = ",
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string(horner(coeff(q,eps,j))),";")),
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c:c+1)
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),
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print(concat(tab3,"break;"))),
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print(" }"),
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'done)$
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maxpow:6$
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compute(maxpow)$
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printcode(maxpow)$
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// Maxima script end
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In the resulting code we should replace each number x by CT(x)
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e.g. using the following scirpt:
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sed -e 's/[0-9]\+/CT(&)/g; s/\[CT(/\[/g; s/)\]/\]/g;
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s/case\sCT(/case /g; s/):/:/g'
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*/
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static inline void evaluate_coeffs_n(CT const& n, CT coeffs_n[])
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{
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switch (SeriesOrder) {
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case 0:
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coeffs_n[0] = CT(2)/CT(3);
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break;
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case 1:
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coeffs_n[0] = (CT(10)-CT(4)*n)/CT(15);
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coeffs_n[1] = -CT(1)/CT(5);
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coeffs_n[2] = CT(1)/CT(45);
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break;
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case 2:
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coeffs_n[0] = (n*(CT(8)*n-CT(28))+CT(70))/CT(105);
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coeffs_n[1] = (CT(16)*n-CT(7))/CT(35);
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coeffs_n[2] = -CT(2)/CT(105);
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coeffs_n[3] = (CT(7)-CT(16)*n)/CT(315);
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coeffs_n[4] = -CT(2)/CT(105);
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coeffs_n[5] = CT(4)/CT(525);
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break;
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case 3:
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coeffs_n[0] = (n*(n*(CT(4)*n+CT(24))-CT(84))+CT(210))/CT(315);
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coeffs_n[1] = ((CT(48)-CT(32)*n)*n-CT(21))/CT(105);
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coeffs_n[2] = (-CT(32)*n-CT(6))/CT(315);
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coeffs_n[3] = CT(11)/CT(315);
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coeffs_n[4] = (n*(CT(32)*n-CT(48))+CT(21))/CT(945);
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coeffs_n[5] = (CT(64)*n-CT(18))/CT(945);
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coeffs_n[6] = -CT(1)/CT(105);
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coeffs_n[7] = (CT(12)-CT(32)*n)/CT(1575);
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coeffs_n[8] = -CT(8)/CT(1575);
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coeffs_n[9] = CT(8)/CT(2205);
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break;
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case 4:
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coeffs_n[0] = (n*(n*(n*(CT(16)*n+CT(44))+CT(264))-CT(924))+CT(2310))/CT(3465);
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coeffs_n[1] = (n*(n*(CT(48)*n-CT(352))+CT(528))-CT(231))/CT(1155);
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coeffs_n[2] = (n*(CT(1088)*n-CT(352))-CT(66))/CT(3465);
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coeffs_n[3] = (CT(121)-CT(368)*n)/CT(3465);
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coeffs_n[4] = CT(4)/CT(1155);
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coeffs_n[5] = (n*((CT(352)-CT(48)*n)*n-CT(528))+CT(231))/CT(10395);
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coeffs_n[6] = ((CT(704)-CT(896)*n)*n-CT(198))/CT(10395);
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coeffs_n[7] = (CT(80)*n-CT(99))/CT(10395);
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coeffs_n[8] = CT(4)/CT(1155);
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coeffs_n[9] = (n*(CT(320)*n-CT(352))+CT(132))/CT(17325);
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coeffs_n[10] = (CT(384)*n-CT(88))/CT(17325);
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coeffs_n[11] = -CT(8)/CT(1925);
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coeffs_n[12] = (CT(88)-CT(256)*n)/CT(24255);
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coeffs_n[13] = -CT(16)/CT(8085);
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coeffs_n[14] = CT(64)/CT(31185);
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break;
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case 5:
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coeffs_n[0] = (n*(n*(n*(n*(CT(100)*n+CT(208))+CT(572))+CT(3432))-CT(12012))+CT(30030))
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/CT(45045);
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coeffs_n[1] = (n*(n*(n*(CT(64)*n+CT(624))-CT(4576))+CT(6864))-CT(3003))/CT(15015);
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coeffs_n[2] = (n*((CT(14144)-CT(10656)*n)*n-CT(4576))-CT(858))/CT(45045);
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coeffs_n[3] = ((-CT(224)*n-CT(4784))*n+CT(1573))/CT(45045);
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coeffs_n[4] = (CT(1088)*n+CT(156))/CT(45045);
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coeffs_n[5] = CT(97)/CT(15015);
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coeffs_n[6] = (n*(n*((-CT(64)*n-CT(624))*n+CT(4576))-CT(6864))+CT(3003))/CT(135135);
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coeffs_n[7] = (n*(n*(CT(5952)*n-CT(11648))+CT(9152))-CT(2574))/CT(135135);
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coeffs_n[8] = (n*(CT(5792)*n+CT(1040))-CT(1287))/CT(135135);
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coeffs_n[9] = (CT(468)-CT(2944)*n)/CT(135135);
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coeffs_n[10] = CT(1)/CT(9009);
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coeffs_n[11] = (n*((CT(4160)-CT(1440)*n)*n-CT(4576))+CT(1716))/CT(225225);
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coeffs_n[12] = ((CT(4992)-CT(8448)*n)*n-CT(1144))/CT(225225);
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coeffs_n[13] = (CT(1856)*n-CT(936))/CT(225225);
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coeffs_n[14] = CT(8)/CT(10725);
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coeffs_n[15] = (n*(CT(3584)*n-CT(3328))+CT(1144))/CT(315315);
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coeffs_n[16] = (CT(1024)*n-CT(208))/CT(105105);
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coeffs_n[17] = -CT(136)/CT(63063);
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coeffs_n[18] = (CT(832)-CT(2560)*n)/CT(405405);
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coeffs_n[19] = -CT(128)/CT(135135);
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coeffs_n[20] = CT(128)/CT(99099);
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break;
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}
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}
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/*
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Expand in k2 and ep2.
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*/
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static inline void evaluate_coeffs_ep(CT const& ep, CT coeffs_n[])
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{
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switch (SeriesOrder) {
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case 0:
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coeffs_n[0] = CT(2)/CT(3);
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break;
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case 1:
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coeffs_n[0] = (CT(10)-ep)/CT(15);
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coeffs_n[1] = -CT(1)/CT(20);
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coeffs_n[2] = CT(1)/CT(180);
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break;
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case 2:
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coeffs_n[0] = (ep*(CT(4)*ep-CT(7))+CT(70))/CT(105);
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coeffs_n[1] = (CT(4)*ep-CT(7))/CT(140);
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coeffs_n[2] = CT(1)/CT(42);
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coeffs_n[3] = (CT(7)-CT(4)*ep)/CT(1260);
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coeffs_n[4] = -CT(1)/CT(252);
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coeffs_n[5] = CT(1)/CT(2100);
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break;
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case 3:
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coeffs_n[0] = (ep*((CT(12)-CT(8)*ep)*ep-CT(21))+CT(210))/CT(315);
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coeffs_n[1] = ((CT(12)-CT(8)*ep)*ep-CT(21))/CT(420);
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coeffs_n[2] = (CT(3)-CT(2)*ep)/CT(126);
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coeffs_n[3] = -CT(1)/CT(72);
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coeffs_n[4] = (ep*(CT(8)*ep-CT(12))+CT(21))/CT(3780);
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coeffs_n[5] = (CT(2)*ep-CT(3))/CT(756);
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coeffs_n[6] = CT(1)/CT(360);
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coeffs_n[7] = (CT(3)-CT(2)*ep)/CT(6300);
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coeffs_n[8] = -CT(1)/CT(1800);
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coeffs_n[9] = CT(1)/CT(17640);
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break;
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case 4:
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coeffs_n[0] = (ep*(ep*(ep*(CT(64)*ep-CT(88))+CT(132))-CT(231))+CT(2310))/CT(3465);
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coeffs_n[1] = (ep*(ep*(CT(64)*ep-CT(88))+CT(132))-CT(231))/CT(4620);
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coeffs_n[2] = (ep*(CT(16)*ep-CT(22))+CT(33))/CT(1386);
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coeffs_n[3] = (CT(8)*ep-CT(11))/CT(792);
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coeffs_n[4] = CT(1)/CT(110);
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coeffs_n[5] = (ep*((CT(88)-CT(64)*ep)*ep-CT(132))+CT(231))/CT(41580);
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coeffs_n[6] = ((CT(22)-CT(16)*ep)*ep-CT(33))/CT(8316);
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coeffs_n[7] = (CT(11)-CT(8)*ep)/CT(3960);
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coeffs_n[8] = -CT(1)/CT(495);
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coeffs_n[9] = (ep*(CT(16)*ep-CT(22))+CT(33))/CT(69300);
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coeffs_n[10] = (CT(8)*ep-CT(11))/CT(19800);
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coeffs_n[11] = CT(1)/CT(1925);
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coeffs_n[12] = (CT(11)-CT(8)*ep)/CT(194040);
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coeffs_n[13] = -CT(1)/CT(10780);
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coeffs_n[14] = CT(1)/CT(124740);
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break;
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case 5:
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coeffs_n[0] = (ep*(ep*(ep*((CT(832)-CT(640)*ep)*ep-CT(1144))+CT(1716))-CT(3003))+CT(30030))/CT(45045);
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coeffs_n[1] = (ep*(ep*((CT(832)-CT(640)*ep)*ep-CT(1144))+CT(1716))-CT(3003))/CT(60060);
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coeffs_n[2] = (ep*((CT(208)-CT(160)*ep)*ep-CT(286))+CT(429))/CT(18018);
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coeffs_n[3] = ((CT(104)-CT(80)*ep)*ep-CT(143))/CT(10296);
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coeffs_n[4] = (CT(13)-CT(10)*ep)/CT(1430);
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coeffs_n[5] = -CT(1)/CT(156);
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coeffs_n[6] = (ep*(ep*(ep*(CT(640)*ep-CT(832))+CT(1144))-CT(1716))+CT(3003))/CT(540540);
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coeffs_n[7] = (ep*(ep*(CT(160)*ep-CT(208))+CT(286))-CT(429))/CT(108108);
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coeffs_n[8] = (ep*(CT(80)*ep-CT(104))+CT(143))/CT(51480);
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coeffs_n[9] = (CT(10)*ep-CT(13))/CT(6435);
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coeffs_n[10] = CT(5)/CT(3276);
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coeffs_n[11] = (ep*((CT(208)-CT(160)*ep)*ep-CT(286))+CT(429))/CT(900900);
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coeffs_n[12] = ((CT(104)-CT(80)*ep)*ep-CT(143))/CT(257400);
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coeffs_n[13] = (CT(13)-CT(10)*ep)/CT(25025);
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coeffs_n[14] = -CT(1)/CT(2184);
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coeffs_n[15] = (ep*(CT(80)*ep-CT(104))+CT(143))/CT(2522520);
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coeffs_n[16] = (CT(10)*ep-CT(13))/CT(140140);
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coeffs_n[17] = CT(5)/CT(45864);
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coeffs_n[18] = (CT(13)-CT(10)*ep)/CT(1621620);
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coeffs_n[19] = -CT(1)/CT(58968);
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coeffs_n[20] = CT(1)/CT(792792);
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break;
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}
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}
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/*
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Given the set of coefficients coeffs1[] evaluate on var2 and return
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the set of coefficients coeffs2[]
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*/
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static inline void evaluate_coeffs_var2(CT const& var2,
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CT const coeffs1[],
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CT coeffs2[])
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{
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std::size_t begin(0), end(0);
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for(std::size_t i = 0; i <= SeriesOrder; i++)
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{
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end = begin + SeriesOrder + 1 - i;
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coeffs2[i] = ((i==0) ? CT(1) : math::pow(var2, int(i)))
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* horner_evaluate(var2, coeffs1 + begin, coeffs1 + end);
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begin = end;
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}
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}
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/*
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Compute the spherical excess of a geodesic (or shperical) segment
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*/
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template <
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bool LongSegment,
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typename PointOfSegment
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>
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static inline CT spherical(PointOfSegment const& p1,
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PointOfSegment const& p2)
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{
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CT excess;
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if(LongSegment) // not for segments parallel to equator
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{
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CT cbet1 = cos(geometry::get_as_radian<1>(p1));
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CT sbet1 = sin(geometry::get_as_radian<1>(p1));
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CT cbet2 = cos(geometry::get_as_radian<1>(p2));
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CT sbet2 = sin(geometry::get_as_radian<1>(p2));
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CT omg12 = geometry::get_as_radian<0>(p1)
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- geometry::get_as_radian<0>(p2);
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CT comg12 = cos(omg12);
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CT somg12 = sin(omg12);
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CT alp1 = atan2(cbet1 * sbet2
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- sbet1 * cbet2 * comg12,
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cbet2 * somg12);
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CT alp2 = atan2(cbet1 * sbet2 * comg12
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- sbet1 * cbet2,
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cbet1 * somg12);
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excess = alp2 - alp1;
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} else {
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// Trapezoidal formula
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CT tan_lat1 =
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tan(geometry::get_as_radian<1>(p1) / 2.0);
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CT tan_lat2 =
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tan(geometry::get_as_radian<1>(p2) / 2.0);
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excess = CT(2.0)
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* atan(((tan_lat1 + tan_lat2) / (CT(1) + tan_lat1 * tan_lat2))
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* tan((geometry::get_as_radian<0>(p2)
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- geometry::get_as_radian<0>(p1)) / 2));
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}
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return excess;
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}
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struct return_type_ellipsoidal
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{
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return_type_ellipsoidal()
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: spherical_term(0),
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ellipsoidal_term(0)
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{}
|
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CT spherical_term;
|
CT ellipsoidal_term;
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};
|
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/*
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Compute the ellipsoidal correction of a geodesic (or shperical) segment
|
*/
|
template <
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template <typename, bool, bool, bool, bool, bool> class Inverse,
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typename PointOfSegment,
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typename SpheroidConst
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>
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static inline return_type_ellipsoidal ellipsoidal(PointOfSegment const& p1,
|
PointOfSegment const& p2,
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SpheroidConst const& spheroid_const)
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{
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return_type_ellipsoidal result;
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|
// Azimuth Approximation
|
|
typedef Inverse<CT, false, true, true, false, false> inverse_type;
|
typedef typename inverse_type::result_type inverse_result;
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|
inverse_result i_res = inverse_type::apply(get_as_radian<0>(p1),
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get_as_radian<1>(p1),
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get_as_radian<0>(p2),
|
get_as_radian<1>(p2),
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spheroid_const.m_spheroid);
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|
CT alp1 = i_res.azimuth;
|
CT alp2 = i_res.reverse_azimuth;
|
|
// Constants
|
|
CT const ep = spheroid_const.m_ep;
|
CT const f = formula::flattening<CT>(spheroid_const.m_spheroid);
|
CT const one_minus_f = CT(1) - f;
|
std::size_t const series_order_plus_one = SeriesOrder + 1;
|
std::size_t const series_order_plus_two = SeriesOrder + 2;
|
|
// Basic trigonometric computations
|
|
CT tan_bet1 = tan(get_as_radian<1>(p1)) * one_minus_f;
|
CT tan_bet2 = tan(get_as_radian<1>(p2)) * one_minus_f;
|
CT cos_bet1 = cos(atan(tan_bet1));
|
CT cos_bet2 = cos(atan(tan_bet2));
|
CT sin_bet1 = tan_bet1 * cos_bet1;
|
CT sin_bet2 = tan_bet2 * cos_bet2;
|
CT sin_alp1 = sin(alp1);
|
CT cos_alp1 = cos(alp1);
|
CT cos_alp2 = cos(alp2);
|
CT sin_alp0 = sin_alp1 * cos_bet1;
|
|
// Spherical term computation
|
|
CT sin_omg1 = sin_alp0 * sin_bet1;
|
CT cos_omg1 = cos_alp1 * cos_bet1;
|
CT sin_omg2 = sin_alp0 * sin_bet2;
|
CT cos_omg2 = cos_alp2 * cos_bet2;
|
CT cos_omg12 = cos_omg1 * cos_omg2 + sin_omg1 * sin_omg2;
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CT excess;
|
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bool meridian = get<0>(p2) - get<0>(p1) == CT(0)
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|| get<1>(p1) == CT(90) || get<1>(p1) == -CT(90)
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|| get<1>(p2) == CT(90) || get<1>(p2) == -CT(90);
|
|
if (!meridian && cos_omg12 > -CT(0.7)
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&& sin_bet2 - sin_bet1 < CT(1.75)) // short segment
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{
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CT sin_omg12 = cos_omg1 * sin_omg2 - sin_omg1 * cos_omg2;
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normalize(sin_omg12, cos_omg12);
|
|
CT cos_omg12p1 = CT(1) + cos_omg12;
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CT cos_bet1p1 = CT(1) + cos_bet1;
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CT cos_bet2p1 = CT(1) + cos_bet2;
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excess = CT(2) * atan2(sin_omg12 * (sin_bet1 * cos_bet2p1 + sin_bet2 * cos_bet1p1),
|
cos_omg12p1 * (sin_bet1 * sin_bet2 + cos_bet1p1 * cos_bet2p1));
|
}
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else
|
{
|
/*
|
CT sin_alp2 = sin(alp2);
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CT sin_alp12 = sin_alp2 * cos_alp1 - cos_alp2 * sin_alp1;
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CT cos_alp12 = cos_alp2 * cos_alp1 + sin_alp2 * sin_alp1;
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excess = atan2(sin_alp12, cos_alp12);
|
*/
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excess = alp2 - alp1;
|
}
|
|
result.spherical_term = excess;
|
|
// Ellipsoidal term computation (uses integral approximation)
|
|
CT cos_alp0 = math::sqrt(CT(1) - math::sqr(sin_alp0));
|
CT cos_sig1 = cos_alp1 * cos_bet1;
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CT cos_sig2 = cos_alp2 * cos_bet2;
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CT sin_sig1 = sin_bet1;
|
CT sin_sig2 = sin_bet2;
|
|
normalize(sin_sig1, cos_sig1);
|
normalize(sin_sig2, cos_sig2);
|
|
CT coeffs[SeriesOrder + 1];
|
const std::size_t coeffs_var_size = (series_order_plus_two
|
* series_order_plus_one) / 2;
|
CT coeffs_var[coeffs_var_size];
|
|
if(ExpandEpsN){ // expand by eps and n
|
|
CT k2 = math::sqr(ep * cos_alp0);
|
CT sqrt_k2_plus_one = math::sqrt(CT(1) + k2);
|
CT eps = (sqrt_k2_plus_one - CT(1)) / (sqrt_k2_plus_one + CT(1));
|
CT n = f / (CT(2) - f);
|
|
// Generate and evaluate the polynomials on n
|
// to get the series coefficients (that depend on eps)
|
evaluate_coeffs_n(n, coeffs_var);
|
|
// Generate and evaluate the polynomials on eps (i.e. var2 = eps)
|
// to get the final series coefficients
|
evaluate_coeffs_var2(eps, coeffs_var, coeffs);
|
|
}else{ // expand by k2 and ep
|
|
CT k2 = math::sqr(ep * cos_alp0);
|
CT ep2 = math::sqr(ep);
|
|
// Generate and evaluate the polynomials on ep2
|
evaluate_coeffs_ep(ep2, coeffs_var);
|
|
// Generate and evaluate the polynomials on k2 (i.e. var2 = k2)
|
evaluate_coeffs_var2(k2, coeffs_var, coeffs);
|
|
}
|
|
// Evaluate the trigonometric sum
|
CT I12 = clenshaw_sum(cos_sig2, coeffs, coeffs + series_order_plus_one)
|
- clenshaw_sum(cos_sig1, coeffs, coeffs + series_order_plus_one);
|
|
// The part of the ellipsodal correction that depends on
|
// point coordinates
|
result.ellipsoidal_term = cos_alp0 * sin_alp0 * I12;
|
|
return result;
|
}
|
|
// Check whenever a segment crosses the prime meridian
|
// First normalize to [0,360)
|
template <typename PointOfSegment>
|
static inline bool crosses_prime_meridian(PointOfSegment const& p1,
|
PointOfSegment const& p2)
|
{
|
CT const pi
|
= geometry::math::pi<CT>();
|
CT const two_pi
|
= geometry::math::two_pi<CT>();
|
|
CT p1_lon = get_as_radian<0>(p1)
|
- ( floor( get_as_radian<0>(p1) / two_pi )
|
* two_pi );
|
CT p2_lon = get_as_radian<0>(p2)
|
- ( floor( get_as_radian<0>(p2) / two_pi )
|
* two_pi );
|
|
CT max_lon = (std::max)(p1_lon, p2_lon);
|
CT min_lon = (std::min)(p1_lon, p2_lon);
|
|
return max_lon > pi && min_lon < pi && max_lon - min_lon > pi;
|
}
|
|
};
|
|
}}} // namespace boost::geometry::formula
|
|
|
#endif // BOOST_GEOMETRY_FORMULAS_AREA_FORMULAS_HPP
|
