text: // Boost.Geometry
  
  // Copyright (c) 2016, Oracle and/or its affiliates.
  // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
  
  // Use, modification and distribution is subject to the Boost Software License,
  // Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  // http://www.boost.org/LICENSE_1_0.txt)
  
  #ifndef BOOST_GEOMETRY_STRATEGIES_SPHERICAL_INTERSECTION_HPP
  #define BOOST_GEOMETRY_STRATEGIES_SPHERICAL_INTERSECTION_HPP
  
  #include <algorithm>
  
  #include <boost/geometry/core/cs.hpp>
  #include <boost/geometry/core/access.hpp>
  #include <boost/geometry/core/radian_access.hpp>
  #include <boost/geometry/core/tags.hpp>
  
  #include <boost/geometry/algorithms/detail/assign_values.hpp>
  #include <boost/geometry/algorithms/detail/assign_indexed_point.hpp>
  #include <boost/geometry/algorithms/detail/equals/point_point.hpp>
  #include <boost/geometry/algorithms/detail/recalculate.hpp>
  
  #include <boost/geometry/arithmetic/arithmetic.hpp>
  #include <boost/geometry/arithmetic/cross_product.hpp>
  #include <boost/geometry/arithmetic/dot_product.hpp>
  #include <boost/geometry/formulas/spherical.hpp>
  
  #include <boost/geometry/geometries/concepts/point_concept.hpp>
  #include <boost/geometry/geometries/concepts/segment_concept.hpp>
  
  #include <boost/geometry/policies/robustness/segment_ratio.hpp>
  
  #include <boost/geometry/strategies/side_info.hpp>
  #include <boost/geometry/strategies/intersection.hpp>
  #include <boost/geometry/strategies/intersection_result.hpp>
  
  #include <boost/geometry/util/math.hpp>
  #include <boost/geometry/util/select_calculation_type.hpp>
  
  
  namespace boost { namespace geometry
  {
  
  namespace strategy { namespace intersection
  {
  
  // NOTE:
  // The coordinates of crossing IP may be calculated with small precision in some cases.
  // For double, near the equator noticed error ~1e-9 so far greater than
  // machine epsilon which is ~1e-16. This error is ~0.04m.
  // E.g. consider two cases, one near the origin and the second one rotated by 90 deg around Z or SN axis.
  // After the conversion from spherical degrees to cartesian 3d the following coordinates
  // are calculated:
  // for sph (-1 -1,  1 1) deg cart3d ys are -0.017449748351250485 and  0.017449748351250485
  // for sph (89 -1, 91 1) deg cart3d xs are  0.017449748351250571 and -0.017449748351250450
  // During the conversion degrees must first be converted to radians and then radians
  // are passed into trigonometric functions. The error may have several causes:
  // 1. Radians cannot represent exactly the same angles as degrees.
  // 2. Different longitudes are passed into sin() for x, corresponding to cos() for y,
  //    and for different angle the error of the result may be different.
  // 3. These non-corresponding cartesian coordinates are used in calculation,
  //    e.g. multiplied several times in cross and dot products.
  // If it was a problem this strategy could e.g. "normalize" longitudes before the conversion using the source units
  // by rotating the globe around Z axis, so moving longitudes always the same way towards the origin,
  // assuming this could help which is not clear.
  // For now, intersection points near the endpoints are checked explicitly if needed (if the IP is near the endpoint)
  // to generate precise result for them. Only the crossing (i) case may suffer from lower precision.
  
  template <typename Policy, typename CalculationType = void>
  struct relate_spherical_segments
  {
      typedef typename Policy::return_type return_type;
  
      enum intersection_point_flag { ipi_inters = 0, ipi_at_a1, ipi_at_a2, ipi_at_b1, ipi_at_b2 };
  
      template <typename CoordinateType, typename SegmentRatio, typename Vector3d>
      struct segment_intersection_info
      {
          typedef typename select_most_precise
              <
                  CoordinateType, double
              >::type promoted_type;
  
          promoted_type comparable_length_a() const
          {
              return robust_ra.denominator();
          }
  
          promoted_type comparable_length_b() const
          {
              return robust_rb.denominator();
          }
  
          template <typename Point, typename Segment1, typename Segment2>
          void assign_a(Point& point, Segment1 const& a, Segment2 const& b) const
          {
              assign(point, a, b);
          }
          template <typename Point, typename Segment1, typename Segment2>
          void assign_b(Point& point, Segment1 const& a, Segment2 const& b) const
          {
              assign(point, a, b);
          }
  
          template <typename Point, typename Segment1, typename Segment2>
          void assign(Point& point, Segment1 const& a, Segment2 const& b) const
          {
              if (ip_flag == ipi_inters)
              {
                  // TODO: assign the rest of coordinates
                  point = formula::cart3d_to_sph<Point>(intersection_point);
              }
              else if (ip_flag == ipi_at_a1)
              {
                  detail::assign_point_from_index<0>(a, point);
              }
              else if (ip_flag == ipi_at_a2)
              {
                  detail::assign_point_from_index<1>(a, point);
              }
              else if (ip_flag == ipi_at_b1)
              {
                  detail::assign_point_from_index<0>(b, point);
              }
              else // ip_flag == ipi_at_b2
              {
                  detail::assign_point_from_index<1>(b, point);
              }
          }
  
          Vector3d intersection_point;
          SegmentRatio robust_ra;
          SegmentRatio robust_rb;
          intersection_point_flag ip_flag;
      };
  
      // Relate segments a and b
      template <typename Segment1, typename Segment2, typename RobustPolicy>
      static inline return_type apply(Segment1 const& a, Segment2 const& b,
                                      RobustPolicy const& robust_policy)
      {
          typedef typename point_type<Segment1>::type point1_t;
          typedef typename point_type<Segment2>::type point2_t;
          point1_t a1, a2;
          point2_t b1, b2;
  
          // TODO: use indexed_point_view if possible?
          detail::assign_point_from_index<0>(a, a1);
          detail::assign_point_from_index<1>(a, a2);
          detail::assign_point_from_index<0>(b, b1);
          detail::assign_point_from_index<1>(b, b2);
  
          return apply(a, b, robust_policy, a1, a2, b1, b2);
      }
  
      // Relate segments a and b
      template <typename Segment1, typename Segment2, typename RobustPolicy, typename Point1, typename Point2>
      static inline return_type apply(Segment1 const& a, Segment2 const& b,
                                      RobustPolicy const&,
                                      Point1 const& a1, Point1 const& a2, Point2 const& b1, Point2 const& b2)
      {
          BOOST_CONCEPT_ASSERT( (concepts::ConstSegment<Segment1>) );
          BOOST_CONCEPT_ASSERT( (concepts::ConstSegment<Segment2>) );
  
          // TODO: check only 2 first coordinates here?
          using geometry::detail::equals::equals_point_point;
          bool a_is_point = equals_point_point(a1, a2);
          bool b_is_point = equals_point_point(b1, b2);
  
          if(a_is_point && b_is_point)
          {
              return equals_point_point(a1, b2)
                  ? Policy::degenerate(a, true)
                  : Policy::disjoint()
                  ;
          }
  
          typedef typename select_calculation_type
              <Segment1, Segment2, CalculationType>::type calc_t;
  
          calc_t const c0 = 0;
          calc_t const c1 = 1;
  
          typedef model::point<calc_t, 3, cs::cartesian> vec3d_t;
  
          using namespace formula;
          vec3d_t const a1v = sph_to_cart3d<vec3d_t>(a1);
          vec3d_t const a2v = sph_to_cart3d<vec3d_t>(a2);
          vec3d_t const b1v = sph_to_cart3d<vec3d_t>(b1);
          vec3d_t const b2v = sph_to_cart3d<vec3d_t>(b2);
          
          vec3d_t norm1 = cross_product(a1v, a2v);
          vec3d_t norm2 = cross_product(b1v, b2v);
  
          side_info sides;
          // not normalized normals, the same as in SSF
          sides.set<0>(sph_side_value(norm2, a1v), sph_side_value(norm2, a2v));
          if (sides.same<0>())
          {
              // Both points are at same side of other segment, we can leave
              return Policy::disjoint();
          }
          // not normalized normals, the same as in SSF
          sides.set<1>(sph_side_value(norm1, b1v), sph_side_value(norm1, b2v));
          if (sides.same<1>())
          {
              // Both points are at same side of other segment, we can leave
              return Policy::disjoint();
          }
  
          // NOTE: at this point the segments may still be disjoint
  
          bool collinear = sides.collinear();
  
          calc_t const len1 = math::sqrt(dot_product(norm1, norm1));
          calc_t const len2 = math::sqrt(dot_product(norm2, norm2));
  
          // point or opposite sides of a sphere, assume point
          if (math::equals(len1, c0))
          {
              a_is_point = true;
              if (sides.get<0, 0>() == 0 || sides.get<0, 1>() == 0)
              {
                  sides.set<0>(0, 0);
              }
          }
          else
          {
              // normalize
              divide_value(norm1, len1);
          }
  
          if (math::equals(len2, c0))
          {
              b_is_point = true;
              if (sides.get<1, 0>() == 0 || sides.get<1, 1>() == 0)
              {
                  sides.set<1>(0, 0);
              }
          }
          else
          {
              // normalize
              divide_value(norm2, len2);
          }
  
          // check both degenerated once more
          if (a_is_point && b_is_point)
          {
              return equals_point_point(a1, b2)
                  ? Policy::degenerate(a, true)
                  : Policy::disjoint()
                  ;
          }
  
          // NOTE: at this point one of the segments may be degenerated
          //       and the segments may still be disjoint
  
          calc_t dot_n1n2 = dot_product(norm1, norm2);
  
          // NOTE: this is technically not needed since theoretically above sides
          //       are calculated, but just in case check the normals.
          //       Have in mind that SSF side strategy doesn't check this.
          // collinear if normals are equal or opposite: cos(a) in {-1, 1}
          if (!collinear && math::equals(math::abs(dot_n1n2), c1))
          {
              collinear = true;
              sides.set<0>(0, 0);
              sides.set<1>(0, 0);
          }
          
          if (collinear)
          {
              if (a_is_point)
              {
                  return collinear_one_degenerted<calc_t>(a, true, b1, b2, a1, a2, b1v, b2v, norm2, a1v);
              }
              else if (b_is_point)
              {
                  // b2 used to be consistent with (degenerated) checks above (is it needed?)
                  return collinear_one_degenerted<calc_t>(b, false, a1, a2, b1, b2, a1v, a2v, norm1, b1v);
              }
              else
              {
                  calc_t dist_a1_a2, dist_a1_b1, dist_a1_b2;
                  calc_t dist_b1_b2, dist_b1_a1, dist_b1_a2;
                  // use shorter segment
                  if (len1 <= len2)
                  {
                      calculate_collinear_data(a1, a2, b1, b2, a1v, a2v, norm1, b1v, dist_a1_a2, dist_a1_b1);
                      calculate_collinear_data(a1, a2, b1, b2, a1v, a2v, norm1, b2v, dist_a1_a2, dist_a1_b2);
                      dist_b1_b2 = dist_a1_b2 - dist_a1_b1;
                      dist_b1_a1 = -dist_a1_b1;
                      dist_b1_a2 = dist_a1_a2 - dist_a1_b1;
                  }
                  else
                  {
                      calculate_collinear_data(b1, b2, a1, a2, b1v, b2v, norm2, a1v, dist_b1_b2, dist_b1_a1);
                      calculate_collinear_data(b1, b2, a1, a2, b1v, b2v, norm2, a2v, dist_b1_b2, dist_b1_a2);
                      dist_a1_a2 = dist_b1_a2 - dist_b1_a1;
                      dist_a1_b1 = -dist_b1_a1;
                      dist_a1_b2 = dist_b1_b2 - dist_b1_a1;
                  }
  
                  segment_ratio<calc_t> ra_from(dist_b1_a1, dist_b1_b2);
                  segment_ratio<calc_t> ra_to(dist_b1_a2, dist_b1_b2);
                  segment_ratio<calc_t> rb_from(dist_a1_b1, dist_a1_a2);
                  segment_ratio<calc_t> rb_to(dist_a1_b2, dist_a1_a2);
                  
                  // NOTE: this is probably not needed
                  int const a1_wrt_b = position_value(c0, dist_a1_b1, dist_a1_b2);
                  int const a2_wrt_b = position_value(dist_a1_a2, dist_a1_b1, dist_a1_b2);
                  int const b1_wrt_a = position_value(c0, dist_b1_a1, dist_b1_a2);
                  int const b2_wrt_a = position_value(dist_b1_b2, dist_b1_a1, dist_b1_a2);
  
                  if (a1_wrt_b == 1)
                  {
                      ra_from.assign(0, dist_b1_b2);
                      rb_from.assign(0, dist_a1_a2);
                  }
                  else if (a1_wrt_b == 3)
                  {
                      ra_from.assign(dist_b1_b2, dist_b1_b2);
                      rb_to.assign(0, dist_a1_a2);
                  }
  
                  if (a2_wrt_b == 1)
                  {
                      ra_to.assign(0, dist_b1_b2);
                      rb_from.assign(dist_a1_a2, dist_a1_a2);
                  }
                  else if (a2_wrt_b == 3)
                  {
                      ra_to.assign(dist_b1_b2, dist_b1_b2);
                      rb_to.assign(dist_a1_a2, dist_a1_a2);
                  }
  
                  if ((a1_wrt_b < 1 && a2_wrt_b < 1) || (a1_wrt_b > 3 && a2_wrt_b > 3))
                  {
                      return Policy::disjoint();
                  }
  
                  bool const opposite = dot_n1n2 < c0;
  
                  return Policy::segments_collinear(a, b, opposite,
                      a1_wrt_b, a2_wrt_b, b1_wrt_a, b2_wrt_a,
                      ra_from, ra_to, rb_from, rb_to);
              }
          }
          else // crossing
          {
              if (a_is_point || b_is_point)
              {
                  return Policy::disjoint();
              }
  
              vec3d_t i1;
              intersection_point_flag ip_flag;
              calc_t dist_a1_a2, dist_a1_i1, dist_b1_b2, dist_b1_i1;
              if (calculate_ip_data(a1, a2, b1, b2, a1v, a2v, b1v, b2v, norm1, norm2, sides,
                                    i1, dist_a1_a2, dist_a1_i1, dist_b1_b2, dist_b1_i1, ip_flag))
              {
                  // intersects
                  segment_intersection_info
                      <
                          calc_t,
                          segment_ratio<calc_t>,
                          vec3d_t
                      > sinfo;
  
                  sinfo.robust_ra.assign(dist_a1_i1, dist_a1_a2);
                  sinfo.robust_rb.assign(dist_b1_i1, dist_b1_b2);
                  sinfo.intersection_point = i1;
                  sinfo.ip_flag = ip_flag;
  
                  return Policy::segments_crosses(sides, sinfo, a, b);
              }
              else
              {
                  return Policy::disjoint();
              }
          }
      }
  
  private:
      template <typename CalcT, typename Segment, typename Point1, typename Point2, typename Vec3d>
      static inline return_type collinear_one_degenerted(Segment const& segment, bool degenerated_a,
                                                         Point1 const& a1, Point1 const& a2,
                                                         Point2 const& b1, Point2 const& b2,
                                                         Vec3d const& v1, Vec3d const& v2, Vec3d const& norm,
                                                         Vec3d const& vother)
      {
          CalcT dist_1_2, dist_1_o;
          return ! calculate_collinear_data(a1, a2, b1, b2, v1, v2, norm, vother, dist_1_2, dist_1_o)
                  ? Policy::disjoint()
                  : Policy::one_degenerate(segment, segment_ratio<CalcT>(dist_1_o, dist_1_2), degenerated_a);
      }
  
      template <typename Point1, typename Point2, typename Vec3d, typename CalcT>
      static inline bool calculate_collinear_data(Point1 const& a1, Point1 const& a2,
                                                  Point2 const& b1, Point2 const& b2,
                                                  Vec3d const& a1v, // in
                                                  Vec3d const& a2v, // in
                                                  Vec3d const& norm1, // in
                                                  Vec3d const& b1v_or_b2v, // in
                                                  CalcT& dist_a1_a2, CalcT& dist_a1_i1) // out
      {
          // calculate dist_a1_a2 and dist_a1_i1
          calculate_dists(a1v, a2v, norm1, b1v_or_b2v, dist_a1_a2, dist_a1_i1);
  
          // if i1 is close to a1 and b1 or b2 is equal to a1
          if (is_endpoint_equal(dist_a1_i1, a1, b1, b2))
          {
              dist_a1_i1 = 0;
              return true;
          }
          // or i1 is close to a2 and b1 or b2 is equal to a2
          else if (is_endpoint_equal(dist_a1_a2 - dist_a1_i1, a2, b1, b2))
          {
              dist_a1_i1 = dist_a1_a2;
              return true;
          }
  
          // or i1 is on b
          return segment_ratio<CalcT>(dist_a1_i1, dist_a1_a2).on_segment();
      }
  
      template <typename Point1, typename Point2, typename Vec3d, typename CalcT>
      static inline bool calculate_ip_data(Point1 const& a1, Point1 const& a2, // in
                                           Point2 const& b1, Point2 const& b2, // in
                                           Vec3d const& a1v, Vec3d const& a2v, // in
                                           Vec3d const& b1v, Vec3d const& b2v, // in
                                           Vec3d const& norm1, Vec3d const& norm2, // in
                                           side_info const& sides, // in
                                           Vec3d & i1, // in-out
                                           CalcT& dist_a1_a2, CalcT& dist_a1_i1, // out
                                           CalcT& dist_b1_b2, CalcT& dist_b1_i1, // out
                                           intersection_point_flag& ip_flag)     // out
      {
          // great circles intersections
          i1 = cross_product(norm1, norm2);
          // NOTE: the length should be greater than 0 at this point
          //       if the normals were not normalized and their dot product
          //       not checked before this function is called the length
          //       should be checked here (math::equals(len, c0))
          CalcT const len = math::sqrt(dot_product(i1, i1));
          divide_value(i1, len); // normalize i1
  
          calculate_dists(a1v, a2v, norm1, i1, dist_a1_a2, dist_a1_i1);
          
          // choose the opposite side of the globe if the distance is shorter
          {
              CalcT const d = abs_distance(dist_a1_a2, dist_a1_i1);
              if (d > CalcT(0))
              {
                  CalcT const dist_a1_i2 = dist_of_i2(dist_a1_i1);
                  CalcT const d2 = abs_distance(dist_a1_a2, dist_a1_i2);
                  if (d2 < d)
                  {
                      dist_a1_i1 = dist_a1_i2;
                      multiply_value(i1, CalcT(-1)); // the opposite intersection
                  }
              }
          }
  
          bool is_on_a = false, is_near_a1 = false, is_near_a2 = false;
          if (! is_potentially_crossing(dist_a1_a2, dist_a1_i1, is_on_a, is_near_a1, is_near_a2))
          {
              return false;
          }
  
          calculate_dists(b1v, b2v, norm2, i1, dist_b1_b2, dist_b1_i1);
  
          bool is_on_b = false, is_near_b1 = false, is_near_b2 = false;
          if (! is_potentially_crossing(dist_b1_b2, dist_b1_i1, is_on_b, is_near_b1, is_near_b2))
          {
              return false;
          }
  
          // reassign the IP if some endpoints overlap
          using geometry::detail::equals::equals_point_point;
          if (is_near_a1)
          {
              if (is_near_b1 && equals_point_point(a1, b1))
              {
                  dist_a1_i1 = 0;
                  dist_b1_i1 = 0;
                  //i1 = a1v;
                  ip_flag = ipi_at_a1;
                  return true;
              }
              
              if (is_near_b2 && equals_point_point(a1, b2))
              {
                  dist_a1_i1 = 0;
                  dist_b1_i1 = dist_b1_b2;
                  //i1 = a1v;
                  ip_flag = ipi_at_a1;
                  return true;
              }
          }
  
          if (is_near_a2)
          {
              if (is_near_b1 && equals_point_point(a2, b1))
              {
                  dist_a1_i1 = dist_a1_a2;
                  dist_b1_i1 = 0;
                  //i1 = a2v;
                  ip_flag = ipi_at_a2;
                  return true;
              }
  
              if (is_near_b2 && equals_point_point(a2, b2))
              {
                  dist_a1_i1 = dist_a1_a2;
                  dist_b1_i1 = dist_b1_b2;
                  //i1 = a2v;
                  ip_flag = ipi_at_a2;
                  return true;
              }
          }
  
          // at this point we know that the endpoints doesn't overlap
          // reassign IP and distance if the IP is on a segment and one of
          //   the endpoints of the other segment lies on the former segment
          if (is_on_a)
          {
              if (is_near_b1 && sides.template get<1, 0>() == 0) // b1 wrt a
              {
                  dist_b1_i1 = 0;
                  //i1 = b1v;
                  ip_flag = ipi_at_b1;
                  return true;
              }
  
              if (is_near_b2 && sides.template get<1, 1>() == 0) // b2 wrt a
              {
                  dist_b1_i1 = dist_b1_b2;
                  //i1 = b2v;
                  ip_flag = ipi_at_b2;
                  return true;
              }
          }
  
          if (is_on_b)
          {
              if (is_near_a1 && sides.template get<0, 0>() == 0) // a1 wrt b
              {
                  dist_a1_i1 = 0;
                  //i1 = a1v;
                  ip_flag = ipi_at_a1;
                  return true;
              }
  
              if (is_near_a2 && sides.template get<0, 1>() == 0) // a2 wrt b
              {
                  dist_a1_i1 = dist_a1_a2;
                  //i1 = a2v;
                  ip_flag = ipi_at_a2;
                  return true;
              }
          }
  
          ip_flag = ipi_inters;
  
          return is_on_a && is_on_b;
      }
  
      template <typename Vec3d, typename CalcT>
      static inline void calculate_dists(Vec3d const& a1v, // in
                                         Vec3d const& a2v, // in
                                         Vec3d const& norm1, // in
                                         Vec3d const& i1, // in
                                         CalcT& dist_a1_a2, CalcT& dist_a1_i1) // out
      {
          CalcT const c0 = 0;
          CalcT const c1 = 1;
          CalcT const c2 = 2;
          CalcT const c4 = 4;
              
          CalcT cos_a1_a2 = dot_product(a1v, a2v);
          dist_a1_a2 = -cos_a1_a2 + c1; // [1, -1] -> [0, 2] representing [0, pi]
  
          CalcT cos_a1_i1 = dot_product(a1v, i1);
          dist_a1_i1 = -cos_a1_i1 + c1; // [0, 2] representing [0, pi]
          if (dot_product(norm1, cross_product(a1v, i1)) < c0) // left or right of a1 on a
          {
              dist_a1_i1 = -dist_a1_i1; // [0, 2] -> [0, -2] representing [0, -pi]
          }
          if (dist_a1_i1 <= -c2) // <= -pi
          {
              dist_a1_i1 += c4; // += 2pi
          }
      }
  
      // the dist of the ip on the other side of the sphere
      template <typename CalcT>
      static inline CalcT dist_of_i2(CalcT const& dist_a1_i1)
      {
          CalcT const c2 = 2;
          CalcT const c4 = 4;
  
          CalcT dist_a1_i2 = dist_a1_i1 - c2; // dist_a1_i2 = dist_a1_i1 - pi;
          if (dist_a1_i2 <= -c2)          // <= -pi
          {
              dist_a1_i2 += c4;           // += 2pi;
          }
          return dist_a1_i2;
      }
  
      template <typename CalcT>
      static inline CalcT abs_distance(CalcT const& dist_a1_a2, CalcT const& dist_a1_i1)
      {
          if (dist_a1_i1 < CalcT(0))
              return -dist_a1_i1;
          else if (dist_a1_i1 > dist_a1_a2)
              return dist_a1_i1 - dist_a1_a2;
          else
              return CalcT(0);
      }
  
      template <typename CalcT>
      static inline bool is_potentially_crossing(CalcT const& dist_a1_a2, CalcT const& dist_a1_i1, // in
                                                 bool& is_on_a, bool& is_near_a1, bool& is_near_a2) // out
      {
          is_on_a = segment_ratio<CalcT>(dist_a1_i1, dist_a1_a2).on_segment();
          is_near_a1 = is_near(dist_a1_i1);
          is_near_a2 = is_near(dist_a1_a2 - dist_a1_i1);
          return is_on_a || is_near_a1 || is_near_a2;
      }
  
      template <typename CalcT, typename P1, typename P2>
      static inline bool is_endpoint_equal(CalcT const& dist,
                                           P1 const& ai, P2 const& b1, P2 const& b2)
      {
          using geometry::detail::equals::equals_point_point;
          return is_near(dist) && (equals_point_point(ai, b1) || equals_point_point(ai, b2));
      }
  
      template <typename CalcT>
      static inline bool is_near(CalcT const& dist)
      {
          CalcT const small_number = CalcT(boost::is_same<CalcT, float>::value ? 0.0001 : 0.00000001);
          return math::abs(dist) <= small_number;
      }
  
      template <typename ProjCoord1, typename ProjCoord2>
      static inline int position_value(ProjCoord1 const& ca1,
                                       ProjCoord2 const& cb1,
                                       ProjCoord2 const& cb2)
      {
          // S1x  0   1    2     3   4
          // S2       |---------->
          return math::equals(ca1, cb1) ? 1
               : math::equals(ca1, cb2) ? 3
               : cb1 < cb2 ?
                  ( ca1 < cb1 ? 0
                  : ca1 > cb2 ? 4
                  : 2 )
                : ( ca1 > cb1 ? 0
                  : ca1 < cb2 ? 4
                  : 2 );
      }
  };
  
  
  #ifndef DOXYGEN_NO_STRATEGY_SPECIALIZATIONS
  namespace services
  {
  
  /*template <typename Policy, typename CalculationType>
  struct default_strategy<spherical_polar_tag, Policy, CalculationType>
  {
      typedef relate_spherical_segments<Policy, CalculationType> type;
  };*/
  
  template <typename Policy, typename CalculationType>
  struct default_strategy<spherical_equatorial_tag, Policy, CalculationType>
  {
      typedef relate_spherical_segments<Policy, CalculationType> type;
  };
  
  template <typename Policy, typename CalculationType>
  struct default_strategy<geographic_tag, Policy, CalculationType>
  {
      typedef relate_spherical_segments<Policy, CalculationType> type;
  };
  
  } // namespace services
  #endif // DOXYGEN_NO_STRATEGY_SPECIALIZATIONS
  
  
  }} // namespace strategy::intersection
  
  }} // namespace boost::geometry
  
  
  #endif // BOOST_GEOMETRY_STRATEGIES_SPHERICAL_INTERSECTION_HPP
  