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libs/geometry/area_on_earth.cpp

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+#include "geometry/area_on_earth.hpp"
+
+#include "geometry/distance_on_sphere.hpp"
+#include "geometry/point3d.hpp"
+
+#include "base/assert.hpp"
+#include "base/math.hpp"
+
+#include <algorithm>
+#include <cmath>
+
+namespace ms
+{
+namespace
+{
+double constexpr kEarthRadiusMetersSquared = kEarthRadiusMeters * kEarthRadiusMeters;
+
+m3::PointD GetPointOnSphere(LatLon const & ll, double sphereRadius)
+{
+  ASSERT(LatLon::kMinLat <= ll.m_lat && ll.m_lat <= LatLon::kMaxLat, (ll));
+  ASSERT(LatLon::kMinLon <= ll.m_lon && ll.m_lon <= LatLon::kMaxLon, (ll));
+
+  double const latRad = math::DegToRad(ll.m_lat);
+  double const lonRad = math::DegToRad(ll.m_lon);
+
+  double const x = sphereRadius * cos(latRad) * cos(lonRad);
+  double const y = sphereRadius * cos(latRad) * sin(lonRad);
+  double const z = sphereRadius * sin(latRad);
+
+  return {x, y, z};
+}
+}  // namespace
+
+// Look to https://en.wikipedia.org/wiki/Solid_angle for more details.
+// Shortly:
+// It's possible to calculate area of triangle on sphere with it's solid angle.
+// Ω = A / R^2
+// Where Ω is solid angle of triangle, R - sphere radius and A - area of triangle on sphere.
+// So A = Ω * R^2
+double AreaOnEarth(LatLon const & ll1, LatLon const & ll2, LatLon const & ll3)
+{
+  m3::PointD const a = GetPointOnSphere(ll1, 1.0 /* sphereRadius */);
+  m3::PointD const b = GetPointOnSphere(ll2, 1.0 /* sphereRadius */);
+  m3::PointD const c = GetPointOnSphere(ll3, 1.0 /* sphereRadius */);
+
+  double const triple = m3::DotProduct(a, m3::CrossProduct(b, c));
+
+  ASSERT(::AlmostEqualAbs(a.Length(), 1.0, 1e-5), ());
+  ASSERT(::AlmostEqualAbs(b.Length(), 1.0, 1e-5), ());
+  ASSERT(::AlmostEqualAbs(c.Length(), 1.0, 1e-5), ());
+
+  double constexpr lengthMultiplication = 1.0;  // a.Length() * b.Length() * c.Length()
+  double const abc = m3::DotProduct(a, b);      // * c.Length() == 1
+  double const acb = m3::DotProduct(a, c);      // * b.Length() == 1
+  double const bca = m3::DotProduct(b, c);      // * a.Length() == 1
+
+  double const tanFromHalfSolidAngle = triple / (lengthMultiplication + abc + acb + bca);
+  double const halfSolidAngle = atan(tanFromHalfSolidAngle);
+  double const solidAngle = halfSolidAngle * 2.0;
+  double const area = solidAngle * kEarthRadiusMetersSquared;
+  return fabs(area);
+}
+
+// Look to https://en.wikipedia.org/wiki/Solid_angle for details.
+// Shortly:
+// Ω = A / R^2
+// A is the spherical surface area which confined by solid angle.
+// R - sphere radius.
+// For circle: Ω = 2π(1 - cos(θ / 2)), where θ - is the cone apex angle.
+double CircleAreaOnEarth(double distanceOnSphereRadius)
+{
+  double const theta = 2.0 * distanceOnSphereRadius / kEarthRadiusMeters;
+  double constexpr kConst = 2 * math::pi * kEarthRadiusMetersSquared;
+  double const sinValue = sin(theta / 4);
+  // 1 - cos(θ / 2) = 2sin^2(θ / 4)
+  return kConst * 2 * sinValue * sinValue;
+}
+}  // namespace ms