Icosphere-impl.hpp

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#include <array>
 
#include "dart/math/Constants.hpp"
#include "dart/math/Icosphere.hpp"
 
namespace dart {
namespace math {
 
//==============================================================================
template <typename S>
std::size_t Icosphere<S>::getNumVertices(std::size_t subdivisions)
{
  std::size_t numVertices = 12;
  for (auto i = 0u; i < subdivisions; ++i)
    numVertices += getNumEdges(i);
  return numVertices;
}
 
//==============================================================================
template <typename S>
std::size_t Icosphere<S>::getNumEdges(std::size_t subdivisions)
{
  return getNumTriangles(subdivisions) / 2 * 3;
}
 
//==============================================================================
template <typename S>
std::size_t Icosphere<S>::getNumTriangles(std::size_t subdivisions)
{
  return 20 * std::pow(4, subdivisions);
}
 
//==============================================================================
template <typename S>
std::pair<typename Icosphere<S>::Vertices, typename Icosphere<S>::Triangles>
Icosphere<S>::computeIcosahedron(S radius)
{
  constexpr S phi = constants<S>::phi();
  const S unitX = 1 / std::sqrt(1 + phi * phi);
  const S unitZ = unitX * phi;
 
  const S x = radius * unitX;
  const S z = radius * unitZ;
 
  std::vector<Vector3> vertices = {{-x, 0, z},
                                   {x, 0, z},
                                   {-x, 0, -z},
                                   {x, 0, -z},
                                   {0, z, x},
                                   {0, z, -x},
                                   {0, -z, x},
                                   {0, -z, -x},
                                   {z, x, 0},
                                   {-z, x, 0},
                                   {z, -x, 0},
                                   {-z, -x, 0}};
 
  static std::vector<Triangle> triangles
      = {{0, 4, 1},  {0, 9, 4},  {9, 5, 4},  {4, 5, 8},  {4, 8, 1},
         {8, 10, 1}, {8, 3, 10}, {5, 3, 8},  {5, 2, 3},  {2, 7, 3},
         {7, 10, 3}, {7, 6, 10}, {7, 11, 6}, {11, 0, 6}, {0, 1, 6},
         {6, 1, 10}, {9, 0, 11}, {9, 11, 2}, {9, 2, 5},  {7, 2, 11}};
 
  return std::make_pair(vertices, triangles);
}
 
//==============================================================================
template <typename S>
Icosphere<S>::Icosphere(S radius, std::size_t subdivisions)
  : mRadius(radius), mSubdivisions(subdivisions)
{
  static_assert(
      std::is_floating_point<S>::value,
      "Scalar must be a floating point type.");
  assert(radius > 0);
 
  build();
}
 
//==============================================================================
template <typename S>
S Icosphere<S>::getRadius() const
{
  return mRadius;
}
 
//==============================================================================
template <typename S>
std::size_t Icosphere<S>::getNumSubdivisions() const
{
  return mSubdivisions;
}
 
//==============================================================================
template <typename S>
void Icosphere<S>::build()
{
  // Reference: https://schneide.blog/2016/07/15/generating-an-icosphere-in-c/
 
  // Create icosahedron
  std::tie(this->mVertices, this->mTriangles) = computeIcosahedron(mRadius);
 
  // Return if no need to subdivide
  if (mSubdivisions == 0)
    return;
 
  // Create index map that is used for subdivision
  using IndexMap = std::map<std::pair<std::size_t, std::size_t>, std::size_t>;
  IndexMap midVertexIndices;
 
  // Create a temporary array of faces that is used for subdivision
  std::vector<Triangle> tmpFaces;
  if (mSubdivisions % 2)
  {
    this->mTriangles.reserve(getNumTriangles(mSubdivisions - 1));
    tmpFaces.reserve(getNumTriangles(mSubdivisions));
  }
  else
  {
    this->mTriangles.reserve(getNumTriangles(mSubdivisions));
    tmpFaces.reserve(getNumTriangles(mSubdivisions - 1));
  }
 
  // Create more intermediate variables that are used for subdivision
  std::vector<Triangle>* currFaces = &(this->mTriangles);
  std::vector<Triangle>* newFaces = &tmpFaces;
  std::array<std::size_t, 3> mid;
 
  // Subdivide icosahedron/icosphere iteratively. The key is to not duplicate
  // the newly created vertices and faces during each subdivision.
  for (std::size_t i = 0; i < mSubdivisions; ++i)
  {
    // Clear the array of faces that will store the faces of the subdivided
    // isosphere in this iteration. This is because the faces of the previous
    // isosphere are not reused.
    (*newFaces).clear();
    midVertexIndices.clear();
 
    // Iterate each face of the previous icosphere and divide the face into
    // four new faces.
    for (std::size_t j = 0; j < (*currFaces).size(); ++j)
    {
      const auto& outter = (*currFaces)[j];
 
      // Create vertices on the middle of edges if not already created.
      for (std::size_t k = 0; k < 3; ++k)
      {
        auto indexA = outter[k];
        auto indexB = outter[(k + 1) % 3];
 
        // Sort indices to guarantee that the key is unique for the same pairs
        // of indices.
        if (indexA > indexB)
          std::swap(indexA, indexB);
 
        // Check whether the mid vertex given index pair is already created.
        const auto result = midVertexIndices.insert(
            {{indexA, indexB}, this->mVertices.size()});
        const auto& inserted = result.second;
        if (inserted)
        {
          // Create a vertex on the middle of the edge where the length of the
          // vertex is equal to the radius of the icosphere.
          const auto& v1 = this->mVertices[indexA];
          const auto& v2 = this->mVertices[indexB];
          this->mVertices.emplace_back(mRadius * (v1 + v2).normalized());
        }
 
        mid[k] = result.first->second;
      }
 
      // Add four new faces.
      (*newFaces).emplace_back(Triangle(outter[0], mid[0], mid[2]));
      (*newFaces).emplace_back(Triangle(mid[0], outter[1], mid[1]));
      (*newFaces).emplace_back(Triangle(mid[0], mid[1], mid[2]));
      (*newFaces).emplace_back(Triangle(mid[2], mid[1], outter[2]));
    }
 
    // Swap the arrays of faces.
    std::swap(currFaces, newFaces);
  }
 
  // Assign faces if needed.
  this->mTriangles = *currFaces;
}
 
} // namespace math
} // namespace dart