h3

What rules are used to define the i j k axes of an icosahedron face?

Hi，I'm reading source code of H3 (v4.1.0) and in the file "faceijk.c" I find these codes on lines 86~130;

According to the comments, the variable "faceAxesAzRadsCII" is used to hardcode the azimuths of i,j,k axes of each icosahedron face.

My question is, given an icosahedron face, how can we determined which vertex out of its three is the "0 vertex" and thus the i,j,k axes are determined? What rules are used to choose the vertex-0 ?

Thanks!

Source codes of line 86~130 in file "faceijk.c":

/** @brief icosahedron face ijk axes as azimuth in radians from face center to
 * vertex 0/1/2 respectively
 */
static const double faceAxesAzRadsCII[NUM_ICOSA_FACES][3] = {
    {5.619958268523939882, 3.525563166130744542,
     1.431168063737548730},  // face  0
    {5.760339081714187279, 3.665943979320991689,
     1.571548876927796127},  // face  1
    {0.780213654393430055, 4.969003859179821079,
     2.874608756786625655},  // face  2
    {0.430469363979999913, 4.619259568766391033,
     2.524864466373195467},  // face  3
    {6.130269123335111400, 4.035874020941915804,
     1.941478918548720291},  // face  4
    {2.692877706530642877, 0.598482604137447119,
     4.787272808923838195},  // face  5
    {2.982963003477243874, 0.888567901084048369,
     5.077358105870439581},  // face  6
    {3.532912002790141181, 1.438516900396945656,
     5.627307105183336758},  // face  7
    {3.494305004259568154, 1.399909901866372864,
     5.588700106652763840},  // face  8
    {3.003214169499538391, 0.908819067106342928,
     5.097609271892733906},  // face  9
    {5.930472956509811562, 3.836077854116615875,
     1.741682751723420374},  // face 10
    {0.138378484090254847, 4.327168688876645809,
     2.232773586483450311},  // face 11
    {0.448714947059150361, 4.637505151845541521,
     2.543110049452346120},  // face 12
    {0.158629650112549365, 4.347419854898940135,
     2.253024752505744869},  // face 13
    {5.891865957979238535, 3.797470855586042958,
     1.703075753192847583},  // face 14
    {2.711123289609793325, 0.616728187216597771,
     4.805518392002988683},  // face 15
    {3.294508837434268316, 1.200113735041072948,
     5.388903939827463911},  // face 16
    {3.804819692245439833, 1.710424589852244509,
     5.899214794638635174},  // face 17
    {3.664438879055192436, 1.570043776661997111,
     5.758833981448388027},  // face 18
    {2.361378999196363184, 0.266983896803167583,
     4.455774101589558636},  // face 19
};
Solution

  • The zero vertex is on the Class II i-axis. If you unfold the icosahedron with the triangles arranged as pictured, it is the direction in each triangle indicated by the orientation of the triangle number label.

    Unfolded Icosahedron:

    Unfolded Icosahedron