Represent the three rhombohedral coordinates by the symbols Rx, Ry, and Rz, and the three hexagonal coordinates by the symbols Hx, Hy, and Hz. Write a set of crystallographic position coordinates as Rx,Ry,Rz or Hx,Hy,Hz.
Convert the rhombohedral x-coordinate (Rx) to the equivalent hexagonal x-coordinate (Hx) by applying the formula: Hx = (2 x (Rx) -- Ry -- Rz)/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral x-coordinate of Rx = 1. The equivalent Hx coordinate is: (2 x (Rx) -- Ry -- Rz)/3 = (2 x (1) -- 0 -- 0)/3 = 2/3.
Change the rhombohedral y-coordinate (Ry) to the equivalent hexagonal y-coordinate (Hy) by applying the formula: Hy = (Rx + Ry -- (2 x Rz))/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral y-coordinate of Ry = 0. The equivalent Hy coordinate is: (Rx + Ry -- (2 x Rz))/3 = (1 + 0 -- (2 x 0))/3 = 1/3.
Transform the rhombohedral z-coordinate (Rz) to the equivalent hexagonal z-coordinate (Hz) by applying the formula: Hz = (Rx + Ry + Rz)/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral z-coordinate of Rz = 0. The equivalent Hz coordinate is: (Rx + Ry + Rz)/3 = (1 + 0 + 0)/3 = 1/3.
Combine the values for Hx, Hy and Hz and write them as a set of position coordinates for the hexagonal cell: Hx,Hy,Hz. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will be positioned at 2/3,1/3,1/3 in a hexagonal unit cell.