Usage

All examples use use mortif and the standard iso_fortran_env integer kinds.

3D Encoding

Pass three integer(int32) indexes to morton3D. It returns an integer(int64) Morton code.

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int64) :: code

code = morton3D(i=0_int32, j=1_int32, k=0_int32)
print '(A,I20)', "Morton code of {0,1,0}: ", code   ! 2

The Z-order code is constructed by interleaving the binary representations of i, j, and k, with i in bit position 0, j in bit position 1, and k in bit position 2.

3D Decoding

Pass a 64-bit Morton code to demorton3D. The three decoded integer(int32) indexes are returned through intent(inout) arguments.

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int32) :: i, j, k

call demorton3D(code=2_int64, i=i, j=j, k=k)
print '(A,3(I0,1X))', "Decoded indexes of code 2: ", i, j, k   ! 0 1 0

TIP

demorton3D is the exact inverse of morton3D — encoding then decoding (or vice versa) returns the original values, provided the indexes stay within the 21-bit limit per axis.

2D Encoding

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int64) :: code

code = morton2D(i=3_int32, j=1_int32)
print '(A,I20)', "Morton code of {3,1}: ", code   ! 11

For 2D, i bits occupy even positions and j bits occupy odd positions of the 64-bit result.

2D Decoding

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int32) :: i, j

call demorton2D(code=11_int64, i=i, j=j)
print '(A,2(I0,1X))', "Decoded indexes of code 11: ", i, j   ! 3 1

Elemental Use on Arrays

Because all four procedures are elemental, you can apply them directly to conformable arrays:

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int32) :: ix(4) = [0_int32, 1_int32, 2_int32, 3_int32]
integer(int32) :: iy(4) = [0_int32, 0_int32, 1_int32, 1_int32]
integer(int64) :: codes(4)
integer(int32) :: di(4), dj(4)

! Encode all 4 points at once
codes = morton2D(i=ix, j=iy)
print *, codes   ! 0 1 4 5

! Decode all 4 codes at once
call demorton2D(code=codes, i=di, j=dj)
print *, di      ! 0 1 2 3
print *, dj      ! 0 0 1 1

Optional b Parameter

All four procedures accept an optional b argument (integer(int16)) that limits encoding to b significant bits per axis. Allowed values: 2, 4, 8, 16, 32.

This is useful when you know your indexes fit within a smaller range and want to keep only the relevant bits:

use, intrinsic :: iso_fortran_env, only : int32, int64
use mortif

integer(int64) :: code

! Encode using only the lowest 8 bits of each index
code = morton3D(i=5_int32, j=3_int32, k=1_int32, b=8_int16)

3D bit limit

For 3D encoding the Morton code fits in 64 bits only if each axis index has at most 21 significant bits (values up to 2 097 151). Exceeding this limit produces a silently incorrect result — no runtime error is raised.

2D bit limit

For 2D encoding each axis may have at most 32 significant bits (values up to 2³² − 1).