ISPRS Workshop on Service and Application of Spatial Data Infrastructure, XXXVI(4/W6), Oct. 14-16, Hangzhou, China 
Figure 5. The correlation between quaternary Morton code and 
row/column number 
The neighbor finding of the diamond is much more easier than 
that of the triangle, this is determined by the properties of the 
diamond mesh. Because the triangle cell in the triangle mesh 
has not uniform orientation and the adjacent neighbor of it 
varies according to its location in the mesh. Neighbor finding is 
different for either different orientated triangle or different 
located triangle in the mesh. Diamond mesh has several 
properties such as radial symmetry, translation congruence and 
uniform orientation, unlike triangles, thus makes some spatial 
operations such as neighbor-finding much more easily. 
There are three cases in the neighbor-finding of the diamond: 
the diamond located in the inside of the base diamond 
(quadrant), the diamond located in the boundary of the base 
diamond and the diamond located in the comer of the base 
diamond. 
Figure 6. Adjacent neighbors of the diamond located on the 
comer of the base diamond 
Algorithm for determining the edge adjacent neighbors is as 
follows (the algorithm can be extended to computed the address 
of the vertex adjacent neighbors as well): 
EdgeDiamondAdjacent (QDcode DM, Direction Dir) 
{ 
D<— PrefixD(DM); // extract the started digit 
M<—DelePrefixD(DM); // delete the start digit 
(i,j) <—F-l(M) // conversion from Morton code to 
row/column numner 
switch(Dir) 
{ 
EN: // the northeast neighbor 
if(i<I) i++; // located in same base diamond 
else // located in different base diamond 
{if(d=3) d=0; else d++ 
k=j;j=0;i=I-k;} break; 
ES: // the southeast neighbor 
f(i>0) i- -; // located in same base diamond 
else // located in different base diamond 
{if(d=0) d=3; else d— 
k=j ;j=I;i=I-k;} break; 
WN: // the northwest neighbor 
if(j>o)j-; // located in same base diamond 
else // located in different base diamond 
(if(d=0) d=3; else d- 
k=i;i=I;j=I-k;} break; 
WS: // the southwest neighbor 
if(j<i)j++; // located in same base diamond 
else // located in different base diamond 
(if(d=3) d=0; else d++ 
k=i;i=0;j=I-k;} break; 
} 
M^F(ij); // conversion from row/column number to 
Morton code 
DM^AppendToD(M,D); // append the quadcode of the 
base diamond 
Retum(DM); 
} 
6. NEIGHBOR FINDING OF TRIANGLE 
Diamond tessellations are compatible with tessellations of 
triangles in nature. The diamond can be regarded as the merging 
of two triangles. These two triangles made up the diamond can 
be distinguished through appending a digit 0 or 1 to the end of 
code of the diamond. As showed in figure 7, the code of triangle 
ended with“0” indicates that it has a south edge-adjacent 
neighbor, and the code of triangle ended with “1” indicates that 
it has a north edge-adjacent neighbor within the same diamond. 
The suffixes of the code of triangle indicate the triangle is either 
up or down triangle (figure 7). The code of triangle traces a 
zigzag course through the triangles at any given level (figure 7). 
Based on the algorithm of neighbor finding of the diamond, we 
develop an algorithm for determining the edge adjacent 
neighbors of triangle. The same strategy can be used to compute 
the address of the vertex adjacent neighbors as well. 
Figure 7. Labeling and indexing of two triangles made up the 
diamond 
Algorithm for determining the edge adjacent neighbors of 
the triangles is as follows: 
EdgeTriangleAdjacent (QTcode TM, Direction Dir)