1SPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
Rule 3:The death errors £ ¡andE , off and tj, respectively, are 
different(let£ ¡<e j be the case, i.e., ti refined before tj, with tj 
possibliy unrefined), and edge e is unfined in the direct 
expansion of tj; 
In fig.3, ruled applies for triangles t 5 -t 6 , rule 2 applies for 
triangles t 0 -ti, and for t r t 2 ; rule 3 applies for t 3 -t 4 , and for t4-t 5 . 
Refinement 
Fig 2 DAG representing of Fig. 1 
3.2. Constructive mutiresolution model based on 
Delaunay rule [Berg and Dobrindt, 95] 
The most simple method of constructive multiresolution terrain 
model is that the triangles are refined by adding new data 
points inside them and retriangulating each triangle with its 
new interior points. Thus each triangle is replaced by a number 
of smaller triangles. This process is repeated until all ata points 
have been added, or some presion criterion is met. Such a 
hierarchy can be modeled as a tree. The nodes in this tree 
correspond to the triangles in the hierarchy, and there is an arc 
from the node corresponding to a triangle t to the node 
corresponding to a triangle t’ if the triangles belong to 
consecutive levels and t’ is contained in t. There is also an 
extra root node which is connected to all triangles of the first 
level. Figure 1 shows an example of a hierarchy and the 
corresponding tree. For tree-like hierarchs it is quite easy to 
combine different levels into one representation (see Section 
Using appropriate data structure, we can easily combine 
different mutiresolution into single representation. But the 
consistency of HTIN is still core problem in multiresolution 
terrain model. Above three rules is only basic rule. For a 
general applications, it is very difficulty to apply above three 
rules. We must look for the methods of how to solve this 
problem. 
incremental refinement/simplification of a triangulation. 
Fig.3 An example of HTIN 
Built on a sequence of four tolerance values for the 
approximation error 
3). Unfortunately, they have a serious initial triangulation 
remain present at more detailed levels. This effect is already 
apparent in the two-level hierarchy of Figure 1. Skinny 
triangles can cause robustness and aliasing problems. 
Based on Delaunay rule, for inserting points retriangulate all 
points applying tiie Delaunay rules can solve above problems. 
For deleting points, we only delete interior no-adjacent points. 
The Delaunay triangulation has the nice property that it 
maximizes the minimum angle of the triangles. Thus 
robustness and aliasing problems are reduced. The 
hierarchies of this category can be represented by a directed 
acyclic graph: the nodes in this graph again correspond to the 
triangles in the hierarchy, and there is an arc from the node 
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