olyhedron A Unit B 
Figure 4. Typical spatial relationships with communication unit 
and the building 
3.2.1.1 Polyhedron Decomposition 
According to our knowledge, the relationship between a point 
and a polyhedron must be decomposed to relationships between 
point and triangles. This accelerates the operation that evaluate 
whether a point belongs to the area surrounded by a collection 
of triangles. The detail implementation of this decomposition is 
demonstrated in figure 5 (The polyhedron A in figure 4 is 
decomposed). After this process, we use a method named as 
“point in triangle normal space evaluation” to solve the point- 
inner-space evaluation problem. 
A 
A a 
237 
Triangle collection of polyhedron A Unit B 
  
  
Figure 5.  Polyhedron-inner side evaluation decomposition 
process 
3212 Point Triangle Relationship Analysis 
As is shown in figure 6, this method uses a reference point P, 
on triangle plane belonged to the polyhedron, the normal vector 
Np pointing to the polyhedron inside and the vector ¥¢ from the 
reference point P, to the considering point P.. If the vector 
product of My and Vg is positive, then the point F; is on the 
normal direction side of the triangle consisting the polyhedron; 
otherwise the point P; on the anti-normal direction side. After 
summing up all the computational result between the point and 
all the triangles of the polyhedron, we can tell whether the 
pointP;is topologically inside the polyhedron. 
     
     
   
     
    
  
  
  
  
     
  
    
  
      
    
  
    
   
     
    
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
  
Point in triangle normal direction space Point in triangle anti-normal direction space 
Figure 6. Polyhedron-inner side evaluation for point by using 
the triangle plane 
3.2.2 Evaluation of Accessible Feature 
After the evaluation of the relationship between the basic unit 
and the polyhedron, there are two possible results. First possible 
result is that the basic unit does not belong to the inner space of 
the polyhedron, thus there is no need to analyse whether the 
unit is accessible. Second result is that the unit belongs to the 
inner space, and this means we need to evaluate the 
accessibility of the unit. 
Figure 7 shows that the accessibility evaluation progress could 
be classified into two parts. The first part is named as the 
evaluation between the unit and bottom triangles of the 
polyhedron. Since the accessibility of a position requires this 
position must be placed on the building floor, the intersection 
between the unit and bottom triangles of the polyhedron need to 
be checked. The second part is mainly about the slope 
evaluation between the plane of unit-intersecting triangle and 
the horizontal plane. This operation insures that the accessible 
position must be placed on the slope that has an acceptable 
angle of inclination, and eliminate any vertical movement 
beyond the moving ability of normal people. 
  
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