P2-5-2 
3. CONCEPT OF POLYGON SHIFT 
Polygon shift can be defined as shift of polygons to a 
certain direction termed as “shift direction’ with a 
small distance termed “shift amount” as shown in Fig. 
2(a). Then, the original polygon A and the shifted 
polygon A ’ will create three parts by Boolean 
operations: 
(1) Overlapping area; A and A ' 
(2) A part of A not included in A ’: A -(A AND A ') 
(3) A part of A ' not included in A : A '-(A AND A ’) 
In this study, the area defined by (2) is called “Fore" 
while the area defined by (3) “Aft”. Visible parts of 
vertical walls will be identified by “Fore”, while 
invisible parts will be detected by “Aft”. 
3D view of a building with the roof and visible vertical 
walls can be generated by a sequence of polygon shifts 
with the shift repetition according to the height of the 
building, as shown in Fig. 2(b). In case of column type 
buildings limited in this study, the roof represents the 
finally shifted polygon, while the visible walls 
represents a series of “Fore” parts. The invisible walls 
are automatically hidden in “Aft” parts in the shift 
process. 
4. DETERMINATION OF HIDING AND HIDDEN 
PARTS 
There would be no problem for determination of 
hiding and hidden parts, if 3D view images of 
buildings were displayed on a monitor sequentially in 
the order from far to close because far buildings would 
be automatically hidden. This is true only if these 
buildings have a convex shape as the simple case 
shown in Fig. 3(a). However, if there are concave 
shape buildings as shown in Fig. 3(b), it is complicated 
to judge which building is farther or closer than the 
other. Therefore, there should be a theoretical criterion 
to judge the visibility. 
In this study, a distance from a visible point on a roof 
or wall to the base termed “Depth Distance” (see Fig. 
4(a)) is used for the determination of hiding and 
hidden parts. If there are overlapping areas among 
“Fore” sides, see example shown in Fig. 4(b) a point 
with a longer “Depth Distance” hides another point 
with a smaller “Depth Distance”. In the shown 
example, Point A on the roof hides Point B on the 
wall, which is originally assigned “Fore” visible side 
according to the definition given in section 3. 
In case of two or more buildings as shown in Fig. 5, 
the depth distance of a point on overlapping polygons 
between roof and wall, between walls (see Fig. 5(a)) 
and between roofs (see Fig. 5(b)) allows the 
determination of hiding and hidden parts. Points R M of 
building No.l and W M of building No.2 in Fig. 5(a) 
and R a of building No.l in Fig. 5(b) hide points W 2I 
and W 22 in Fig. 5(a) and R b in Fig. 5(b) respectively of 
building No.2. 
5. COMPARISON BETWEEN THE POLYGON 
SHIFT METHOD AND EXISTING 
ALGORITHMS. 
5.1 Algorithm of the polygon shift method 
The polygon shift method is composed of the 
following steps: 
Step 1 : input building plan of buildings in vector 
mode and the height. 
Step 2 : set up a raster of the study area with the 
resolution as specified by users and convert the input 
building plans into raster based polygons. 
Step 3 : input shift direction as a function of the 
looking angle and the depression angle, a unit of shift 
amount as a function of the building height. 
Step 4 : clear the buffer of the depth distance of each 
polygon as an initial value. 
Step 5 : shift each polygon with a pitch of the unit 
shift amount to the given shift direction and add I to 
each shifted polygon as the depth distance. 
Step 6 : compare the value of the former buffer with 
the one of the shifted buffer at each pixel of the 
polygons and select the longer depth distance. 
Step 7 : repeat step 5 and step 6 until all polygons are 
shifted with respect to the shift amount given as a 
function of the height. 
Step 8 : output the depth distance of all pixels. 
Step 9 : assign grayscale or color in consideration of 
the shading and the shadow effect. 
In case of a convex shaped building as shown in 
Figure 6(a), the polygon shift method results in a 
visible roof as the shifted polygon and visible walls as 
the repeated “Fore” parts as defined in section 3. All 
“Fore” parts are visible in case of a convex shaped 
building, while some of “Fore” parts are invisible in 
case of a concave shaped building as shown in Fig