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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004 
WtA = TP/ (TP+TQ+TR) (1) 
WIB = TQ/ (TP+TQ+TR) 
WtC = TR/ (TP+TQ+TR) 
Where 
ABC = Three corner points of the triangle 
T = Any point inside the triangle 
Wt- Weight-age of the point as per the displacement vector 
This method satisfies criteria 1 & 2. This method produced 
good results except near some edges of the triangle. It was 
found that the weight distribution of displacement vectors is not 
uniform on both sides of those edges where triangles are not 
uniformly shaped. 
Hence there was a need to device a transformation, which 
would ensure smooth distribution of weight on both sides of the 
edges. Transformation using weight proportional. to the area of 
triangle formed with two other vertices satisfies the same. 
2.3 Transformation using weight proportional to the area 
of triangle formed with two other vertices 
A 
  
Figure 2. Weight proportional to the area of triangle formed 
with two other vertices 
The weight of displacement vectors are calculated as follows 
WtA = Area of TBC/ Area of ABC (2) 
WtB = Area of TCA/ Area of ABC 
"WIC = Arca of TAB/ Area of ABC 
Where 
ABC = Three corner point of the triangle 
T = Any point inside the triangle 
Wt- Weight-age of the point as per the displacement vector 
This method acts like a transformation based on inverse 
distance method (controlled by displacement vectors at both 
ends) at the edges of triangle. Thus it satisfies the criteria 3 in 
addition to 1 & 2. This method produce good results 
throughout the map except some angular changes especially 
Where vertices are too close to each other. 
  
Fig. 3. Weight distribution based on distance to opposite sides 
on left. Weight distribution based on area of triangle on right. 
2.3.1 Final PAI Transformation Equations: 
Equation 2 was used for controlling transformation inside the 
triangles, which were formed based on Delauney Triangulation. 
The resultant displacement at each point is calculated using 
weight factors as below. 
Displacement at point T = 
WtA * [Displacement Vector at A] 
+ WtB * [Displacement Vector at B] 
+ WtC * [Displacement Vector at C] (3) 
Where 
WtA, WtB, WtC - the weight factor of the displacement 
vectors at A, B, C respectively. 
As the total area is divided into triangles and this transformation 
applied inside each triangle ensures that displacement at corners 
is same as required, in order that absolute accuracy inside each 
triangle is achieved. Better geometry of triangle formed based 
on Delauney Triangulation ensures relative accuracy. It is 
important to identify displacement in such positions which will 
form better geometry for Delauney Triangulation. 
Thus it ensures that at the edges of triangle only two 
displacement vectors (specified at both ends) control the 
displacement, which in turn ensures smooth displacement across 
edges of the triangle. This method also ensures smooth 
transition of weight factors, but still produces angle error at 
places where feature spans across several triangles. Hence 
special auto-correction logic was introduced to ensure geometric 
fidelity is perfectly restored for all the features.