Drap, Pierre 
  
4.3 Calculation of intrinsic parameters 
In the case of an arch we have five parameters to compute : the 
cylinder axis, the intrados radius, the extrados radius and the 
two circle centers (who should be on each top of the cylinder 
axis); After the determination of the blocks, the points are easily 
located by geometric criteria. 
Here we need to develop the architectural model : by 
construction the intrados and extrados centers have to be at the 
same location. 
  
4.4 Extrude of blocs measured in theintrados 
Once we have computed all the geometric arch parameters we 
can extrude the bloc which has been measured only by their 
face visible in the intrados. (internal part of the arch). This 
extrude is radial from the main axis of the intrados axis 
cylinder. Each point has its own extrude vector define by the 
projection of the point onto the main axis and the point itself. Figure 4. Arc intrinsic parameters scheme. 
Each computed stone of the arch is represented as an arc of a 
rectangular torus centered on the intrados main axis. 
  
Extradoscircle in Y min 
5 RESULTS: A NEW INTERPRETATION OF THE BUILDING 
5.1 Geometric primitive least square optimization 
We made two kinds of least square optimization in order to make the numerical results explicit. We present in figure 5, 
the two sets of points measured on the intrados and extrados front stones. On the figure, each point at the angle are on 
the same stone. On figure 6 we can see an intrados representation on the developed cylinder. The cylinder is the least 
square cylinder from all the indrados measured points. On the X axis we have the curvilinear abscissa of each points, 
the Y component have not changed and the Z one is the residual distance from the optimized cylinder. 
  
  
  
Figure 5. Intrados and extrados Figure 6. Intrados cylinder least square optimization with 20x anamorphosis 
circle least square optimization of residuals. : iso-deformation curve (each 5 cm) from the computed 
with 20x anamorphosis of residuals. Cylinder. In green out of the cylinder, in black inside the cylinder. 
This type of approach is of considerable interest, as it permits to verify the exactitude of the stone cutting and the 
deformations undergone by the built structure. According to the architectural properties of the arch, as defined by the 
comparison with other arches of the building itself and other contemporary examples of the same technical quality, the 
arch should ideally be a precise hemicycle. The magnified irregularities (20x anamorphosis of residuals) tend to 
demonstrate a partly close approximation of the intrados to this theoretical curve, and confirm the high standard of the 
workmanship of the edifice. The comparatively slight deformations in the lower parts and at the top might be due to 
movements subsequent to the settling down of the masonry after the removal of the sustaining auxiliary wooden 
structure, and the increase of the weight of the surrounding masonry, the arch being - according to archaeological 
evidence - normally built before. 
  
192 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
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