International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
The following drawing represents the calculated points and the 
3D polyline describing the best cone based on these points. 
(Figure 4) 
[6] r 3D polyline 
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Figure 4. Theoretical cone drawn from the 
calculated 3D points 
For better 3D visualization, there also appears on this layout one 
of the polylines joining the homologous points between two 
shots. 
Thanks to this vectorial representation of the column, we are 
now able to develop the obtained 3D polyline on a plane, so as 
to survey the hieroglyphic signs on a two-dimensional surface. 
3. EPIGRAPHIC SURVEY OF THE COLUMN 
In order to make a survey as accurate as possible, the column's 
inscriptions will be drawn by means of Bezier curves on the 
developed surface of the shaft, in which the orthorectified 
photographs will be inserted. This two-dimensional drawing 
leads to a direct publication of the hieroglyphic texts. 
3.4 Development of the cone 
As mentioned above, the best cone approximating the points 
can be at an angle. So, the first step of the development is to 
make the previously obtained cone vertical, for the simplicity 
and the accuracy of the development's calculation. 
A change of reference has been carried out so as to move the 
summit of the original cone to the zero point and to put the axis 
in a perfect vertical position. The transition from one 
coordinates’ system to the other is possible through three non- 
coplanar homologous points in each system. These three points 
are chosen at random among the seven points of the previous 
3D polyline. Knowing that the summit must be translated to the 
point (0, 0, 0), the coordinates of the two other needed points 
are easy to deduct. 
A matrix has been calculated to enable the transfer of all the 
cone's points from one system to the other. These object points 
are replaced exactly « on » the theoretical surface of the new 
vertical cone by means of an orthogonal projection. 
The vertical cone can thus be developed, that is to say the points 
of the 3D polyline will be put on a same plane. 
The cone can also be described developing its surface, which 
consists in five coplanar points : 
the summit [1] 
the cone's generatrix going through the intersection 
between (OX) and the circle [LC] (see above) [1-2] 
the arc describing the development of the circle's 
circumference, intersection between [LC] and the cone's 
surface [2-3] 
the generatrix representing the end of the unwinding [3-4] 
the arc describing the development of the circle's 
circumference, intersection between [HC] and the cone's 
surface [4-5] 
(Figure 5) 
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summit and zero point 
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Figure 5. Link between the 3D model and 
the development of the cone 
The points previously projected on the theoretical surface of the 
vertical cone have been located on the development by 
multiplying their (X, Y) coordinates by a rotation matrix. Their 
Z coordinate is the distance between the real and the projected 
points. 
To be able to draw the hieroglyphic inscriptions, it is now 
necessary to orthorectify the original photographs of the column 
and to insert these pictures in the development. 
3.2  Orthorectification of the photographs 
To put the shots in the development, these must be transformed 
into orthophotographs. The epigraphy will thus be surveyed on 
this « front view » of the column. 
   
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