4 
The plane transformations for the study of the 
deformations 
In the study of the view, the planimetric part was 
separated from the altimetric part, so that the information 
regarding the elevations were studied and evaluated 
separately. Even considering the view as a perspective 
construction based on geometric data surveyed directly or 
extracted from previous charts, the data on the elevations 
would result in not being easily worked with together with 
the planimetric data. 
Considering, rather, that the attachment of the buildings to 
the ground, the canal side walkways and the canals 
themselves (places where the reference points have been 
positioned) lie practically on the same plane, the view of 
de’ Barbari, relative to this plane, can be considered as 
the transformation of a planimetry of the city. Or rather, it 
is possible to obtain a view which corresponds 
geometrically to the city in the 1500’s, taking an 
orthogonal projection of the city and modifying it according 
to a certain rule; that rule is a plane transformation. 
The main problem was to understand what type of 
transformation it was better to use. The plane 
transformations can be divided into two large families: the 
global ones and the local ones. The first ones apply the 
same rule of modification to the entire chart, completing a 
transformation on the basis of different parameters. In 
other words, the first group of transformations 
corresponds to a rigid deformation while the second to an 
elastic deformation. 
In the first case, on the basis of control points 
(approximately 120 have been identified throughout the 
city), those parameters have been estimated to the least 
squares which best are adapted to the points used. The 
study of the dimension and the distribution of the residuals 
of points is fundamental, as these allow for the 
identification of the zones of the map with the greatest 
deformations. Among the various applicable and 
presented transformations (for example, conform, 
polynomial, affine) the projective transformation with eight 
parameters has been chosen. This transformation is equal 
to the central projection of one plane on another, that is, in 
our case, to the placing in prospective of a planimetry. 
From this application, it was possible to evaluate the 
shifting of the view of Jacopo de’ Barbari from a rigorous 
central projection. 
Therefore, we proceeded with the inverse operation: 
applying a projective transformation to the prospective 
view in order to obtain the hypothetical plan of Venice in 
1500. Even in this case, the result obtained emphasised 
the great differences between the historical chart and the 
current map. 
In the case of the local transformations, rather, each point 
of the map is transformed according not to general fixed 
parameters, but to variable parameters which are 
calculated on the basis of nearby control points. To this 
second type, the transformations for finite elements 
belong as well as the local transformations based on 
points or lines (usually called point based warping and 
feature based warping). In plain words, the local 
transformation allows to transfer the geometry of the 
reference chart to the one to be transformed. 
Superimposing a regular grid to the photoplane of Venice 
and mapping it onto the view through a transformation to 
finite elements, a representation of the current city of 
Venice is obtained with the geometric characteristics of 
the 1500’s produced work. From the comparison with the 
correct prospective of the same photoplane (previously 
obtained from the projective transformation) the 
modifications by the author can then be seen. 
In an analogous way, it was possible to reconstruct the 
planimetric image of Venice from the 1500’s, or rather, by 
mapping the view onto the photoplane. 
From the combined application of the transformation to 
finite elements and of the projective transformation, the 
elaborations have been realised which have led to modify 
the view, returning it, for the planimetric part, to a correct 
central projection. 
fig 7 Finite elements transformation: applying this transformation, the new image of the photoplane obtained has the same geometry of 
the de' Barbari one. The irregular deformation of the grid underlines that the view is not a correct perspective. 
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