Historical records show that besides Alberti’s
construction, there were other methods for constructing *pavimenti*.
One of them was known as the distance point construction, and was found in
the treatise of Jean Pelerin (1445 – 1522), also known as the Viator.
Entitled *De Artificiali Perspectiva*, it was first published in Toul
in 1505 and later pirated at Nuremberg in 1509. It produces the same
results as Alberti’s construction, but constructs the *pavimenti*
differently.
Fig. 8 –
Choosing the distance point *D*.
As before, the ground line *AB* is divided
equally, and each of these division points are joined to the centric point
*C*. Next, the distance point *D* is chosen. The distance *
CD* is the viewing distance.
Fig. 9 – The
distance point construction.
The line *AD *will intersect all the orthogonals.
These intersection points are used to draw the transversals.
Fig. 10 –
The plan and vertical section corresponding to
the distance point construction.
We will next formulate a geometrical proof for the
distance point construction. Fig. 10 shows the floor plan for the
distance point construction. This is similar to Fig. 7. For Alberti’s
construction, it is the distance from the viewer to the last transversal
which is of great importance. We can picture the line *ER* as the
line *HP* rotated 90° anticlockwise about the point *X*. Hence,
we show that *MP* is equal to *NR*.
For the distance point construction, we can picture
the line *MD* as the line *MP* rotated 90° anticlockwise about
the point *M*. *D* is the distance point. If one was to stand
at *D* instead of *P*, one can easily see that the distances *
MP* and *MD* are equal. It follows that *MD* is also the
viewing distance.
We have shown that *MD* and *NR* are the
correct viewing distances. Hence, *MD* and *NR* must be the
same length. It follows that Alberti’s and the distance point
construction are equivalent.
To obtain a three-dimensional “proof” of the distance
point construction, consider a square tile shown in the diagram below.
Fig. 11 –
Three-dimensional setup of the distance point construction.
Suppose we
have a “cardboard peephole”, which is a piece of cardboard with the
trapezium cut out. Now we wish to position the square grid behind the
cardboard peephole, in such a way that the entire grid can be seen through
the trapezium hole.
Fig.12 –
Viewing the grid through the cardboard peephole.
The viewing distance is *x*. Now imagine that
the cardboard peephole is rotated about its axis line, as shown in Fig.
13.
Fig. 13 –
Notice that the viewing distance *x*
is due to the distance point construction.
Now imagine that the cardboard peephole is displaced
to the right, such that it sits exactly on top of the grid (Fig. 14).
Fig. 14 –
Notice that the viewing distance *x*
is due to Alberti’s construction.
After the
displacement, notice that *x* is now the viewing distance obtained
from Alberti’s construction. Though hardly a rigorous proof, it suggests
a way of obtaining viewing distances from both methods using physical
models, and provides an intuitive “feel” for the distance point
construction. Most importantly, the model shows that Alberti’s and the
distance point construction are equivalent, as they yield the same viewing
distances. |