Distance Point Construction

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 MD 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.