Perspective in Mathematics and Art

A physical model was built, according to Figs. 11, 12, 13 and 14. The square grid is 12cm by 12cm, with each of the squares measuring 1cm by 1cm. The cardboard peephole measured 12cm by 15cm, with the central vanishing point 10cm above the base. We pick the viewing distance to be 26.3cm. The top of the trapezium is then 3.5 cm from the bottom.

Fig. 15 – A physical model of Alberti’s vs. the distance point construction.

The model is designed such that two cardboard peepholes can be mounted simultaneously to show the relation between Alberti’s and the distance point construction.

Fig. 16 – Model with two cardboard peepholes.

Placing the eye at the correct viewing point, 10 cm above the base, will show the square grid in the correct perspective. This is shown in Fig. 17.

Fig. 17 – The square grid viewed in perspective.

This idea of investigating perspective is at least a few hundred years old. In fact, Albrecht Durer (1471 – 1528), too, substituted sight lines with string when he was constructing the image of a lute in perspective. This is shown in Fig. 18.

Fig. 18 – The use of strings to construct a perspective image of a lute. Taken from Albrecht Durer’s “Treatise on measurement with compasses and straightedge (Underweysung der Messung mit dem Zirkel und Richtscheyt, Nuremberg, 1525).

In Fig. 18, the pointer held by the man on the left marks where the light ray ends. The man on the right marks the position where the ray intersects the picture plane by adjusting strings stretched across the rectangular wooden frame fixed vertically to the table. This is seen more clearly in Fig. 19. Once the crossing strings are in the appropriate position, the string LNGH is moved away and the panel with the drawing paper is swung into the picture plane so that the new point can be marked on it. In this way, all the necessary coordinate points for drawing the lute in perspective can be obtained.

Fig. 19 – The instrument Durer used for drawing a lute.