Natural Perspective

Francesc Casanellas

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8 pages

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Abstract

The first studies on perspective were made in Europe surely by Brunelleschi, and Alberti was the first to write them. Before them, the Arabic author Alhazen had already written “Perspective”, showing that the light arrives at the eye in conical rays. Classical perspective theory was rapidly used by the artists from the 15thXV century to nowadays. But the strict geometry and straight lines of cavalier perspective has always been in contradiction with the intuition of many painters which curved the building edges and the horizon according to what they viewed. In this paper, we intend to demonstrate that the intuition of these artists was based on a more accurate perspective theory than the classical one. That, if it is true that classical perspective gives a quite good representation of vision for small angles, the errors increase when the vision angle widens.

Figures (11)

According to elemental geometry, the plane projection, according to cavalier perspective, keeps the size proportion, so A” = B”. But over the spherical plane, the more distant object B gives a smaller image B’ than the nearest object A. The exact ratio between the images and the objects will be seen later.

Fig. 3 shows two objects A and B of the same size, placed at the distances a and b from the optical centre, in the case of a spherical projection. The images A’, B’ are proportional to the vision angles. As A = B and the arcs are proportional to the radius it is deducted that A’/B’ = b/a. This is the classical law of 1 image size being at the i inverse proportion of the distances. A 4 >, oN fst - 1 41 4° | tae . TNs A 9 The size of an image according to the distance.

The observer in “A” sees the wall “m” of the previous example from the same angle as the drawing of the wall m’. So the ends of the wall will be seen smaller than the centre of the wall and with the same reduction. But the observer in “B” will see the ends of the drawn wall with much less height reduction, because his viewing angle to the drawing m’ is much smaller than his viewing angle to the real wall “m Of course, we know from experience that objects do not change dimension by rotating our eyes or our head. So, what does cavalier perspective represent? Is it a fake, are all people who have been using it for centuries wrong? No. In fact it is a real representation of what we see if we look at an image from the same angle as the object was projected (fig. 7).

So cavalier perspective does not intend to represent what we see in a particular viewing position Nor can it give us any hint of the distance from which the object was seen. It is a convenient bu somewhat abstract representation of reality. It is what we see if the distance to the object tends tc infinite (viewing angle tending to 0). So the representation it gives, not depending on the viewer distance, is most appropriate for architects, engineers, etc. Rit what dn we da if ushat ic intended 1c ta renrecent what we cee ata narmal dictancre and inder « In the figure 8, A and B are columns which are projected to the plane in A' and B'. B' which is the projection of the farthest column, is bigger than A’.

It is interesting to look at the result of (1) for several angles:

LUIS THUILIIe UMildasees We see. ilidt Is UIL LedsUL WY UU HUL See LHe UMN spol OL LUG Letitia, To start with, we come back to the long wall of fig. 5 and we will try to draw it as we see it, or better said, following the mathematical rules of vision. We do not want the drawing to be very large, just the size to be seen under a reasonable angle (where practically there is no natural perspective distortion), as the width of this sheet. The ends must be seen smaller than the centre as they are further away. The centre zone has to be quite flat, without any disrupting point as in w’ in fig. 9, as this would be contrary to our daily experience. So it can be deduced that the long wall should be represented approximately as w’’. Just applying the law of size inversely proportional to distance:

Applying equation (1) to a straight line, it results that the ratio h/ho is constant and independent of the distance to the centre of vision. But the absolute value ho — h, or the curvature, increases with the distance from the centre (fig 14) The simple mathematical rules of natural perspective were found in an attempt to explain why dimensions do not change when the angle of vision changes. The theory of relativity first called by Einstein the “invariance” theory, was elaborated to explain why dimensions of objects do not change in spite of their relative velocity from the observer. For the low speeds of everyday life, the equations fend to the classical Newtonian ones. For normal viewing angles, the natural perspective equations give results similar to the classical perspective. i Sy a a i ee a, ee ee eS a eT ek i Se ee ee, ee eee Sn, ee 5 ee ee emery ie iy eee

Fig. 17. Canaletto Many painters had followed intuitively the rules of natural perspective and painted buildings streets and so on with the edges slightly curved in the right sense. Paintings following strictly th classical perspective, as those of Canaletto, give an impression of artificiality. An interesting exampl is the painting of fig. 17, where Canaletto has drawn the perspective lines with a ruler, according t cavalier perspective and probably against his intuition. But the old columns, according to Greel practice and natural perspective intuition, have the top thinner than the bottom: a striking example o: 1000 vears of obscurantism and loss of most of Greek knowledge.

Fig. 18. Pisarro, Bvd. Montmartre Painters who have followed their instincts more than learned rules, curved lines in the “correct” direction. The same painters may have used straight lines when the viewing angle was small, all according to natural perspective. It is not easy to find examples: objects with straight lines and viewec form a very wide angle are rare in paintings. Furthermore artists learned cavalier perspective at schoo and tried to represent reality accordingly. But impressionists tried to follow their instincts. They avoided straight lines and curvature is normally in the “correct” sense, as fig. 18 shows.