Francesc Casanellas C. Sant Ramon, 5 08591 Aiguafreda - Spain +34 677 00 00 00 francesc@casanellas.com - www.casanellas.com NATURAL PERSPECTIVE Introduction Te first studies on perspective were made in Europe surely by Brunellesci, and Alberti was te first to write tem. Before tem, te Arabic autor Alazen ad already written Perspective, sowing tat ligt arrives at te eye in conical rays. Classical perspective teory was rapidly used by artists from te 15t century to nowadays. But te strict geometry and straigt lines of cavalier perspective as always been in contradiction wit te intuition of many painters wo curved te building edges and te orizon according to wat tey viewed. In tis paper, we intend to demonstrate tat te intuition of tese artists was based on a more accurate perspective teory tan te classical one. Tat, if it is true tat classical perspective gives a quite good representation of vision for small angles, te errors increase wen te vision angle widens. First paradox of classical perspective Figure 1 sows te projection of two objects A and B, of te same size and on te same plane, over a flat and a sperical screen. Te projection over a flat surface is made according to te cavalier perspective or te result of a potograpy made wit an ortoscopic optic or a pin ole camera. Te projection over a sperical surface is similar to te one on te retina. In fact, in te case of cavalier perspective, te projection plane would be placed between te centre of vision O and te object, but apart from te image inversion, te proportions remain te same. According to elemental geometry, te plane projection, according to cavalier perspective, keeps te size proportion, so A = B. But over te sperical plane, te more distant object B gives a smaller image B tan te nearest object A. Te exact ratio between te images and te objects will be seen later. Wat appens if te screen rotates around te optical centre O, as fig. sows? In te case of te plane projection te images cange absolutely: now A is bigger tan B. In te sperical projection, images keep teir size. If our vision was made according to te cavalier perspective, objects would cange size just by rotating our ead. Fortunately, we do not ave ortoscopic crystalline lenses, or a normal crystalline lens wit a flat retina: we would always be seasick! Natural perspective 1 / 1
Te size of an image according to te distance. Fig. 3 sows two objects A and B of te same size, placed at te distances a and b from te optical centre, in te case of a sperical projection. Te images A, B are proportional to te vision angles. As A = B and te arcs are proportional to te radius it is deducted tat A /B = b/a. Tis is te classical law of image size being at te inverse proportion of te distances. An object C of te same size as A and at te same distance gives te same image C = A. Fig. 4 sows te same objects but in te case of cavalier perspective. If te angles are small, tanα sinα α and te previous law is approximately kept. But if te object moves away from te perpendicular to te plane, keeping te same distance, te image increases C > A ). If our eye beaved according to classical perspective, objects would cange size by rotating te ead, as in te experiment wit a camera equipped wit an ortoscopic lens mentioned before. Note: A simple not corrected camera lens wit a diapragm in te front beaves approximately like te uman eye ( barrel distortion, peraps better called natural distortion). A pin ole camera beaves like one wit an ortoscopic lens (fig. 1, projections A, B ). Te long wall paradox. Take te case of an observer in A facing a very long vertical wall (fig. 5). Te distance to te ends d, d3, are longer tan te distance d1 from A to te wall. According to te law of te size being inversely proportional to te distance, te observer would see te eigt of te wall muc smaller tan te eigt of te centre of te wall: / = d1/d. According to classical perspective we would represent te wall as in fig. 6a and if te observer moves is ead towards te left or te rigt, as in fig. 6b and 6c. Natural perspective /
Of course, we know from experience tat objects do not cange dimension by rotating our eyes or our ead. So, wat does cavalier perspective represent? Is it a fake, are all people wo ave been using it for centuries wrong? No. In fact it is a real representation of wat we see if we look at an image from te same angle as te object was projected (fig. 7). Te observer in A sees te wall m of te previous example from te same angle as te drawing of te wall m. So te ends of te wall will be seen smaller tan te centre of te wall and wit te same reduction. But te observer in B will see te ends of te drawn wall wit muc less eigt reduction, because is viewing angle to te drawing m is muc smaller tan is viewing angle to te real wall m. Te column paradox Leonardo da Vinci was te first to remark a still worse paradox of te cavalier perspective, an example were te object wic is farter from te observer sould be represented bigger tan te one near te observer. In te figure 8, A and B are columns wic are projected to te plane in A' and B'. B' wic is te projection of te fartest column, is bigger tan A'. So cavalier perspective does not intend to represent wat we see in a particular viewing position. Nor can it give us any int of te distance from wic te object was seen. It is a convenient but somewat abstract representation of reality. It is wat we see if te distance to te object tends to infinite (viewing angle tending to 0). So te representation it gives, not depending on te viewer distance, is most appropriate for arcitects, engineers, etc. But wat do we do if wat is intended is to represent wat we see at a normal distance and under a wide viewing angle? Tis is a real problem for painters and potograpers. Natural perspective 3 / 3
Natural perspective. So our aim is to find ow to represent wat we see over a plane following te matematical rules of vision: size inversely proportional to te distance and invariance of size wit viewing angle, inexorable rules tat, as we ave seen, classic perspective does not follow. In cavalier perspective, te viewing angle of te drawing as to be te same as te viewing angle of reality if we want to see te same. In natural perspective, te viewing angle of te drawing tends to 0. Of course, in many cases te viewing angle is small and te difference between cavalier perspective and real vision is also small. Our eyes cover about 170º of wic about 60º (±30º) correspond to binocular vision. But only a small angle of about 1º is covered by te fovea were te vision is clear. Te rest is blurred. But our eyes move constantly and te image is reconstructed by te brain. It corrects and integrates te multiple images we see. Tat is te reason we do not see te blind spot of te retina. To start wit, we come back to te long wall of fig. 5 and we will try to draw it as we see it, or better said, following te matematical rules of vision. We do not want te drawing to be very large, just te size to be seen under a reasonable angle (were practically tere is no natural perspective distortion), as te widt of tis seet. Te ends must be seen smaller tan te centre as tey are furter away. Te centre zone as to be quite flat, witout any disrupting point as in w in fig. 9, as tis would be contrary to our daily experience. So it can be deduced tat te long wall sould be represented approximately as w. If we try to calculate rigorously te law according to wic tese curves are elaborated, te solution is straigtforward: Just applying te law of size inversely proportional to distance: 0 = d d 0 = cosα So te function we look for is: = 0 cosα It is interesting to look at te result of (1) for several angles: α /0 10º 0,985 0º 0,94 Natural perspective 4 / 4 (1)
30º 0,866 45º 0,707 Unlike cavalier perspective tis representation carries te information of te angle of vision, and if we know te size of te object, we can deduce te viewing distance. Fig. 10 sows te representation of te previous wall under 10º (±60º) and 60 º (±30 º) vision angles. Note tat even for a quite wide angle as (±30º) te curvature is still small. Of course, te law is te same for te vertical or any oter direction. Fig 11 sows te case of a reference plane inclined wit respect to te observer. It is clear tat just by prolonging te plane and drawing te perpendicular to it from te observer, we may apply te rules deducted from fig. 9, were te caracters refer to te vertical eigts (not drawn): 1 = 0 cosθ 1, = 0 cosθ 1 cosθ = cosθ 1 () wen θ 1 =0 it is reduced to equation (1). We sall skip te demonstration tat te reduction coefficient given te plane angle β and te vision angle α is: 1 sinα = cosα tan β wic if β = 90º becomes (1). Te result is quite similar to te cavalier perspective, except tat edges are sligtly curved (see comparison in fig. 1). Natural perspective 5 / 5
Experimenting wit natural perspective. Place yourself in front of a long wall, a long self, window, corridor, etc., or a vertical building. Use a pencil or a rule in vertical position, wit your arm fully extended and measure vertical distances (orizontal ones if you look at a vertical object), in te way painters use to do. You can ceck te rules of natural perspective comparing dimensions in te centre, ends and intermediate points (Fig. 13). Curvature of an isolated straigt line Applying equation (1) to a straigt line, it results tat te ratio / 0 is constant and independent of te distance to te centre of vision. But te absolute value 0, or te curvature, increases wit te distance from te centre (fig 14) Relativity (or invariance) and natural perspective. Te simple matematical rules of natural perspective were found in an attempt to explain wy dimensions do not cange wen te angle of vision canges. Te teory of relativity first called by Einstein te invariance teory, was elaborated to explain wy dimensions of objects do not cange in spite of teir relative velocity from te observer. For te low speeds of everyday life, te equations tend to te classical Newtonian ones. For normal viewing angles, te natural perspective equations give results similar to te classical perspective. If te observer in fig. 13 takes a straigt ruler and puts it orizontally to ceck te curvature of te lines, te edge of te ruler will also be seen curved (because viewed under te same angle) and it will matc te line someting like mental experiments in te curved space. Tis is sown ere as a curiosity, not to be taken too seriously, of course. Perspective and illumination In classical perspective, objects parallel to te projection plane do not cange dimension, regardless of teir distance. Using classical perspective, te square A te reticule of fig. 15 as te same dimensions as te square B wic is far away (fig. 15, rigt side). Natural perspective 6 / 6
But less ligt reaces te eye or te camera from te distant square B tan from square A. Te intensity of ligt received from A is related to te one from B by E A d = A cosα EB d B so a projection over a plane in classical perspective gives a centre of te screen brigter tan te borders, and tis effect is easily observed using a pin ole camera. In natural perspective, te surface of B is already corrected by te factor ' S A d A = cosα (were S means te represented areas) and as surface decreases in te same ' S B d B proportion as illumination, te surface brigtness is te same and it is kept in all te plane. Natural perspective in arts Te edges of te Partenon are not straigt lines, but ave subtle curves (see fig. 16). Columns are tinner at te top. It as been said tat tis was done to correct supposed visual aberrations. But we can give a more plausible explanation. Greek arcitects knew intuitively te rules of natural perspective and te orizontal curves make te Partenon look wider tan it is, curved columns look taller tan tey are. Many painters ad followed intuitively te rules of natural perspective and painted buildings, streets and so on wit te edges sligtly curved in te rigt sense. Paintings following strictly te classical perspective, as tose of Canaletto, give an impression of artificiality. An interesting example is te painting of fig. 17, were Canaletto as drawn te perspective lines wit a ruler, according to cavalier perspective and probably against is intuition. But te old columns, according to Greek practice and natural perspective intuition, ave te top tinner tan te bottom: a striking example of 1000 years of obscurantism and loss of most of Greek knowledge. Fig. 16. Te Partenon. Potograpy by te autor. Fig. 17. Canaletto Natural perspective 7 / 7
Painters wo ave followed teir instincts more tan learned rules, curved lines in te correct direction. Te same painters may ave used straigt lines wen te viewing angle was small, all according to natural perspective. It is not easy to find examples: objects wit straigt lines and viewed form a very wide angle are rare in paintings. Furtermore artists learned cavalier perspective at scool and tried to represent reality accordingly. But impressionists tried to follow teir instincts. Tey avoided straigt lines and curvature is normally in te correct sense, as fig. 18 sows. Fig. 18. Pisarro, Bvd. Montmartre Summary: Cavalier perspective gives a representation tat is viewed as te real object only if it is viewed from te same angle as it was drawn. Natural perspective gives an image tat it is viewed as te real object if te angle is small enoug as distortion being not appreciable. Natural perspective 8 / 8