Have You Ever Used Two Picture Planes to Draw a Single Perspective View?

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ABSTRACT ©2019 ISAST What if the PP were a real three-dimensional plane? Could it before; based on my research, it seems not. We wonder, then: considered three-dimensionalattribute itsbeen hasif mine current the reflecting in definitionson deter to is PP the of objective My forms. various in applied and known well is be formed on the PP where such rays cross it. This principle to perspective itsallows interestpointsof object an of all to observer the fromrays visual tracing (a), fore,accordingto tributes a–c suffice to understand how the PP works. There- such as size and shape, are less well understood [2].) The at- sightline intersects the PP perpendicularly. (Other attributes, plane onto which objects are projected; and (c) the observer two-dimensional a is PP The (b) viewed; being object the and observer the between placed plane imaginary an is PP The (a) following: the are definitions current in to referred window [1]. Thethree open fundamental attributesan of thethrough PP most seeing of metaphor the using describes, byobserver, the seen objects todepictAlberti essence,as in the know,ofwe As function prime the The caseofperspective? projection beaparticular on anaxonometricview. Thisraisesaquestion:Couldaxonometric upto100timesitssidelength,ittakes cube recedesfromtheobserver change shapeastheymoveawayfromtheviewer. Forexample,ifa use asecondpictureplane.Thisleadstoconsiderationofhowobjects building onthefirstpictureplanewouldbeimprecise,itmaywiseto building canbeseenabout190maway, wheretheprojectionofsuch demonstrates thattodrawtheinteriorofabuildingfromwhichanother ground leveltoestimatethedistancebetweentwoobjects?Thisarticle draw asingleview—forexample,givenlackofspatialreferencesat to use indualpositions.Whatiftwopictureplanesarenecessary take forgrantedtheuseofasinglepictureplane,disregardingitslikely not beenattemptedbefore.Mostperspectivemethods,afterAlberti, Apparently, theuseoftwopictureplanestodrawasingleviewhas to DrawaSinglePerspectiveView? Have Y l a r e n e G with thisissue. See www.mitpressjournals.org/toc/leon/52/2 forsupplementalfilesassociated Cd. de México, México. Email: [email protected]. University of Mexico, Av. Santa Fe 462, Dep. 401A, Cuajimalpa 05348, Tomás García Salgado (researcher), Faculty of Architecture, The National Autonomous H oly Pi oly á m o T doi:10.1162/LEON_a_01603 ct A e l c i t r u r e Pl e s ou EverUsedTwo Picture Planes G a ne C R A í a S pictureis plane(PP) a l D A G

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real dimensions? Such a plane, in in plane, a Such dimensions? real measure to serve also would objects 3D of appearance the representing to addition in that plane versatile more a be ing at just one vanishing point, or more? If it vanished at vanished it If more? or point, vanishing one just at ing vanish sight your Is are: you wherever yourself, ask this, proveTo one. at but points vanishing three or two at not vanishes vision human of representation geometric as tive point,Here,modular perspective. in by definition,- perspec to observer’s the ofvanishing out meansby solely draw to easy turns be views these of Any PPL. the of center the at in space, the server sightline and vanishing point are always Fordat the Foundation, orientationwhatever the ob - ofthe or station space the on whether cases, both In 1). Fig. (see buildings both between air butnothing with Building, tion New York’s Millennium Hotel from inside the Ford Founda- places—for instance,seeing unexpected when in occur also can space in objects seeing of experience similar A down. ground the and up is sky the place; in is everything Earth, on body.back ownOnce their and sightline their are ences refer spatial only the feet, their at Earth with down upside sketch the shuttle along with the space station, while floating to wants astronaut an if There, objects. orbital both tween - be distance the clueofno down, orupestablishing iswhat ground visible no them, between space of void the station, PPL is necessary. Imagine a space shuttle approaching a space my knowledge, included been intraditional methods. tion might explain why adual position of PP the has not, to measurementsondirectly taken it. limitabe This - technical PP, because its size cannot be established second beforehand a place nor to can impractical be might it contrary, the On PPL. second a placing of idea workable the attributesmake These scale. representational their of regardless blueprints, in drawn and read two-dimensional. being actually despite plane, three-dimensional true a is PPL the words, other In space. in object given any of interest of points all measurethree and drawthe directly can be called the Let us now suppose a case in which the use of a second second a of use the which in case a suppose now us Let In addition, the PPL can be of any size, since all points are perspective plane LEONARDO, Vol. 52,No.2, pp.117–122,2019 ), as coordinates are read on read coordinatesare as modules(m), (PPL). On the PPL one can Modular Perspective Modular modularcoordinates of [3], [3], - -

117 The photo’s visual field appears to be in plain format, not cropped. Even if it were cropped, the image formation would not change [7]. Hence, it can be assumed that the observer’s sightline is at the center of the photo (where their diagonals cross). 2 The height and orientation of the observer’s sightline suggest the photo was taken indoors at the Ford Foundation Building, standing on the 12th floor. Here, on the open gallery’s southwest side, the camera was laid above the handrail. From this position one could peer into the sky between the Millennium towers, as in the photo. 3 To infer the type of camera lens that was used in Fig. 2, we first analyzed the breadth of visual field in the photo. Then we drew up a schematic plan includ- ing visual rays from the observer up to the limits of the steel structure as they appear in the photo. This gives us an angle of ≈ 18º, which corresponds to that Fig. 1. Aerial view from Manhattan’s east side (based on Google Maps info), of a zoom lens of 135 mm focal length (a moderate showing Ford Foundation Building (bottom left) and the Millennium Hotel telephoto). (above right). (© Tomás García Salgado)

two, that would be uncanny; if it vanished at three, then you would have to be from another planet. Consider: When you see a cube vanishing at two vanishing points, or three, those vanishing points belong to the cube, not to your sight. This is understood differently in the well-known methods of two- point and three-point perspective. An artist using, let’s say, a camera lucida (1807) uncon- sciously knows that their vanishing point is at the center of their vision. With a camera like this, old but effective, while one is comfortably seated on 12th floor of the Ford Foundation Building, everything in view can be easily drawn; Frederick Catherwood drew splendid illustrations of Mayan monuments (1842), stone by stone, with his camera Fig. 2. Interior view of Ford Foundation Building. lucida [4]. (Photo © David W. Dunlap/The New York Times)

Ford Foundation Building, New York At the Architecture in Perspective 29 meeting (2014) [5], I 4 As in the photo, the transverse steel bars of the sky- saw applied the addition of hand-drawn strokes to digital light clearly vanish toward a distant point (to the images, to give them a handmade touch. Thinking in reverse, right); this means that the observer’s sightline is not digital layers can also be added to handmade drawings. My perpendicular to them. To estimate the bars’ fore- assistants and I applied this idea to draw a view of the Ford shortened angle, from the point already mentioned, Foundation Building [6], with the Chrysler Building in the we drew the observer’s sightline up to the central background. But for another view, we wanted something dif- diamond-shaped window in plan. This gives us an ferent, being prompted by a photo seen online (Fig. 2) depict- angle of ≈ 30º (see Fig. 3a). ing the Millennium Hotel (1 United Nations Plaza, New York) 5 The glass facades of the Millennium Hotel converge seen through the skylight of the Ford Foundation Building. slightly upward to a distant vanishing point, which The contrast of the skylight structure against the Millennium means that the observer’s sightline also has a vertical facade on the background, both seemingly hovering, was in- angle. To estimate the angle, we line up the diamond- deed strikingly different. We knew that drawing such a view shaped steel window with the windows of the Millen- would not be easy because of the lack of spatial references at nium (Fig. 2), along a cross-section of both buildings. ground level. As the photo was the only source we had, we This gives us a vertical angle of ≈ 15º (see Fig. 3b). retrieved as much data as we could. As shown below, most 6 As can be seen in either Fig. 4c or the photo, the hori- of the information was in the image. zontal lines of both buildings vanish at a common 1 The photo image suggests that the camera was care- point. Which makes sense, since the layouts of both fully placed and oriented beforehand to take the shot. buildings are orthogonal one to another (see Fig. 3a).

118 García Salgado, Have You Ever Used Two Picture Planes to Draw a Single Perspective View?

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Fig. 3a. Plan view of Ford Foundation Building and the Millennium Hotel (One United Nations Plaza, New York). (© Tomás García Salgado)

b Fig. 4a. Perspective Plane 1 (PPL1), showing the skylight’s initial sketch. (© Tomás García Salgado)

Fig. 3b. Street cross-section showing the Perspective Plane 1 placed indoors at the Ford Foundation Building and the Perspective Plane 2 placed at the southwest facade of the Millennium Hotel. (© Tomás García Salgado)

Fig. 4b. Perspective Plane 2 (PPL2), showing the Millennium’s initial sketch. (© Tomás García Salgado)

Fig. 3c. Cross-section detail of the Ford Foundation Building, showing the observer standing on the 12th floor, looking at the skylight. (© Tomás García Salgado)

According to point 3, we first constructed a specialmodu - not an option. The same problem would occur in traditional lar scale for a visual field of 18º. Then we drew schematic methods, since projection lines to a distant object turn out blueprints (plan and elevation) of both buildings according imprecise as well. At this point we realized that a second to points 4 and 5 (see Figs 3a and 3b). Next, we read out PPL was necessary to accurately draw the Millennium. It modular coordinates of points of interest on the blueprints Fig. 4c. Perspective drawing superimposing Figs 4a and 4b. by overlaying onto them the planes of symmetry (X and Y). As (© Tomás García Salgado) usual, we placed the PPL near the first object, in this case the skylight (Fig. 3c). Otherwise, if the PPL were placed near the was something new that we had never attempted before; but Millennium, the skylight would necessarily have to be retro- somehow we knew it might work, since both PPL have the projected, creating a nightmare. As expected, we faced the same modular coordinates. problem of reading more than 262 modules of depth when This is how we decided to trace separately the skylight attempting to draw the Millennium on the skylight’s PPL. on PPL1 and the Millennium on PPL2, as shown in Figs 4a In theory we can read much more than that, at the price and 4b. Having finished the sketches, we applied color with of making the drawing imprecise. For us, imprecision was colored pencils on both drawings. Thereupon, we scanned

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Fig. 5a. Plan and elevation of a cube moving away from the observer. All measurements are given in modules (m). (© Tomás García Salgado)

Fig. 5b. Six views of an identical cube, with the Pn-scale placed at different depths on the PPL. (© Tomás García Salgado)

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01603 by guest on 30 September 2021 the drawings first and then composed the final image on the of the cubes. Looking closely at the cubes, the first appears computer screen by overlapping one upon another (see Fig. clearly in perspective, whereas the second and third seem to 4c). Space does not allow explanation in detail of each sketch’s lose perspective. But the fourth appears in axonometric view. construction. However, two scales of P (where P stands for As for the fifth cube, we can tell that it is in axonometric pro- depth) could have been used on a single PPL, as we realized jection. Here our sight cannot discern whether the vertical shortly after finishing Fig. 4c, when we learned how to zoom lines are parallel or not. in the PPL. This procedure consists of placing Pn = 262 m, To understand the fifth and sixth cubes in Fig. 5b, note the as the initial value of the P-scale from which to start reading following principle: Either in perspective or axonometric pro- depths, on PPL1. In this manner, two scales of P (one for PPL1 jection, a cube holds its three vanishing points regardless of its and another for PPL2) would be manageable in a single PPL. position in space or the distance from which it is seen. Vanish- As in theory the P-scale can be placed at any given value of P ing points don’t play hide and seek. A three-point perspective (expressed by the notation Pn), this raises another question: of a cube can disguise itself as a one-point perspective when Could Pn be placed to outline remote objects? For example, two of their parallel systems are parallel to the PPL. Suppos- if Pn = 384,000 Km, could we draw the Lunar Module land- edly, lines parallel to the PPL do not have vanishing points, ing on the Moon? See the answer in the following section. but actually they do. The question here is: Where are these points? It’s simple; these points lie anywhere outside the PPL, Perspective vs. Axonometric at infinity. Thus, it is not surprising to see all parallel systems In the early nineteenth century, a breakthrough in geometry of a cube vanishing at infinity, as in the fifth cube. Notice that was introduced by William Farish’s paper on “isometrical the fifth cube recedes from the observer 2000 times the length perspective” [8]. Later, Jules de la Gournerie [9] laid out the of its side (5 m), which is why we do not even attempt to draw fundamentals of axonometric projection, profusely applied it in plan and elevation (in Fig. 5a). Instead, in perspective, this by Choisy to illustrate his treatise Histoire de L’Architecture cube was easily drawn like the others. At this point we realized (1899). Today, all axonometric projection is considered three- that even with the fifth cube placed at remote distance, its ap- dimensional but without perspective, which is why parallel pearance would not change. Thus, the Lunar Module landing projections and perspective are classified in two different on the Moon can be drawn in the same manner in which we branches. In planar geometry, according to Carlbom and drew the fifth cube. Yet we do not know what ratio between Paciorex [10], “when the center of projection is at infinity, object size and observer distance would produce a perfect so that the projectors are all parallel, the projection is known axonometric projection. This ultimate question remains open as a parallel projection. When the center of projection is at a here, since there is a simple way to overcome this problem. It finite distance from the projection plane, a perspective pro- consists in taking all P values of the cube (in plan and eleva- jection results.” But if the center of projection (of planar ge- tion) as P = 0.00 m, under the following principle: When there ometry) were the observer (of modular perspective), then the is a slight deviation in Pn values (as in the fifth cube), which observer would not be at infinity; rather, the object would be. tend to zero while Pn tends to infinity, then all Pn values can This conceptual discrepancy seems to have a simple explana- be substituted with P = 0.00 m. By this principle both axo- tion. While the center of projection is that which moves away nometric and perspective can be drawn on the PPL, as the from the projection plane (in planar geometry), objects are sixth cube is depicted simultaneously in both projections (see that which move away from the PPL (in modular perspec- the final cube of the series in Fig. 5b). Finally, the sixth cube, tive). This means that thedistance between the observer and viewed from an ideal distance, proves that the observer is not the PPL is constant for any given angle of visual field, no the one at infinity; it is rather the cube. matter whether objects are seen closer or farther. This was the key to drawing the two PPLs of the Ford Foundation Building Conclusions at the same size, since modularly they are equal. The advantage of a 3D perspective plane over a 2D picture As mentioned above, we can make workable the PPL in plane lies not only in adding one more dimension; it rather is dual position by setting the P-scale at any value of depth a new way to perform perspective. One example is the Ford (Pn) on the PPL. This procedure allows us to move objects Foundation Building perspective, traced on two PPLs. Place- far away from the observer and still be able to draw them in ment of more than two PPLs can be considered to visualize perspective. To test this idea, in the following visual analysis architecture in a different way. This idea might be handy for we can see how the parallel lines of a cube behave in both either those who still use pencils or who prefer digital draw- perspective and axonometric. ing. In painting, for instance, once you have a scene on the Let Fig. 5a be a cube represented in plan and elevation PPL, it would be interesting to make it rotate slightly onto a (PLSX and PLSY) at progressive distances (Pn = 29 m, Pn second PPL. Thus when both planes come together, differ- = 50 m, Pn = 64 m and Pn = 1000 m). Here the one rule is ent perception of space might result. It all depends on one’s keeping the orientation of the observer sightline constant in imagination to explore the P-scale possibilities to give the both plane of symmetry X and Y. Now let Fig. 5b be the per- pictorial space a new sense of depth. I also have learned that spective drawings of the cubes depicted in Fig. 5a (plus two an object projected at ideal distance on the PPL looks the other cubes, explained below). Here, it is clear how the shape same in both perspective and axonometric. However, further of the cubes changes gradually. In Fig. 5b, you may identify insights on the models of both projections, axonometric and the scale of Pn by the Pn value assigned at the bottom of each perspective, are necessary.

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01603 by guest on 30 September 2021 Acknowledgments Glossary Special thanks to my assistants at Sistema Nacional de Investigadores distance—the modular interval between the observer (SNI), Viridiana Hernández and Brenda Macias, for their collaboration in the images’ preparation. and the PPL ideal distance—a theoretical infinite distance between the observer and the object, from which all convergent References and Notes visual rays become parallel rays See Glossary for technical terms italicized in the text. modular coordinates—the three dimensions (X, Y, and P)

1 Leon Battista Alberti, Della Pittura, edizione critica a cura de Lu- of any given point within the observer’s visual field igi Malle (Firenze: G.C. Sansoni Editore, 1950) p. 70: “Scrivo uno modular perspective—the author’s method of perspective quadrangolo di retti angoli quanto grande io voglio, el quale reputo in which all points of interest of an object are measured essere una finestra aperta per donde io miri quello che quivi sarà dipinto.” and drawn, in modules, directly on the PPL modular scale—the ruler for measure modular coordi- 2 www.thefreedictionary.com/picture+plane. Here, the shape of the picture plane is interpreted as “an unbounded two-dimensional nates of any given point on the PPL shape.” module (m)—a conventional unit by which the object’s real dimensions are read on the plane of symmetry x, 3 Tomás García Salgado, Perspectiva Modular, Método para Arquitectos y Disciplinas Afines (ed. FA, UNAM, 2018). plane of symmetry y and perspective plane perspective plane (PPL)—the plane onto which objects, as 4 John L. Stephens and F. Catherwood, Incidents of Travel in Yucatan (México: Ed. San Fernando, 2000) p. 161. perceived by the observer, are projected in perspective picture plane (PP)—an imaginary plane placed between 5 Architecture in Perspective 29, Dallas, 2014, hosted by the American Society of Architectural Illustrators (ASAI). the observer and the object being viewed plane of symmetry X (PLSX)—the plane onto which 6 Kevin Roche, John Dinkeloo and Associates (1963–1967), architects. In 1997 the Landmarks Preservation Commission of New York des- modular coordinates (X, P) are read and measured ignated the Ford Foundation Building as an interior landmark. plane of symmetry Y (PLSY)—the plane onto which 7 Tomás García Salgado, “Modular Perspective and Vermeer’s Room,” modular coordinates (Y, P) are read and measured Bridges London (Conference Proceedings 2006, R. Sarhangi and Pn—any value of depth (n) at which P-scale can be placed J. Sharp, eds.) p. 381. on the PPL 8 William Farish, “On Isometrical Perspective,” Transactions of the Cambridge Philosophical Society 1 (1822). Manuscript received 23 February 2016.

9 Jules de la Gournerie, Traité de Géométrie Descriptive (Paris: Mallet- Bachelier, 1860). Tomás García Salgado has a PhD (1987) in Architec- 10 Ingrid Carlbom and Joseph Paciorex, Planar Geometric Projections ture. He is a tenured researcher at the School of Architecture and Viewing Transformations Computing Surveys 10, No. 4 (1978) of UNAM (Mexico City) and holds the distinction of Level p. 476. III National Researcher (SNI). He works professionally as an architect.

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