G e n e r a l a r t i c l e have you ever used Two Picture Planes to Draw a Single Perspective View? T o m á S G arc í A S A l gad o Apparently, the use of two picture planes to draw a single view has be a more versatile plane that in addition to representing not been attempted before. Most perspective methods, after Alberti, the appearance of 3D objects would also serve to measure take for granted the use of a single picture plane, disregarding its likely real dimensions? Such a plane, in Modular Perspective [3], use in dual positions. What if two picture planes are necessary to ABSTRACT draw a single view—for example, given a lack of spatial references at can be called the perspective plane (PPL). On the PPL one can ground level to estimate the distance between two objects? This article measure and draw directly the three modular coordinates of demonstrates that to draw the interior of a building from which another all points of interest of any given object in space. In other building can be seen about 190 m away, where the projection of such words, the PPL is a true three-dimensional plane, despite building on the first picture plane would be imprecise, it may be wise to actually being two-dimensional. use a second picture plane. This leads to consideration of how objects In addition, the PPL can be of any size, since all points are change shape as they move away from the viewer. For example, if a read and drawn in modules (m), as coordinates are read on cube recedes from the observer up to 100 times its side length, it takes on an axonometric view. This raises a question: Could axonometric blueprints, regardless of their representational scale. These projection be a particular case of perspective? attributes make workable the idea of placing a second PPL. On the contrary, it might be impractical to place a second PP, because its size cannot be established beforehand nor can The holy PictuRe PlAne measurements be taken directly on it. This technical limita- As we know, the prime function of the picture plane (PP) is tion might explain why a dual position of the PP has not, to to depict objects seen by the observer, in essence, as Alberti my knowledge, been included in traditional methods. describes, using the metaphor of seeing through an open Let us now suppose a case in which the use of a second window [1]. The three fundamental attributes of the PP most PPL is necessary. Imagine a space shuttle approaching a space referred to in current definitions are the following: (a) The station, the void of space between them, no visible ground PP is an imaginary plane placed between the observer and establishing what is up or down, no clue of the distance be- the object being viewed; (b) The PP is a two-dimensional tween both orbital objects. There, if an astronaut wants to plane onto which objects are projected; and (c) the observer sketch the shuttle along with the space station, while floating sightline intersects the PP perpendicularly. (Other attributes, upside down with Earth at their feet, the only spatial refer- such as size and shape, are less well understood [2].) The at- ences are their sightline and their own body. Once back on tributes a–c suffice to understand how the PP works. There- Earth, everything is in place; the sky is up and the ground fore, according to (a), tracing visual rays from the observer down. A similar experience of seeing objects in space can to all points of interest of an object allows its perspective to also occur in unexpected places—for instance, when seeing be formed on the PP where such rays cross it. This principle New York’s Millennium Hotel from inside the Ford Founda- is well known and applied in various forms. My objective tion Building, with nothing but air between both buildings in reflecting on the current definitions of the PP is to deter- (see Fig. 1). In both cases, whether on the space station or mine if its three-dimensional attribute has been considered at the Ford Foundation, whatever the orientation of the ob- before; based on my research, it seems not. We wonder, then: server in space, the sightline and vanishing point are always What if the PP were a real three-dimensional plane? Could it at the center of the PPL. Any of these views turns out to be easy to draw solely by means of the observer’s vanishing point, in modular perspective. Here, by definition, perspec- Tomás García Salgado (researcher), Faculty of Architecture, The National Autonomous University of Mexico, Av. Santa Fe 462, Dep. 401A, Cuajimalpa 05348, tive as geometric representation of human vision vanishes Cd. de México, México. Email: [email protected]. not at two or three vanishing points but at one. To prove See www.mitpressjournals.org/toc/leon/52/2 for supplemental files associated this, ask yourself, wherever you are: Is your sight vanish- with this issue. ing at just one vanishing point, or more? If it vanished at ©2019 ISAST doi:10.1162/LEON_a_01603 LEONARDO, Vol. 52, No. 2, pp. 117–122, 2019 117 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/leon_a_01603 by guest on 30 September 2021 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 sight- line 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 cam- era 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.