Foreshortening gives way to forelengtheningWe are all familiar with foreshortening in perspective pictures. But students of per-ception should also know about what we might call perspective `forelengthening'.Many probably think foreshortening halts at a finite limitöwhen the object : imageproportions reach 1 :1. Far from it!

Figures 1 and 2 are perspective pictures of square tiles on flat ground recedingtowards the horizon. Consider the bottom row of tiles in figure 1. The rightmost tileis nicely foreshortened in the sense that the oblique and vertical showing its recedingsides are shorter than its front edge (marked by dots). But now consider the leftmosttile. It is `forelengthened'. Indeed, if the front edge of each tile is length L , the leftoblique of the tile is almost 2L.

Now consider the bottom row of tiles in figure 2. Every oblique is longer than L. Therow above is also forelengthened. But the row above that has some foreshortened tiles.

Why do foreshortened tiles in figure 1 give way to forelengthening as we move to theleft? Why do forelengthened tiles in figure 2 give way to foreshortening as we move up?

To begin to explain, let us note that both pictures contain two dots on the horizontalline. The left one (call it dot C) is a centre of converging lines that show the recedingtile sidesöthe ones orthogonal to the picture plane. The lines converging to the right one(dot D) depict the diagonals of the tiles.

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Perception, 2002, volume 31, pages 893 ^ 894

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Figure 1. A perspective picture of square tiles receding to the horizon, showing foreshorteningand forelengthening. Handily, it has within it a picture of the tiles projecting to a point to oneside of a picture surface.

Figure 2. The forelengthening in figure 1 is increased in this figure by shortening the viewingdistance, shown by the small gap between the two dots in the top line.

It is very helpful to switch viewpoint, and to realise the vertical through dot Ccould be treated as a cross section through a picture surface (PS). In this account, wecan let dot D be the location of the observer's eye. If so, the lowest horizontal couldbe a ground line with square tiles width L receding to the left. Now, the lines converg-ing on dot D conveniently show us the tiles projecting onto the picture surface PS.

In figure 2, the tile on the ground nearest PS projects a line on PS lots largerthan L. The leftmost tile in the ground row projects quite a small line on PS. Logically,inbetween the projection has to be the same size as L.

To simplify, it may help to imagine a large tile, width W, with one edge near enoughto PS to touch it, and a far edge projecting a line to dot D at 458 to the ground.The projection line will also be at 458 to PS. It must form an isosceles triangle with PSand the ground. The tile must project a line on PS equal to W.

Consider a tile width L, smaller than W. The width dimension of a tile farther tothe left than the 458 line is foreshortened in familiar fashion when it projects on PS.The further the tile is from PS, the more it is foreshortened on PS, to a limit of zerofor a tile on the horizon. But a tile closer to PS than the 458 line is forelengthened.If PS is very large, the dot D is very close to dot C, and in the limit the foreshorteningis infinite. Truly forelengthened!

Intersections of the lines from the tile edges with PS show the proper projectionsof the tile for an observer placed at dot D, to one side of PS. What can we make ofhorizontal lines through PS at those intersections? Interestingly, if we switch viewpointagain, these horizontals show the proportional foreshortening or forelengthening oftiles receding towards the horizon (receding in the z dimension) for an observer likeyou the reader, in front of figures 1 and 2 (Willats 1997, page 62). In each picture, theproportions are correct for anyone on the orthogonal from dot C, looking from the dis-tance of dot D from dot C. Handily, therefore, the horizontals give us the correctprojections of the near and far edges of the tiles. The other sidesöleft and right sidesöare shown by obliques converging to dot C.

The z-dimension of all the tiles in a horizontal row is foreshortened to the sameextent, as shown in figures 1 and 2. But obliques converge at more acute angles thefurther they are from PS, and become ever longer. They become infinite, in the limit.Again, truly forelengthened! A tile in the column of tiles just below dot C may haveforeshortened obliques, but a sibling in its row, far from dot C, has hugely forelength-ened obliques.

In short, forelengthening occurs in two ways. In the z-dimension it is shown by thedistance between horizontal lines in perspective pictures like figures 1 and 2. And fororthogonals, forelengthening occurs for the obliques in figures 1 and 2 showing thereceding sides of tiles. In both cases, the limit approached by forelengthening is infinite.

Few commentators on perspective, constancy, and drawing development acknowledgethere is anything more than foreshortening to challenge vision in perspective pictureperception. A reading of the rare exceptions, such as Pirenne (1970) and Kubovy (1986),may be a lot more rewarding and provocative once `forelengthening' is understood.

John M Kennedy, Igor JuricevicDepartment of Psychology, University of Toronto, 1265 Military Trail, Toronto, Ontario M1C 1A4, Canada;e-mail: kennedy@utsc.utoronto.ca

ReferencesKubovy M, 1986 The Psychology of Perspective and Renaissance Art (Cambridge: Cambridge

University Press)Pirenne M H, 1970 Optics, Painting and Photography (Cambridge: Cambridge University Press)Willats J, 1997 Art and Representation (Princeton: Princeton University Press)

ß 2002 a Pion publication printed in Great Britain

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