Images in Distortions: From Anamorphosis to Conformal Mappings

*The original article can be consulted at the following adress: https://www.academia.edu/8079832/Images_in_Distortions_From_Anamorphosis_to_Conformal_Mappings

Since antiquity, the question of deformation has been of interest to many artists. Deformation has been largely used to restore the perspective of paintings aimed to be shown in large architectural spaces. The science of deforming the image was largely developed during the Renaissance when the theory of perspective was more precisely defined. Anamorphosis became the popular term to describe such deformations. Many public presentations of exotic anamorphosis needed the use of mirrors to be seen properly. The art of deforming the image led to new artistic forms, and such images are prominent trend in contemporary art. A question arising from these images concerns how certain images are deformed but still allow a readability of the original image. This analysis contends that a certain type of mathematical functions, known as the conformal mappings, is a key concept to answer this question and examines how these functions can be applied to transformed images while implementing neuroscience to confirm the importance of conformity in deforming the image. Finally, further explorations of space representation based on these observations will be discussed.

The word anamorphosis is formed with the prefix ana, which means going back, and the suffix morphosis, meaning the form. The word’s etymology refers to the ability to find an original image that has been transformed. Jurgis Baltrušaitis (1995, 7) defines it as the reconstruction of the image by standing in a peculiar point of view. For instance, to fully read the painting The Amassadors by Holbein, the spectator needs to stand near the bottom left part of the painting to perceive a skull painted diagonally. Historically, this point of view has not always been sufficient. The extended use of mirrors is an example of a tool needed to access the original image. A contemporary example of an anamorphosis with the use of a cylindrical mirror is found in The Mysterious Island by Istvan Orosz. The valley in the middle of this painting mutates into a portrait when observed through its reflection on the cylindrical mirror.

One of the reasons anamorphosis were developed was to permit paintings on high ceilings, like domes, creating a pleasant effect on the spectator looking from the ground. It shows one of the interesting possibilities in forming anamorphosis. The process is done not only by changing the way we draw the images on a canvas, but as well by changing the form of the canvas itself. This choice of deformed canvas is presently popular within graffiti artist communities, particularily in Europe. For instance, the Paper Donut Collective, the TSF Crew and Vincent F have produced many anamorphic graffiti works.

All these creations could not have been so perfectly done without the discovery of linear perspective attributed to the architect Filippo Brunelleschi during the Renaissance (Stillwell (b), p. 128). Leon Alberti was the first to formally document the concept in his painting manual Della Pittura, realizing that in perceiving parallel lines reaching far distances, these lines would seem to gather at a point, called the vanishing point. Mathematicians of the time were working closely with artists in order to find a suitable theory for perspective. Following Alberti’s idea, the French mathematician Desargues added a point at infinity in his geometric model (Dahan-Dalmedico and Pfeiffer 1984, 130). Both within the visual arts and the mathematics, it was now realized more than one point could be used simultaneously; therfore different models for perspective, with two and three points at infinity, appeared. Finally, all these points at infinity could be linked together to form the line at infinity. Poncelet’s conception of this line amounts to the same as having two parallel planes meeting at infinity (Coxeter 1974, 3). These observations led to different geometric models, in particular to non-Eucledian models by Lobachevsky and Polya. (Dal’Bo-Minolet, 2012, 38-45)

At this point, the highly abstract nature of geometric models found in mathematics could no longer fulfill the purpose of the art of rendering a visual simulacrum of the world. Centeries later, it took the imagination of artist Maurelius Cornelius Escher to conciliate both fields. When working on infinite tiling of the plane, he realized the plane tiling could not grasp the concept of infinity as much as he wanted; it could only lead to an imaginary infinity, not one that could be seen all at once (1972, 44-45) which was a concept already discovered in mathematics. The two endings of the line at infinity could be joined together to form a closed system, the horocyle. This model has already proven to be useful for hyperbolic geometry or the projective plane model. Escher used the line at infinity to frame his infinite tiling and all patterns would then vanish near that line. More recently, Jos Leys used the same idea in some of his works, even adding an extra point at infinity or a whole new line at infinity inside the plane contained within the framing line at infinity. On his website, 13 images offer extra points at infinity added to the horocycle and 10 are using two lines at infinity.

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Figure 1: Jos Leys – hyp073ring

Source: http://www.josleys.com/show_gallery.php?galid=332

Stereographic Projection and Photography

Knowing the extended possible utility of points and lines at infinity, the question of the canvas can be analyzed once again. Vanishing points can be embedded on surfaces other than the plane. A simple case of a surface different from the plane is the sphere. The artist Dick Termes constructed spheres, dubbed termespeheres, on which he paints using multiple vanishing points. Before starting his piece, he chooses how many vanishing points will be used; this number can be as high as six. In this case, the six points represent six vanishing points that could, in theory, be seen from inside a sphere in the diegetic world (1994, 244-245). These six points can be chosen to represent the points in front, behind, left, right, the nadir and the zenith of the painting. It is equivalent to creating an atlas of the sphere; in other words, all the points on the sphere are represented in the space created by one of the six perspectives (Pressley, 2001, 65).

Furthermore, embedding a vanishing point happens to be useful for the cration of a one-to-one mapping between the sphere and the plane, meaning that each point of the plane is linked to one on the sphere and vice versa. One way to proceed is to use stereographic projection, a projection utilized since antiquity (Snyder 1987, 154). Setting the sphere on the plane and assigning the North Pole as the starting point of straight rays directed to the plane creates this relation. Each ray crosses the sphere at a point and assimilating the extension of that ray to the plane provides the one-to-one mapping (Pressley, 2001, 109-111). In this case, the North Pole is equivalent to the vanishing point since going to infinity in any direction on the plane leads to approaching the North Pole on the related sphere. Therefore it is a one-to-one mapping between the sphere and the plane plus the point at infinity. Also, one encounters a similar idea with the circle at infinity, the horocycle: by shrinking a circle near the North Pole on the sphere, it creates a circle of infinite radius on the plane. The stereographic projection also possesses the particularity to map circles on the sphere to circles on the plane, with the exception of circles passing through the North Pole being mapped to straight lines on the plane. (Gamelin, p. 13)

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Figure 2: Stereographic Projection

Source: http://www.dimensions-math.org/Dim_CH1_E.htm

The idea of stereographic projection is what leads to Alexandre Duret-Lutz’s anamorphic work. (Duret-Lutz, http://www.flikr.com/photos/gadl/sets/72157594279945875/) His photographs show ‘wee planets’ on which objects take curvy and cartoonesque shapes. Although it is not clear at first sight, these photographs are a result of stereographic projection. To create these pictures, Duret-Lutz takes pictures 360 degrees horizontally and 180 degrees vertically. Using a computer, he then patches them together to create a virtual sphere. Finally, applying stereographic projection, he projects this virtual sphere on a flat virtual canvas. As a result, the circle on the virtual sphere created by the horizon photographed 360 degrees horizontally is mapped onto a circle on the plane, creating the circumference of the wee planet. Further, the camera’s nadir, the highest point in the sky, is mapped to the wee planet’s center. Objects around the camera rising to the sky are rising to the North Pole on the virtual sphere and therefore mapped as objects diverging to the infinite frame when localized on the plane. This explains why trees or buildings in the pictures seem to grow towards the frame since the North Pole, the vanishing point on the sphere, is mapped to the horocyle represented by the frame.

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Figure 3: Wee Planet by Alexndre Duret-Lutz

Source: http://www.flickr.com/photos/gadl/463596797/

Using projections to deform images appears therefore as a legitimate option. There exist various types of projections on different surfaces like the sphere, the cone for Lambert projection and the cylinder in the case of Mercator’s, Miller’s and Cassini’s projections (Snyder, 1987, 37-38).To understand these deformations one needs to address a similar question articulated by Alberti centuries ago: what are the geometric characteristics preserved by the projection in between two surfaces? (Dahan-Dalmedico and Pfeiffer 1984, 128) If length is preserved, the mapping is said to be isometric; if the area is the same, it is equivalent, and finally, if it preserves the numerical values of intersecting angles between lines, it is conformal (Pressley 2001, 106-121).

In his Theorema Egregium, Carl Friedrich Gauss proved isometries to be impossible between the sphere and the plane (Pressley 2001, 229, 238). Nevertheless, conformal mappings are possible between the two surfaces, but only if area is not preserved. The stereographic projection is only one example of conformal mapping (Pressley, 109-111); others, like the Mercator’s rhumb-lines mapping, also exist. Conformity explains why objects keep their overall shape on Duret-Lutz’s wee planets. Buildings typically stand perpendicular to the ground and have right angled corners.

Looking carefully at Duret-Lutz’s portfolio, one can find other interesting pictures. Some of them look similar to wee planets, but show a strange tunnel at the end of which the sky stands. To understand what is happening in these pictures, it is imperative to dig deeper into mathematics.

Complex Numbers and Conformal Mappings

Complex numbers have been avoided by mathematicians for a long time since they allowed square roots for negative numbers, deemed an impossibility in mathematics at the time. They were solely considered as a tool to solve some polynomial equations (Mumford, p. 36-38). A Norwegian-Danish cartographer named Caspar Wessel found a way to represent these strange numbers on the plane by assigning the x axis to the real numbers and the y axis to the value of the negative square roots (Mumford, 2002, 40). In this model, many of the usual transformations of the plane still hold on the complex plane. The translations, rotations and dilation work on the complex plane in a similar fashion as their equivalent on the Cartesian plane. Another interesting one, the inversion, is of note. This inversion has the property of inversing the inside and outside of a circle and to rotate the result. All these four transformations, and any compositions of these are gathered under the name of Möbius transforms, after August Ferdinand Möbius, who studied them in the 19th century (Stillwell (a), p.182).

These transformations are important for four reasons. First, the inversion seems to be giving exactly the same result as the Duret-Lutz’s photographs with the sky in the center. The reverse circle used in his pictures is the wee planet’s circumference. The sky that previously stood outside that circle is now in the middle and the center of the planet and the nadir of the camera is now taking place as the horocycle, the circle at infinity. Second, these transformations of the plane are conformal, they preserve angles (Saff and Snider 2003, 389). Third, the Möbius transforms are related to the stereographic projection in an unexpected way. Every Möbius transformation can be perceived as the result of motions of the sphere paired with the stereographic projection. For instance, the inversion can be constructed by projecting the plane on the sphere, spinning the sphere vertically 180 degrees and mapping it to the plane again stereographically. This can be observed in Duret-Lutz’s photography when the camera’s zenith is mapped at the center of the picture. A video by Arnold and Rogness illustrates their paper about the equivalence between motions of the sphere and the Möbius transforms (2008).

Figure 4: Stereographic Projection and Möbiuss transforms

Source: Arnold and Rogness, 2009.

The fourth reason is a result of the three point theorem for Möbius transforms stating that any triplet of points can be mapped to any other triplet of points (Gamelin 2001, 63). Since the Möbius transforms are conformal mappings, the operation can be done by preserving all angles of intersecting lines. Indeed, this can also be done via the stereographic projection; in other words, three points on the sphere can be chosen and mapped to any other three points. The choice of the triplet might include the South Pole and North Pole of the sphere, or equivalently in Duret-Lutz’s wee planet, the nadir and the zenith.

A video by Ryubin Tokuzawa exemplifies this technique. In Stereographic Projection- Sample 02 (motion picture), the image starts by showing two black discs near the middle. As the motion picture continues, one recognizes these as being the nadir and the zenith of a camera placed on the roof of a car. These points then move up and down the image as the video advances. The black dot containing the zenith concludes by merging with the frame, turning the whole image into a wee planet, just as in Duret-Lutz’s work.

Another interesting example is the photograph Double Spiral by Paul Nylander. By projecting the North and South Pole and a certain type of curves called loxodromes, or rhumb-lines we can construct spirals on the plane. Loxodromes are curves on the sphere crossing meridian lines at constant angles. This property implies that these curves spiral indefinitely around the Poles and therefore spiral around the projected points on the plane (Mumford, Series and Wright 2002, 81-82; Pressley, 83). By tilting the sphere before applying the projection, we obtain a double spiral on the plane, this result can be observed on the picture Playground by Josh Summers.

There are clear links between image distortion, projections and conformal mappings. Conformal mappings allow images to preserve the angles of intersections, therefore maintaining the integrity of the overall shapes. Some projections allow this property to be kept in between two surfaces.

Perception and Conformity

In the field of neuroscience, Irving Biedermann’s article Recognition-by-Components: A Theory of Human Image Understanding is a prominent publication about the importance of line intersections (1987). The article shows an experiment in which he erased 50% of the lines in drawings of easily recognizable objects in two different ways: first by allowing any lines’ parts to be erased, and then by leaving intact all lines’ intersections. Results showed patients were clearly faster in recognizing objects when the intersections were intact. This underscores the importance of line intersection in a person’s manner of perceiving images. Further, it partly explains why objects in conformally deformed images are still relatively easy to identify and determines why some optical illusions, like the strange architecture created by Escher, are so powerful. Even though the overall picture is odd, the lines of intersection defining the objects lead the reader to accept the image as normal (1987, 135-140).

Having confirmed the importance of conformity in image distortion and image recognition, it is natural then to explore new paths of research. If an image is deformed but keeps an overall readability, it makes sense to verify if the transformation used is conformal. It also leads to explore further possibilities offered by conformal mapping outside the Möbius range of transformations.

In the first case, Printing Gallery by Escher is a perfect object of study. The overall pattern is hardly recognizable, so intricate Escher himself could not finish the work and had to leave a blank space in the middle. In 2003, a team of mathematicians led by H. Lenstra was able to find this pattern and finish the work. What is now known as the Droste effect in tribute to an old recursive publicity by Droste has been defined mathematically as a series of transformations including exponential and logarithmic functions (Smit and Lenstra, 2003, 449-450). As expected, these functions were all conformal. Interestingly enough, the result can be obtained via projections over an infinite cylinder as well (Carphin and Rousseau 2009, 2-5). The Droste effect is now used by many photographers to create spiraling self-included pictures. An excellent example is offered by Josh Sommers in The Obligatory Droste Self Protrait.

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Figure 5: Obligatory Droste Self-Portrait by Josh Sommers

Source: http://joshsommers.smugmug.com/Art/The-Droste-Effect/i-38TkFvr

The second consideration brought by conformal mapping is the exploration of conformal functions to create highly deformed but still readable images. The photographer Seb Pzbr has extensively used many different conformal mappings in his photographs, from trigonometric functions to polynomial functions. He was originally the artist who inspired Duret-Lutz to work with stereographic projection. Even if in many of his photographs the function is not easily recognizable, the result is always very pleasing, especially  when presented with a symmetrical environment. ((Duret-Lutz, www.flikr.com/photos/gadl/sets/72157594279945875/ )

Conclusion

This paper sheds light on some properties of deformed images. In the wide range of possibilities, it focuses on images that can be deconstructed while keeping the original image recognizable. More specifically, it describes the photographs obtained by use of stereographic projections in the work of Alexandre Duret-Lutz. The paper shows certain links existing between the stereographic projections and some conformal mappings, the Möbius transforms, and uses them to explain some other types of photography. By looking at the three point theorem on Möbius transforms, the mechanism behind one of Ryobin Tokuzawa’s video is clarified. Biedermann’s experiment confirmed the importance of angles of intersection, therefore underlying the utility of using conformal mappings in visual art. This leads to two paths of discussion. First, it highlights the legitimacy of looking at the conformity of a deformation when an image is still recognizable and then paves the way to further explorations of this class of functions in search of more interesting visual results. It would be interesting to explore the results of conformal mappings obtained on sculpture, either virtual or real. Three dimensional printers enable such possibilities. Further, since the group of isometric functions is already well known, it could be interesting to explore the ways in which the class of equivalent functions can find applications in visual arts.

 

REFERENCES

Baltrušaitis, Jurgis. 1995. Aberrations : Les Perspectives Dépravées-1. Paris: Flammarion.

Biederman, Irving. 1987. ‘’Recognition-by-Components: A Theory of Human Image Understanding.’’ Psychological Review, 94 (2):115-147.

Carphin, Philippe et Christianne Rousseau. 2009. ‘’Finir une gravure d’Escher.’’ Acromath, 4:20-25.

Coxeter, H.S.M. 1974. Projective Geometry. 2nd Ed. Toronto: Univerity of Toronto Press.

Dal’Bo-Minolet, Françoise. 2012. ‘’La géométrie des horizons.’’ Pour la Science, 411 :38-45.

Dahan-Dalmedico, A. et J. Pfeiffer. 1984. Une Histoire des Mathématiques : routes et dédales. Paris : Éditions du Seuil.

Delahaye, Jean-Paul. 2004. ‘’Calculer dans un monde hyperbolique?’’ Pour la Science, 316 :90-95.

Duret-Lutz, Alexandre. ‘’Answers to Some Frequently Asked Questions‘’. Accessed December 22, 2013. http://www.flickr.com/photos/gadl/sets/72157594279945875

Escher, M. C. 1972. ‘’L’approche de l’infini’’ Dans Le Monde de M.C. Escher : L’œuvre de M.C.Escher, commenté par J.L. Locher, H.A. Broos, M.C.Escher, G.W. Locher et H.S.M.Coxeter. J.L. Locher Editor. Paris : Éditions du Chêne.

Gamelin, Theodore W.. 2001. Complex Analysis. New York: Springer-Verlag.

Gray, Jeremy. 2007. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. New York: Springer.

Andersen, Kirsti. 2007. The Geometry of an Art: The history of the mathematical theory of perspective from Alberti to Monge. New York: Springer.

Mumford, David, Caroline Series and David Wright. 2002. Indra’s Pearls: The Vision of Felix Klein. Cambridge: Cambridge University Press.

Pressley, Andrew. 2001. Elementary Differential Geometry. Springer Undergraduate          Mathematics Series. London: Springer.

Saff E.B. and, A.D. Snider. 2003. Fundamentals of Complex Analysis with Applications to Engineering and Science, 3rd Ed. New Jersey: Prentice Hall and Pearson education.

Smit, B. de and H.W. Lenstra Jr. 2003. ‘’Artful Mathematics: The Heritage of M.C. Escher.’’ Notices of the American Mathematical Society, 50 (4): 446-451.

Snyder, John. P. 1987. Map Projections-A Working Manual. Washington: United States Government Printing Office.

Stillwell, John (a). 2010. Four Pillars of Geometry. New York: Springer-Verlag.

Stillwell, John (b). 2010. Mathematics and its History. New York: Springer.

Termes, Dick. 1994. ‘’The Geometries Behind my Spherical Paintings.’’ In The Visual Mind: Arts and Mathematics. Michele Emmer Editor. Cambridge: MIT Press, 243-248.

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