# Cylindrical Mirror Anamorphosis and Urban-Architectural Ambience

- 500 Downloads
- 3 Citations

## Abstract

Cylindrical mirror surfaces fall into the group of reflecting surfaces that give a distorted image of an object. However, if the object is designed according to the laws of optical geometry, in a way that its mirror image is conceived in advance, then this is anamorphosis. The objective of the present study is to emphasise the potential of cylindrical mirror anamorphosis, in the context of change in the urban-architectural ambience. In this respect it is necessary to obtain a constructive, geometrically correct solution of the 3D model of cylindrical-mirror anamorphosis, whereby the mirror surface is a vertical rotating cylinder. This topic is the primary focus of the present research. In addition, the conditions for change in the anamorphic form were analysed, and its possible functions in architecture were identified. Various examples of existing buildings with cylindrical mirror elements, in respect of which it was possible to construct and apply these types of anamorphoses, were used. The method of constructive perspective and the laws of optical geometry were applied. Analyses were made on the basis of experiments and using AutoCAD 3D methods to analyse the mirror anamorphosis of a cube and octahedron.

## Keywords

Optics Perspective Representation of architecture Design analyses CAD Descriptive geometry Geometric analyses Modeling Perspective geometry Polyhedron Ratio Shapes Transformation Virtual architecture Urban design## Introduction

Anamorphosis is a type of *projection form,* the meaning of which is understood only when observed from the viewpoint in respect to which it has been constructed, and in a wider sense, it is a form to which specific projections give more meaning. Cylindrical mirror (catoptric) anamorphoses are a subcategory of mirror anamorphoses, the identification of which requires a cylindrical mirror.

Given that the hitherto published constructions of cylindrical mirror anamorphosis^{1} relate to plane anamorphosis, the main objective of the present study is to obtain a constructive, geometrically correct solution for the 3D model of cylindrical mirror anamorphosis, whereby the mirror surface is a vertical rotating cylinder.

Jean-François Nicéron (1663: tab. 44, 45) was the first to describe, by way of geometrical methods, cylindrical mirror plane anamorphosis, which was popular during the 17th century. Later on, many authors published different constructive explanations of this phenomenon (Goddijn 1992; Füsslin and Hentze 1999; Leeman 1976), but due to the fact they were only approximate, they cannot be applied to 3D cylindrical mirror anamorphosis. Construction by means of a nephroid grid, with its flaw being a viewpoint that is positioned at infinity, is also not adequate, as section two of this paper demonstrates, when a 3D model is in question. The constructive solution of the 3D models of Jonty Hurwitz, the British artist, have not been published anywhere.^{2} In this context, the aim of the present paper was to check all the constructions analysed in our study, in order to determine cube and octahedron mirror anamorphosis using an experimental method. In addition to the constructive perspective method, optical geometry laws were also applied.

## Geometry of Nephroids and the Catacaustics Grid

*radiant point*,

^{3}which is obtained by their reflection from an established mirror surface, it is then known as a catacaustic (Fig. 2).

- (a)
the position of the radiant point is at infinity (parallel rays) and the catacaustic curve is a nephroid;

- (b)
the position of the radiant point outside the circle is finite, and the catacaustic curve is an irregular nephroid.

*r*) (base curve of a cylindrical mirror) a pencil of parallel rays falls, the envelope of reflected rays is a nephroid. For a concave mirror, the nephroid envelope is also a nephroid which, relative to the initial nephroid, is orthogonally oriented and inscribed in the base circle (Fig. 4a). A nephroid is an epicycloid

^{4}with two vertices. In respect to kinetic geometry, it is the trajectory of the circle point, the radius of which is

*r*, which rolls around the fixed circle of a twofold larger radius (2

*r*).

## Cylindrical Mirror Anamorphosis Obtained Using Methods of Optical Geometry

On the basis of the analyses conducted, the aim was to obtain cube anamorphosis, using the method of optical geometry, that is, by constructing its characteristic points. The idea was to construct every point in space by means of the three^{5} basic principles of optical geometry, given that, by the construction of the anamorphosis of the cube’s horizontal square, by means of the nephroid curves grid, no satisfactory solutions had been obtained.

## Procedure for Constructing Point Anamorphosis

*law of the reflection,*the image in the mirror is obtained from the intersection of virtual rays, which are geometrical extensions of the reflected rays, then the anamorphosis of point A (i.e., point A2), can be solved in a simple manner. Point A2 is equally distant from the tangent plane, as is its mirror image, point A, which can be clearly seen in the first orthogonal projection (Fig. 7). In order to obtain point A2, it is necessary that the mirror image of the required point is situated on the visual ray from the viewpoint O. The angle of incidence

*α*, which covers the mentioned visual ray with the tangent plane (T) of the mirror, must be equal to the angle of reflection—angle

*β*, so that the consequence of such a construction is a point in space—point A2. This can be seen best in the orthogonal projection of the

*plane of reflection*

^{6}(Fig. 8), in which the visual ray reflects from the mirror. The point of intersection of the visual ray through the cylindrical surface of the mirror, point A1, is the point of reflection of the visual ray from point A2, and is also the cylindrical perspective of point A.

## Procedure for Constructing an Anamorphosis of a Cube by Means of its Horizontal Sections

^{7}are not straight lines, as is the case in a nephroid—they coincide only for the directions that lie in the axial plane of the cylindrical mirror.

## Construction of an Octahedron Anamorphosis Using Characteristic Sections

For the previous construction, the cube edge was divided optimally into five parts, in order to obtain a 3D grid, for obtaining the octahedron anamorphosis. Accordingly, for this construction it was necessary to add another three mutually perpendicular planes, which were orthogonal to the faces of the cube, with a common point in its centre, the sections of which were the three axes of the octahedron (given that without these planes, the construction of the octahedron vertices would have been approximate).

## Change of Architectural-Urban Ambience by Means of Cylindrical Mirror Anamorphosis

Reflecting surfaces—both those that give the ideal and those that give a curved distorted image—are often used in modern urban-architectural practice. Both types of surfaces provide opportunities for the transformation of the urban-architectural ambience by means of anamorphosis. Cylindrical mirror surfaces fall into the group of reflecting surfaces which generate a distorted image, so that changes in urban-architectural ambience may be made in respect to existing cylindrical elements which create the effect of a mirror. These might include glass-clad façades, circular rotating doors, chrome pipes and chrome street columns.

An example of this type of reflective surface is found in the mirror-clad façade of the Cairns Botanic Gardens Visitors Centre (Cairns, Australia, Charles Wright Architects, 2011); an exceptional example of a form that offers multiple opportunities for the application of mirror anamorphosis. According to the design, the surrounding tree canopy is reflected onto the façade, but by means of anamorphosis (which has the effect of surprise), it also has, in addition to camouflaging the structure, a marketing function. British artist Jonty Hurwitz has also used street columns as cylindrical mirrors for his 3D free-form anamorphosis.^{8}

There is a similar composition in the structures of the Westin Bonaventure Hotel, a 112 m tall tower in Los Angeles (Fig. 18b), which has been designed by the same architect. This complex expands the geometrical possibilities of anamorphosis, given that the central cylindrical-shaped building in the complex is symmetrically surrounded by four cylindrical buildings of a smaller diameter and lower height. It is possible to achieve different forms of anamorphoses and their mirror images, in respect to a larger number of cylindrical mirrors.

In addition to the functions of aesthetics, utility, marketing and communication, mirror anamorphosis may also include the function of energy efficiency. Sunlight rays, which are insufficiently utilised in the context of energy efficiency, when reflected from a cylindrical mirror surface, produce a catacaustic effect.^{9} If an anamorphosis were designed so as to absorb these sunrays more efficiently, its function would be multiple, and its surface would be dependent on this function. In this way, it would be able to accumulate light energy, and also emit it. Thus, it can be concluded that the position, appearance and size of an anamorphosis is determined, on one hand, by the ideal position of the viewer, and on the other, by the catacaustic effect of sunlight reflection.

## Conclusion

A detailed analysis of the existing construction methods showed some flaws in applying the construction of plane anamorphoses to the creation of 3D models.

The experimental method and geometrical solving of the construction of a 3D model of catoptric anamorphosis, in respect to a mirror surface—vertical rotating cylinder—as well as on the basis of specific points, resulted in models of the cube and octahedron anamorphoses.

For the definition of anamorphosis, it is necessary to have both a virtual object, positioned between the tangent planes of a cylinder that contains a viewpoint, as well as specific points and sections of that virtual object.

For achieving free-form anamorphosis, it is practical to use the isohypse points of that free form, that is, its horizontal sections. In this respect, each isohypse has its anamorphosis, and their union is an anamorphosis of that free form. A form achieved in this way will play a significant role in the practical realisation of a model of anamorphosis.

Analysis of the change in the form of anamorphosis, in respect to the change of different parameters, led to the conclusion that the greatest influence on the appearance of the anamorphosis, in addition to the ratio between the mirror size and the virtual object, is the position of the principal optical axis.

Throughout this study, all of the examples indicated the various potential applications of anamorphosis, whereby its five functions are emphasised: aesthetic, utilitarian, marketing, communication and energy efficiency.

## Footnotes

- 1.
- 2.
In the interview Jonty Hurwitz explains that he used software to create 3D anamorphosis http://www.youtube.com/watch?v=KcTp5Q3-8Ng (accessed 15 August 2013). De Comité (2011: 33) used a chain of software and programming languages to make the spherical mirror anamorphosis and Soddu (2010: 39) designed the software for cylindrical mirror plane anamorphosis.

- 3.
The point of illumination for a caustic. Wolfram Mathworld, http://mathworld.wolfram.com/Catacaustic.html (accessed 15 August 2013).

- 4.
An epicycloid, the trajectory of the circle point, which without slipping rolls around the second circle, is the catacaustic curve that emanates as an envelope of rays reflected off the circle.

- 5.
Geometrical optics is based on the four fundamental laws:

*the law of rectilinear propagation of light; the law of the independence of light propagation; the law of light reflection; and the law of light refraction*, of which the former three relate to mirrors and are included in the experiment. - 6.
It is determined by the view ray A, A1, O and A2.

- 7.
The principal optical axis is a line that passes through

*mirror vertex*T (the most convex, or most concave, point) and the centre of the curvature of the base circle S (Fig. 15). - 8.
Hand anamorphosis, Rejuvenation, Copper and Chrome | 60 × 60 × 45 cm, 2008, http://www.jontyhurwitz.com/rejuvenation#e-8.

- 9.
The case study, conducted at the University of Applied Sciences in Germany, analyses in details, by simulation and experiments, catacaustic effects produced by the sunlight reflected from some curved façades. (Vollmer and Möllmann 2012).

## Notes

### Acknowledgments

Authors express their gratitude to the Ministry of Science and Technological Development of the Republic of Serbia, for supporting our research, which is the part of the project No. TP 36008 entitled: Development and application of scientific methods in design and building of high-economic structural systems by application of new technologies.

## References

- Baltrušaitis, J. 2004.
*Anamorfosi o Traumaturgus opticus.*Milano: Adelphi Edizioni.Google Scholar - De Comité, F. 2010. A General Procedure for the Construction of Mirror Anamorphoses. In:
*Bridges Pécs*—*Mathematics, Music, Art, Architecture, Culture*, Bridges Pécs Conference Proceedings 2010, ed. George W. Hart and Reza Sarhangi, 231–239. Pécs: Tessellations Publishing.Google Scholar - De Comité, F. 2011. A New Kind of Three-Dimensional Anamorphosis. In:
*Bridges Coimbra*—*Mathematics, Music, Art, Architecture, Culture,*Bridges Coimbra Conference Proceedings 2011, ed. Reza Sarhangi and Carlo Séquin, 33–39. Coimbra: Tessellations Publishing.Google Scholar - Füsslin, G. and E. Hentze. 1999.
*Anamorphosen*. Stuttgart: Füsslin Verlag.Google Scholar - Goddijn, A. P. 1992.
*Anamorphosen*. Frankfurt am Main: Explora Museum.Google Scholar - Hamngren, H. 1981. My Anamorphoses: Types That Produce Three Kinds of Images in Circular Cylindrical Mirrors.
*Leonardo*14 (3): 198–201.Google Scholar - Heeke, M. 2003. Anamorphosen und Luftspiegelungen aus dem Blickwinkel des Physikunterrichts. http://didaktik.physik.fu-berlin.de/~nordmei/PhysikKunstMusik/Software/Anamorphosen.pdf (accessed 22 November 2009).
- Hickin, P. 1992. Anamorphosis.
*The Mathematical Gazette*76 (476): 208–221.CrossRefGoogle Scholar - Hunt J.L., Nickel, B.G., and C. Gigault. 2000. Anamorphic Images.
*American Journal of Physics*68 (3): 232–237.Google Scholar - Leeman, F. 1976.
*Hidden Images*. New York: Harry N. Abrams, Inc., Publishers.Google Scholar - Nicéron, J. F. 1663.
*La Perspective Curieuse […]: Avec L’optique et la Catoptrique du R. P. Mersenne*. Paris: Chez Jean Du Puis, http://www.bvh.univ-tours.fr/Consult/consult.asp?numtable=B372615206_7163&numfiche=225&mode=3&ecran=0&offset=444 (accessed 15 October 2013). - Soddu, C. 2010. Perspective, a Visionary Process: The Main Generative Road for Crossing Dimensions.
*Nexus Network Journal*12 (1): 33–46.Google Scholar - Vollmer, M. and K-P. Möllmann, 2012. Caustic Effects due to Sunlight Reflections from Skyscrapers Simulations and Experiments.
*European Journal of Physics*33: 1429–1455.Google Scholar