# Anamorphic Projection: Analogical/Digital Algorithms

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## Abstract

The study presents the first results of a wider research project dealing with the theme of “anamorphosis”, a specific technique of geometric projection of a shape on a surface. Here we investigate how new digital techniques make it possible to simplify the anamorphic applications even in cases of projections on complex surfaces. After a short excursus of the most famous historical and contemporary applications, we propose several possible approaches for managing the geometry of anamorphic curves both in the field of descriptive geometry (by using interactive tools such as Cabrì and GeoGebra) and during the complex surfaces realization process, from concept design to manufacture, through CNC systems (by adopting generative procedural algorithms elaborated in Grasshopper).

## Keywords

Anamorphosis Anamorphic technique Descriptive geometry Architectural geometry Generative algorithms Free form surfaces## Introduction

Anamorphosis^{1} is a geometrical procedure for representing shapes on surfaces: the projected image is only recognizable from a predetermined point, and appears deformed from all other positions.

This specific topic—which dates back to the fourteenth century, but was refined in its scientific rigor in Europe between the fifteenth and sixteenth centuries—offers many opportunities for interdisciplinary research in the field of geometry applied to architecture, figurative arts, visual perception, history of architecture and industrial design.

The application of the anamorphic process requires mastery and skill in the use of different techniques of the science of representation aimed at finding the rule for deforming and regenerating represented images. Our work in progress continues the research conducted by researchers of the Drawing School of the Engineering Faculty at the University of Palermo (Inzerillo 2008), which has always promoted experimentation and innovation in of the application of descriptive geometry to topics based on the relationship between geometry and architecture/engineering.

Here we investigate new digital techniques through which the applications of anamorphisms are simplified even in the case of projections on complex surfaces. This paper is organized as follows. Section 2 gives a short overview of the most famous historical and contemporary applications. Section 3 we introduce some possible approaches that allow management of the geometry of anamorphic curves in the field of descriptive geometry (by using interactive tools such as Cabrì and GeoGebra). Section 4 presents experimental outcomes dealing with the process of producing complex surfaces from concept design to manufacture through CNC systems (by adopting generative procedural algorithms elaborated in *Grasshopper*). We close the presentation with a discussion of the advantages of the method and directions for future work.

## From the Origin to Digital Representation

*Pala di Brera*by Piero della Francesca (1472),

*The*

*Ambassadors*by Hans Holbein the Younger (1533), and the collection of unpublished anamorphic drawings of the

*Codex Atlanticus*by Leonardo da Vinci (1478–1519). The technique, later referred to as optic anamorphosism, arose as a consequence of an intuition that re-elaborates the theoretical foundations of perspective, already codified between the fourteenth and fifteenth centuries by eminent artists such as Ambrogio Lorenzetti, Filippo Brunelleschi, Masaccio, Leon Battista Alberti, and Albrecht Dürer. The optic artifice allows projection of a shape onto a surface such that it can only be read as its real, undistorted form from a specific point of view, placed wherever the author decides. To realize an optic anamorphism to the artist must carry out a mechanical-optical procedure that allows him to transfer the preparatory drawing from the paper to the wall; once realized, the anamorphic image can be read, in its real shape, from a specific point without using special devices (Fig. 1).

*La perspective curieuse*(Nicéron 1638; Lomazzo 1585; Schott 1657) (Fig. 2).

*Sacra Conversazione*(1475), where the object of anamorphosis is a suspended egg that turns in a sphere if observed from a particular point of view (Fig. 3); the German painter Hans Holbein the Younger, in his famous painting

*The Ambassadors*(1533), in which the distorted shape in the bottom of the frame transforms into a skull (Fig. 4); the two Friars Minor confreres, Jean François Nicéron and Emmanuel Maignan, in the frescos of the galleries of the cloister in the monastery of SS. Trinità dei Monti depicting San Giovanni (1639–1640) (Fig. 5) and San Francesco di Paola (1642) (Fig. 6).

In both these latter cases, the visitor placed at the center of the galleries observes two scenes, that are apparently confused and reached by landscape elements (lakes, rivers, grape harvest scenes, mountains, palm groves and exotic flowers, vessels and far away cities). Furthermore, in both frescos, a big olive tree is used as perspective background.

When the observer arrives at the end of both galleries he discovers, with great surprise, an astonishing effect that is perceived only from that view point: he can observe the optically correct image of the represented saint from the *punctum* *optimum*, planned by the author (De Rosa and Cristian 2012: 595–602).

Scientific research in the Renaissance era in the perspective field, in projective geometry and in optics helped to determine, in the Baroque period, new illusionistic spatial effects and mastery of the tricks of stage design (Accolti 1625).

Between the sixteenth and eighteenth centuries, the kinds of painting named “quadraturism” and “trompe-l’oeil” arose and developed; these techniques aimed at the composition of faux architectural perspectives realized at full scale to create illusory effects on walls and ceilings.

The painters used the concepts of anamorphosis in the creation of their works, with extreme skill and mastery, creating wonderful examples on the curved surfaces of apses and niches, over large areas of aristocratic salons or on the articulated vaults of churches.

We can cite, among the examples of anamorphosis on large scale, the apse of the church of Santa Maria in San Satiro in Milan created by Donato Bramante (1483), the corridors of Palazzo Spada in Rome by Francesco Borromini (1540), and the trompe-l’oeil scenography of Palladio’s Teatro Olimpico in Vicenza designed by Vincenzo Scamozzi (1584).

Beyond the famous and historical scientific experiments in optical-perceptive field, in our own day interest in this aspect of representation has been reawakened in the fields of media and advertising, industrial design, decorative art, theatrical scenic design, architectural and urban planning fields, where we find several examples that demonstrate the fertility of experimentation. One of the practical uses of anamorphic effects in daily life is found in text written on highways where an optical correction that allows drivers to read the road signs in the correct proportions. Another example is found in the advertising texts drawn on sporting stadiums, which are distorted on the playing surface but are clearly visible in correct proportion when viewed from the angle of the television cameras.

The development of informatics technologies in graphical-digital representation contribute to experiment the anamorphosis techniques creating new magnificent and engaging scenarios able to surprise and touch the audience. Nowadays companies specialized into the creation and managing of shows for events and special occasions use innovative techniques such as digital anamorphic projection and special effects.

^{2}

## Digital Dynamic Applications of Descriptive Geometry

Anamorphosis represents an interesting field of investigation that includes numerous theoretical studies in both analytic-geometrical and graphical-geometrical keys.

The constructions proposed here explicate the potential of anamorphic procedures applied to a shape projected on plane, ruled or polyhedral surfaces by using dynamic geometry software such as Cabrì Géomètre and GeoGebra.

The use of these tools allowed us to strengthen the interaction between the theoretical-conceptual and figurative component in applications of descriptive geometry, deepening cultural and educational knowledge in the field of the science of representation. The commands allow us to analyze the intrinsic relations between geometric entities, through the manipulation (moving, rotations, enlargements) of control points during the constructions. Thus, it is possible to interactively validate an initial hypothesis, and to presuppose and verify the properties and relations that remain unchanged or vary during geometrical transformations.

Compared to the traditional tools for drawing and CAD, the interactive and dynamic interface allows us to modify the mutual position of the constrained geometrical entities, setting up several case studies. Furthermore, we can highlight important and effective interactive procedures inside these software programs which are useful to comprehend the sequence of graphical operations that generate the geometrical shapes, such as step-by-step reproduction of graphical procedures, visualization of the construction protocol by means of the list of executed commands, and the construction of “geometrical loci” of points that satisfy determined conditions (Di Paola 2010).

### Anamorphosis on a Plane Surface

The shape in true form and dimension is represented on picture plane (in earlier times, gridded canvas was used), placed at a known distance from the observer, orthogonal to the wall surface. Once made, the anamorphic picture is in inverse perspective (by means of auxiliary tools such as taut wires that materialize the visual rays). In order to observe the drawing in the correct proportions, the observer must be positioned obliquely to the wall, placing himself in the same position of the point of view of construction. This optical phenomenon occurs because it materializes the picture projection that was ideally positioned where the painter had placed the drawing in true form before projecting it on the wall (Di Lazzaro and Murra 2013).

### Specular Anamorphosis on Cylindrical Ruled Surface

Given a cylinder whose height and diameter of the circular directrix are known, and which is placed on the horizontal picture plane π, we define V as the observer’s point of view. We then represent an auxiliary vertical plane β that contains the cylinder axis and whose direction is orthogonal to the picture plane π. On this plane β we can trace a curve (a circle in the example). A physical model would consider a cylinder made of a highly reflective material and a chosen virtual curve. Having chosen a point P belonging to the assigned curve, we trace the visual ray λ that connects V with P and intersects the cylinder in S and the picture plane in the virtual point Q′. From S′, projection of S on π, we draw the tangent to the cylinder directrix, t_{p}; this last is the trace of γ vertical plane tangent to the cylinder and passing for S.

According to the laws of reflections, the slope of the reflected ray λ* can be determined considering the point V*, symmetric of V respect to γ; the intersection of λ* ray, V*S, with picture plane determines S*. We can observe that the straight lines that connect the points VQ′ and V*S* are symmetric in relation to γ plane and their points of intersection with π, Q′ and S*, are in axial symmetry with respect to t_{p}. Furthermore, the rectangle triangles VF′_{nπ}Q′ and PP′Q′ are similar, hence we have the relation h: k = F′_{nπ}Q′: P′Q ′, where F′_{nπ} is the vanishing point of the perpendicular straight lines respect to π (Martinez et al. 2008).^{3}

If we move P on the virtual circumference, it describes two geometrical loci: c*, which represents the anamorphic projection of the assigned curve and c**, the skew curve belonging to the cylindrical surface. To see the curve without any distortion, the observer must look at the anamorphic curve reflected on the cylindrical mirror from V, the point of view; thus, the virtual and the skew curve will overlap.

It is possible to construct the anamorphic curve directly on the picture plane π. It is sufficient to revolve the plane β into the picture plane π around the straight line t_{β}, determining the point (P) and the point (V), thus establishing an homology with center (V), axis t_{β} and corresponding points Q′(P). Given a point P, it locates the point P′ orthogonally on π, and then, through homological construction, it determines Q′ and finally S*, through axial symmetry respect to t_{p}.

### Applications of Anamorphosis in Descriptive Geometry

#### Anamorphosis in Perspective

We represent directly in projection two anamorphic figures (a eight-pointed star and a circle), perceived from the chosen point of view V in real shape (that is, not deformed) and belonging to a ruled surface σ.

To construct this surface, we chose any vertical plane γ, turning over the outline t_{γ} on π; the generatrix q is determined by arranging sinusoidal arches in sequence along it. The surface is generated by a vertical line translated along the generatrix q. The surface and the plane are then turned over on π, resulting in the projection of the curves and of σ on the plane γ. The ruled surface is developed on the plane γ by determining true size and shape of the surface and the two anamorphic figures.

The distance Δ of V from the picture plane π, corresponds to the distance from V to F’_{nπ}, where F’_{nπ} is the vanishing point of the perpendicular straight lines to π (Inzerillo 2008, 2012). The vertical oblique plan γ is uniquely determined by marking the trace lines t_{γ} and the vanishing lines f_{γ}. The ruled surface q, with amplitude and wavelength assigned, rests over γ.

In this paper, we leave out the description of the geometric-constructive steps that lead to the determination of the overturning of the wavy surface. It is observed that, for development of the surface, the known relationship between the length of the elliptic curve and the sine wave has been exploited.

The development of the latter curve is calculated using the formula of the Indian mathematician Ramanujan that determines with approximation the length of the ellipse as the development in series of an integral curvilinear (Villarino 2008).

#### Anamorphosis in Direct Axonometry

In order to define the axonometric projection layout, we have to trace two orthogonal straight lines on the sheet: the z′ vertical straight line (projection on the picture plane of the z axis of the three-orthogonal triad) and the horizontal straight line as t_{xy} (intersection of the xy plane with π picture plane). Then on the z′ axis we define O′ [projection of O, the origin of (x, y, z) triad, on π picture plane] and (O) (turned over of O on the π picture plane) by setting up an homological ratio H(O)/HO′ (H is the intersection between xy maximum slope straight line and t_{xy}) (Inzerillo 2012: 50–55). When we turn over the xy plane on the picture plane we establish an affine orthogonal homology whose center is at infinity in the direction O′ (O), the axis is the straight line t_{xy} and a pair of corresponding points is O′ (O). Once the cone is built (Inzerillo 2012: pp 69–73), the anamorphic letter is obtained by drawing, on the overturned plane, the real shape and size of the letter (in our example, a W), in a position that it is inside the cone directrix. Choosing a point P on the letter, we find (P) on the overturned xy plane, then we trace the generatrix of the cone where (P) lies and, following homological procedures, we find the corresponding generatrix of the cone in projection and the corresponding point P′. In GeoGebra, the anamorphic letter on the ruled surface is determined by the geometrical locus described by point P′ according to its constraint to P*.

#### Anamorphic Curve by Two Projection Views

^{4}

## Anamorphic Curves in Freeform Architecture

It is now common practice in contemporary architecture to design and construct buildings whose architectural expression relies on complex forms. The implementation of the initial concept can be carried out by two methods: (1) by using continuous curve surfaces, described by mathematical function (NURBS, *Non*-*Uniform Rational B*-*Splines*), and executing them as either a single solution or as a composition of several panels suitably molded (according to the characteristics of the material used); (2) by using “polygon mesh” polyhedral surfaces, composed of small polygonal plane faces that contribute, to the definition of a variously articulated form (Wallner and Pottmann 2011).

For some years now a convergence of diverse sectors has resulted in a new field of study called “architectural geometry” (Pottmann et al. 2007). This discipline involves all those aspects concerning complex geometries used in designing and constructing buildings. The panelling of complex surfaces is one of the hottest topics in architectural geometry. According to a classification of the major trends (Tonelli 2012), there are two discretisation methods. The first, based on polygonal plane panels comprises, among others: triangular meshes (which, however, show issues related to an high number of links, resulting in a heavy structure, and in case of glazed areas, a reduced transparency), quadrilateral meshes (which simplify and lighten the structure, but do not automatically guarantee the panels’ flatness, requiring a finishing of the calculations methods) and hexagonal and polygonal hybrid meshes (generated by even more complex algorithms). The second method is based on ruled surfaces, and involves the acquisition of concept surfaces and a consequent geometric analysis, aimed at a possible approximation with a set of straight lines, which would make it possible to create portions of concept surfaces using ruled surfaces (Flöry and Pottmann 2010; Eigensatz et al. 2010).

The case studies that appear here show the anamorphic curve as if it were meant to be perceived from a privileged point of view, such as a raised platform, a nearby building or a specific position below the roof. Curves may represent signs, used to identify the building or some of its portions, passage openings or skylights, or even for decoration.^{5}

While in applications of regular anamorphosis, discussed earlier, the figures were reproduced on flat surfaces or, at least, on surfaces with well-known characteristics, such as small-scale cones and cylinders, applications in contemporary architecture presuppose other dimensions and surfaces, with different levels of geometrical complexity. Even in cases of polyhedral surfaces, the high number of attitudes may necessitate a laborious procedure for creating the anamorphic images through the use of standard methods, given that would be necessary to deal with each element separately depending on the attitude of the projection plane. In these cases it is essential to use procedures that allow the greatest possible degree of automation in the process of creating the projected images, while also making interactive single-element interrogation possible during each phase of design: this purely graphic-geometrical approach requires procedural generative algorithms in *Rhinoceros* environment through *Grasshopper* plug-in for visual programming. This plug-in makes it possible to generate a workflow by linking several components allowing input data elaboration through geometrical, mathematical or Boolean processes, typical of mesh and NURBS modeling. It also allows data interrogation for an accurate verification of their geometrical properties (curvature, continuity, etc.). The general procedure involved the acquisition of a concept surface and its panelization by the two most popular methods used by architecture professionals (Tonelli 2012): discretization by polyhedral polysurfaces, composed of triangular faces, and approximation by surfaces with known geometrical properties and simple geometrical genesis.

### Case Study 1: Projection on a Freeform Surface Divided into Triangles Along the uv Directions

### Case Study 2: Projection on a Freeform Surface Anyhow Divided in Triangles

### Case Study 3: Projection on a Surface Approximated by Portions of Double-Curved Surfaces

The preliminary design can feature, as a concept, a NURBS surface to be made either of materials with a certain level of reflectivity (such as a metal or opaque glass cladding) or with transparent materials, showing refraction phenomena. If the surface is particularly extended or complex, it is extremely expensive or even impossible to manufacture it as a single piece; instead, applying one of the methods previously described, approximating it to a polyhedral surface, can produce an unsatisfactory overall outcome, thanks to the different directions of the normals to the faces (Baglioni 2012). In fact, the surface would show discontinuous reflection and refraction phenomena, quite different from the outcome predicted by the designer (Pottmann et al. 2008).

^{6}

*Rhinoceros*environment, is imported in

*Grasshopper*and compared with the known surface through an evolutionary approach, made possible by the Galapagos component, by David Rutten. It uses a sequence of numerical sliders as input, the “gene pool”: each slider describes a certain value related to the comparison surface, such as the coordinates of the vertices of the hyperbolic paraboloid or the radii of the torus; an addition three sliders present on each surface describe its angle of revolution around the three coordinate axes, in order to maximize the chances for a better correspondence. In other words, the sliders describe an

*n*-dimensional “landscape”, where

*n*is the number of the “genes” in the gene pool. The component is also linked to an output value, the “fitness”, dependent on the sliders, which can be maximized or minimized depending on the purpose of the evolutionary simulation. For the case study at hand, the surfaces, obtained bycutting of the original NURBS, were compared to the approximation surfaces, calculating, through a high number of samples, the mean deviation: this value was used as the fitness of the component and prearranged for minimization. The evolutionary approach initially entails the attribution of a random sequence of values to the sliders, which will generate certain values of fitness. The the determination of the initial values is followed by a sequence of iterations (so-called “generations”) resulting in an elimination of those combinations different from the desired outcome and the selection of the best ones, which are cross-referenced for the following generation. The output values of the “sons” are contained in the range of the values of the “parents”, thus approaching the best possible value (Fig. 39).

The task, however, may not allow a unique solution or any solution at all. It is possible to establish some regulation values which will halt the simulation, deciding, for example, that it must stop when the values found from a generation to the next differ by a certain percentage from the “parent” values after a certain number of iterations; the only limitation to this approach is the calculator’s computing capacities.

^{2}, we determined, in the best cases, a deviation between the concept NURBS and the approximating one with a maximum of 0.2 cm and a nearly-zero minimum, and a mean deviation on the panel’s edge of 0.1 cm, with a maximum of 0.2 cm and a nearly-zero minimum. As an additional control method, we carried out an accurate analysis of the deviation of the surfaces extended over the entire panel, which allowed a prompt visual feedback of the portions that showed a greater or smaller deviation compared to the concept (Figs. 41, 42).

## Conclusions and Future Work

Our study has more deeply developed techniques for digital anamorphic projection. The different case studies led us to unexpected and surprising results: we demonstrated that the application of anamorphic principles onto complex architectural surfaces has been slimmed down and facilitated by using digital techniques and procedural algorithms.

The “3D architectural mapping” technique, among the applications most used today, provides evidence that anamorphic projections constitute one of the new frontiers in multimedia communication.

The graphical determination of reflected anamorphic images according to methods of descriptive geometry is an attractive field of research open to new exploration, particularly in the presence of different types of quadric surfaces: furthermore, the anamorphic projection of forms and shapes onto free-form complex surfaces is a very important topic in architectural geometry, as part of more general issues relating to the monitoring, investigation and analysis by means of geometric computer modelers.

The case studies of anamorphic curves we have proposed were suggested by real situations of architectural building elements with complex covering surfaces. By using the methodology explained here it is possible to achieve a balance between the costs of execution and the geometrical-formal adherence to the design model. Future developments of our research will aim at refining the design of algorithms useful for optimizing the production process with CNC systems, and we plan to deepen our knowledge of this topic by cooperating with specialized companies in the field.

## Footnotes

- 1.
Neologism of the seventeenth century, from the Greek άναμÓρϕωσις (“reforming”), derivation of άναμορϕÓω (“to form again”).

- 2.
The realization of a 3D mapping project forecasts an advanced hardware/software equipment (3D modeling, liquid simulation, creation of special effects, high bright projectors) and takes advantage of knowledge that span different scientific fields.

- 3.
Unlike our proposed method, the one used by seventeenth-century treatise writers is based on transformation theory: the drawing is divided into some square cells that give a system (in this case according to cylindrical symmetry) of distorted cells, up to becoming parts of a circular crown.

- 4.
The algorithm used exists in the command set “Tools curve” within the work environment of the software

*Rhinoceros*. The command “curve from two views” creates a third curve from two planar curves in different construction plans. The new curve is the same as both curves when viewed from their original construction plans. - 5.
Our ongoing research also deals with the definition of the necessary procedures for texture anamorphism, a matter not treated here but which will be the subject of further developments.

- 6.
In particular, the ruled surfaces have a curvature value equal to zero on each of their points, given that it is always possible to determine a direction where the section is a straight line –in other words, where the osculator circle has infinite radius, thus the curve has a null curvature.

## References

- Accolti, Pietro. 1625.
*Lo inganno degli occhi*: Firenze (free e-book. http://www.books.google.com). - Baglioni, Leonardo. 2012.
*I poliedri e le tecniche di tassellazione delle superfici continue: un nuovo punto di incontro*. In:*Attualità della Geometria Descrittiva*, eds. Carlevaris Laura, De Carlo Laura, Migliari Riccardo, Rome: Gangemi editore. pp. 297–314.Google Scholar - Baltrušaitis, Jurgis. 1969.
*Anamorfosi o magia artificiale degli effetti meravigliosi*. In: Bertolucci, P. (traduzione a cura di), Milano Adelphi.Google Scholar - De Rosa, Agostino, Alessio Bortot, Cristian Boscaro, Cosimo Monteleone, and Elena Trevisan. 2012. Memoria e oblio. Scoperta e rilievo digitale dell’anamorfosi murale di J.-F. Nicéron.
*Atti 16° Conferenza Nazionale ASITA*—*Fiera di Vicenza 6*-*9 novembre 2012*: 595–602.Google Scholar - Di Lazzaro, Paolo, and Daniele Murra. 2013.
*L’Anamorfismo tra arte, percezione visiva e “Prospettive bizzarre”*. Roma: ENEA.Google Scholar - Di Paola, Francesco. 2010.
*Le Curve di Apollonio. Tradizione ed Innovazione nei processi risolutivi.*Roma: Aracne.Google Scholar - Eigensatz, Michael, Martin Kilian, Alexander Schiftner, Niloy J. Mitra, Helmut Pottmann, and Mark Pauly. 2010. Paneling architectural freeform surfaces.
*ACM SIGGRAPH*2010(45): 1–10.CrossRefGoogle Scholar - Eigensatz, Michael, and Alexander Schiftner. 2012: http://www.glassonweb.com/articles/article/765. Accessed 10/2013.
- Flöry, Simon, and Helmut Pottmann. 2010. Ruled Surfaces for Rationalization and Design in Architecture.
*LIFE information. On Responsive Information and Variations in Architecture. Proc. ACADIA 2010*: 103–109.Google Scholar - Heumann, Andrew. 2011. http://heumanndesigntech.wordpress.com/2011/03/26/unfolding-surface-strips/. Accessed 10/2013.
- Inzerillo, Michele. 2008.
*Fondamenti e applicazioni di Scienza della Rappresentazione*-*Geometria del Disegno*. http://www.inzerillomichele.eu. Accessed 10/2013. - Inzerillo, Laura. 2012.
*Assonometria e futuro*. Roma: Aracne.Google Scholar - Lomazzo, Giovanni Paolo. 1585.
*Trattato dell’arte della pittura, scoltura et architettura*. Milan: per Paolo Gottardo Pontio.Google Scholar - Martinez, Annalisa, Marcello Pergola, Marco Turrini, and Carla Zanoli. 2008.
*Perspectiva artificialis*, the digital catalog of the conference sponsored by the Association Macchine Matematiche, Cremona, Museo Ugulani-Dati.Google Scholar - Nicéron, Jean François. 1638.
*La perspective curieuse, ou magie artificielle des effets merveilleux de l’optique par la vision directe, la catoptrique, par la réflexion des miroirs plats, cylindriques & coniques, la dioptrique, par la réfraction des crystaux*: Chez la veufue F. Langlois, dit Chartres, 1652, Public Library of Lyon, Paris (free e-book). http://www.books.google.com. - Piacentino, Luigi. 2013. http://www.giuliopiacentino.com/weaverbird/. Accessed 10/2013.
- Pottmann, Helmut, Andreas Asperl, Michael Hofer, and Axel Kilian. 2007.
*Architectural Geometry*. Benteley: Bentley Institute Press.Google Scholar - Pottmann, Helmut, Alexander Schiftner, Pengbo Bo, Heinz Schmiedhofer, Wenping Wang, Niccolo Baldassini, and Johannes Wallner. 2008. Freeform surfaces from single curved panels.
*ACM SIGGRAPH*2008(76): 1–10.CrossRefGoogle Scholar - Schott, Gaspard. 1657.
*Magia universalis naturae et artis*Partis I Liber III Pars I: Würtzburg (free e-book). http://www.books.google.com. - Tonelli, Davide. 2012.
*Sinossi sull’ingegneria delle forme libere*. In G.R.I.F.F.-Free Forms Research Group at the University of Pisa founded and coordinated by Maurizio Froli. http://www.dic.unipi.it/davide.tonelli/files/Sinossi.pdf. Accessed 10/2013. - Villarino, Mark B. 2008.
*Ramanujan’s Perimeter of an Ellipse*, Escuela de Matematica, Universidad de Costa Rica. http://arxiv.org/abs/math/0506384v1. Accessed 10/2013. - Wallner, Johannes, and Helmut Pottmann. 2011. Geometric computing for freeform architecture.
*Journal of Mathematics in Industry 1**4*: 1–18.Google Scholar