Origins

In this section, we explore the origins, history and sources of perspective.

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Introduction

A growing historical awareness means that there have been many more books on Renaissance perspective published in the twentieth than in the fifteenth and sixteenth centuries combined. Renaissance perspective focussed on geometrical space. Twentieth century versions of perspective focus on visual space and hence entail a variety of alternative picture planes: spherical, cylindrical and even polygonal. New goals of art have influenced these developments as have computer algorithms, cartographical methods, new awareness of illusions, the development of virtual reality and fractals.

Arnason, in his standard History of Modern Art described Renaissance art as “imitations of nature” and claimed that:

Perhaps the greatest revolution of early modern art lay in the abandonment of this attitude and the perspective technique that made it possible.

Arnason in History of Modern Art, 1998.

Such claims have led to a widespread view that interest in perspective died in the twentieth century. Even a quick glance at the number of books published on perspective since the fifteenth century reveals that this is not the case (fig. 1).

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Time Period	Perspective Books
1400 -1499	1
1500 -1599	456
1600 -1699	732
1700 -1799	849
1800 -1899	2714
1900 -1989	2801

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This partly reflects a growing historical awareness. There have been more editions of fifteenth and sixteenth century authors on perspective in the Twentieth Century than ever before, including first critical editions of Alberti, Filarete, Francesco di Giorgio and now Piero della Francesca.

Antiquarian and nostalgic interest comprise but one aspect of modern concern for perspective. Partly because there are now a number of different goals of art, artists are exploring a whole range of alternative methods. Some of these experiments relate perspective to what might be termed a personalized optics. Others reflect a new awareness of transformations brought about by mapping, computers and virtual reality. Meanwhile the advent of fractals has raised but not answered new philosophical problems concerning the parameters within which perspective is valid. It will be useful to review briefly the context for and characteristics of Renaissance perspective before considering each of these topics in turn.

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Renaissance Perspective

In Antiquity, Euclid’s Optics focussed on how things appear to the eye: a concern which is now termed psychological optics. Euclid’s treatise also contained four propositions on surveying. This introduced a quantitative dimension into an otherwise qualitative discussion.

In the course of the Middle Ages, particularly thanks to thinkers such as Ibn al-Haytham (Alhazen) in the Arabic tradition, this concern with potentially quantitative, objective aspects of sight gained greatly in importance. By the 1270’s, when Witelo wrote his great compendium on optics, (which was termed perspectiva in Latin), he was concerned not just with the eye but also with instruments “for the certification of sight” such as quadrants and astrolabes.

A nexus thus evolved which linked optics, surveying, astronomy and instrumentation. This concern with certification of sight led to practical demonstrations which involved representation. Brunelleschi’s demonstration (of perspective) was in this tradition. In the fifteenth and sixteenth centuries this became known as practical (linear) perspective, for which Euclidean optics was assumed to provide the theory (incorrectly).

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Renaissance or One-Point Perspective

Renaissance or One-Point Perspective is thought to have been devised about 1415 by Italian Renaissance architect Filippo Brunelleschi and later documented by architect and writer Leon Battista Alberti in 1435 (Della Pittura).

One-Point Perspective is a type of Graphical Perspective (ref. perspective drawing) created on a 2D plane that employs a central (or single) viewpoint for the depicted scene, (ostensibly) with one Vanishing Point in the distance which is placed on the Horizon Line, and from which everything in the drawing is set out (or converges). 

Alan Radley

Linear Perspective

Linear Perspective is sometimes taken to be identical to One-Point (1-point) Perspective in meaning and refers to an approximate representation, generally on a flat surface, of an image as it is seen by the eye. As an alternative Two-Point (2-point) and Three-Point (3-point) perspective types are also sometimes referred to as Linear Perspective. All forms of Linear Perspective (1,2, 3-point) are recognised because they map rectilinear (straight) lines existing in the orthogonal and lateral directions within the object space to a correspondingly rectilinear (straight) form in the image space. This is contrasted with curved lines in Curvilinear Perspective (for example).

Alan Radley

Perspective Nomenclature

Renaissance Perspective is also known as: One-Point, Central (linear form), Front, Linear, Albertian, Brunelleschian Perspective; and is sometimes referred to generally as: Plane, Artificial, Practical and Academic Perspective (although the latter names are sometimes used for other types/classes of perspective).

Central Perspective (Central Projection)

In a strict mathematical sense, the term Central Perspective refers to all forms of Graphical Projection in which an object/scene is ‘viewed’ from a frontal direction or along the central line of the projection (or line-of-sight); whereby objects in the scene are scaled according to one or more vanishing point locations. Accordingly, the depicted scene appears much as it is produced by the eye, the camera etc, and in terms of what is known as Central Projection. Central Perspective, therefore, mimics what the eye sees when it looks at a three-dimensional scene from a fixed vantage point. In a strict mathematical sense, Central Perspective includes One-Point, Two-Point, Three-Point projections, and Curvilinear Perspective. All of these projection methods are termed Central Perspective because in each case, the scene is viewed from a frontal direction but with scene rectilinear lines in the orthogonal, lateral and vertical directions (in object space) appearing to recede towards one or more vanishing points (in the image space).

Central Perspective creates the impression that the represented objects behave (visually) as in reality under visual conditions. It is important to note a common misconception in relation to the term Central Perspective as it is sometimes incorrectly employed; whereby it is equated to One-Point Perspective in the mistaken belief that the name derives from the use of a single ‘central’ vanishing point (i.e. One-Point Perspective). However, Central Perspective refers (in a strict mathematical sense) to any form of perspective representation that depicts a scene in a frontal fashion and employs one or more vanishing points to create a systematised and homogeneous image space. Another way of stating this fact is to say that Central Perspective is a form of Optical or Technical Perspective that employs frontal projection plus vanishing points to create a ‘uni-directional’ image or view of a three-dimensional scene using logically consistent projective principles.

Alan Radley

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Perspective, Representation and Vision

Links between optics and representation go back at least to Antiquity. Greco-Roman efforts were linked with scenography (skaenographia). This resulted in a sense of depth usually through convergence along an axis of points (fish-bone perspective) and occasionally towards a central vanishing point. Fictive columns were typically used to close a visual space within a distance of less than 50 meters. Although seemingly realistic, perspective was linked with imitation: with (visual/painted) narratives rather than with physical reality. 

While the mediaeval period largely abandoned geometrical depiction of pictorial space, it introduced a series of projection methods for systematic representation of space, notably, planisphere and astrolabe projections, as well as experimenting with the effects of camera obscuras. Such projections were a starting point of late 14th century scholars such as Prosdocimo da Beldomandi at Padua. 

Renaissance perspective, while vaunted to be a rebirth of ancient methods, transformed its methods and goals, from imitation of general things to matching of specific objects. Brunelleschi’s first demonstrations using the Baptistery of San Giovanni and the Piazza dell Signoria in Florence are famous examples. Paradoxically, this did not lead immediately to a copying of the natural world. It led to idealised views (e.g. Baltimore, Berlin and Urbino panels).

In Florence, the method was applied primarily to marquetry (intarsia) typically entailing regular solids, musical instruments and some idealised scenes. In the 16th century, when the method spread to Germany, Nürnberg goldsmiths and jewellers were particularly enthusiastic about depicting regular and semi-regular shapes. By contrast, French and Netherlandish artists depicted ruins and idealised buildings. 

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One of the unforeseen consequences was the development of views (vedute). Some were effectively topographical studies in the manner of surveyor’s maps and pictures. These led to neo-classical art, with seemingly realistic architecture. Others were completely fictive (capriccios or architectural fantasy). Some, such as Ruisdael’s landscapes of Norwegian waterfalls convincingly depicted places he/they had never visited. Others, such as Turner’s paintings, showed events in ancient Carthage or Rome at which he could not possibly have been present. Indeed, this led to a whole school of Romanticism, which took man to the peaks of mountains, the centre of storms and other places of danger at the limits of the imaginable. 

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A veduta (Italian for “view”; plural vedute) is a highly detailed, usually large-scale painting or, more often, print of a cityscape or some other vista.

This genre of landscape originated in Belgium, where artists painted vedute as early as the 16th century. In the 17th century, Dutch painters produced detailed and accurate cityscapes and landscapes. As the itinerary of the Grand Tour became standardized, vedute of familiar scenes like the Roman Forum or Grand Canal were produced for aristocratic Englishmen. By the mid-18th century, Venice became the centre of the painters of vedute who were referred to as vedutisti. In the later 19th century, topographical accuracy became the focus of this movement, which was satisfied by painted, and later photographed, panoramas.

Alan Radley

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Piero della Francesca and Leonardo da Vinci were among the pioneers in exploring anamorphic effects, cases where extreme positions of a picture plane relative to objects caused dramatic distortions. Leonardo was also one of the early artists to explore the effects of applying the new method to cylindrical and spherical surfaces. In the 16th century, this was further explored by Carlo d’Urbino and Cigoli. In the 17th century, this theme became a regular feature of printed books: e.g. Dubreuil, 1640s and Abraham Bosse, 1648. One reaction was to avoid near distance cases where distortion was extreme. This led to a narrowing of the field of vision. 

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Meanwhile, the eighteenth century increasingly applied perspective to gardens, and large spaces stretching as far as the horizon. Some spaces were calculated to make distances look closer, others to make it look further away. Space became something to be manipulated: the eye a sense to be tricked. This often led to a widening of the field of vision, and occasionally to a narrow peephole as in Piranesi’s gate of Knights of Malta in the Santa Sabina hill in Rome. 

The 19th century saw new attention to properties of geometry and optics. Von Helmholtz explored the possibility that optical space within the retina might entail Riemannian rather than Euclidean space. In the latter 19th century, theoreticians such as Herdman (1863) outlined (cylindrical) curvilinear perspective. 

Rhetorically, there has been an attack on perspective in the 20th century, to the extent that some have spoken of the death of perspective. In practice, there have been more publications than ever, along with a radical increase in alternative methods. Jobin (1932) was interested in curvilinear methods in connection with skyscrapers. Barre & Flocon (1967) were interested in where it corresponded to perception of the eye. 

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Painters such as Termes have compared 1, 2, 3, 4, 5, and 6-point perspective on spheres. Architects such as Correia (2015, 2015) have developed software that allows users to go from linear to cylindrical and spherical perspective as part of a spectrum of choices. Solutions include a simple perspectival machine by the twins Ryan and Trevor Oakes (2011). More fundamental is the work of Paul Debevec (1996, 1998), whose reflection mapping has led to new ways to record both real and fictive spaces and whose light stage includes a total-surround solution to recreation of spaces, which is being adapted in the field of virtual actors in order to bring no longer living actors back to the life on the stage. The recent interest of the Fovolab group in perspective methods (Barrett et al, 2016) in Britain relates to a richer tradition than might be immediately apparent.

Today, new perspective theories, methods, and forms are being applied to solve multiple problems within the human vision, modelling, and representation arenas. Solutions such as Cinerama, Circle Vision, IMAX, Google Earth, Virtual and Augmented Reality systems etc; have all demonstrated how we can use perspective to dramatically expand the field, scope, detail, and relevance of human vision.

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Progress in Perspective Methods, Techniques and Applications

How, you may ask, did we jump from the early perspective experiments of Brunelleschi to the advanced work of Paul Debevec, IMAX, Google Earth and the Fovolab? Patently, we have raced through centuries of dramatic progress in terms of new perspective categories, forms, methods, products, applications, etc.

To better understand how this progress has occurred, it is helpful to take a few steps backward and look at how a range of theoretical ideas and practical technologies have contributed to perspective techniques as they exist today.

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Ptolemy and the Origins of Linear Perspective 

To account for the (re-)discovery of Renaissance linear perspective there exist at least five explanations:

• A change in ‘world view’
• Workshop practice
• Architectural tradition using ground plan/elevation methods
• Surveying
• Cartography

The purpose of this section is to reconsider briefly the importance of cartography and suggest the importance of a sixth context; astronomy which introduced the use of planisphere and astrolabe. 

Ptolemy in his Geography discussed three projection methods. If these were significant in the Renaissance one would expect to find careful attention to the diagrams illustrating these methods. Examination of Codex Urbinas Graecus 82–to cite one important example–reveals, however, that the diagram for methods one and two are not elaborately produced and that the diagram for method three is omitted entirely. 

To question that geography was responsible for the discovery of perspective is not to deny, however, that there existed close links between cartography and perspective in the latter fifteenth and sixteenth centuries: cf, for example, Leonardo’s sketches (fig. 36); Dürer’s woodcut for the 1525 edition of Ptolemy’s Geography – albeit the principles here relate primarily to astronomy; Christoforo Sorte, who was active in geography as well as perspective; or Egnazio Dante, editor of Vignola’s Due regole della prospettiva (1583), who also produced maps of Perugia and later of all the papal states.

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A passage from Accolti’s Lo inganno degli ‘occhi (1625, p. 125) is particularly interesting for this, theme – although we must beware of post hoc ergo propter hoc arguments: 

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Whence if we wish to constitute a measure or a scale as the geographers say in order to be able to measure any and every member of the said drawing, we shall do it in this way most easily and most expeditely, also without needing to use other instruments of quandrants with scales of heights and similar mathematical instruments. 

Accolti’s Lo inganno degli ‘occhi (1625, p. 125)

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Hence for Accolti there were obvious links between cartography and geography. How these links developed, what their connection was, in turn, with topography and surveying deserve future attention. 

Concerning the origins of perspective Doesschate alluded to another influence: astronomy. A regular planisphere such as that described by Ptolemy imagines the eye to be at the south pole, uses the equator as picture plane and projects onto it the tropics of Cancer and Capricorn plus the ecliptic. Here one has the three basic elements of linear perspective: eye at a fixed point, picture plane and object. The astrolabe is one step more complex. It begins with the same projection as the planisphere, which may be termed the standard point of view. It then adds projection of latitudes (almucanthars) and longitudes (azimuths) relative to the viewer’s position (fig. 37). 

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Hence it combines systematically two points of view: one “standard”, the other apparent. This, it is suggested, marks an important step in the abstraction process necessary for the discovery of linear perspective which, it will be recalled, also combines systematically two distinct points of view: usually ground-plan/elevation, but in fact any two points of view at right angles to one another, e. g. frontal plus lateral etc. 

Evidence of close links between astronomy and perspective abounds. Brunelleschi, the discoverer of perspective cooperated with Paolo dal Pozzo Toscanelli in constructing an astronomical clock in Santa Maria del Fiore. Uccello, famous for annoying his wife in bed with his greater love for perspective, was also concerned with astronomical clocks. Alberti, Melozzo da Forli, and Dürer were all active in both perspective studies and astronomy. 

In Gregor Reisch’s Margarita Philosophica (eg. editions of 1512, 1515) Pélerin’s treatise on linear perspective is found between a treatise on architecture and one on astrolabe construction. Commandino’s edition of the planisphere of Ptolemy (1552) contains a commentary that is a treatise of perspective. Commandino also wrote on analemmas and sundials. D. Barbaro dedicated book VI of his Pratica della prospettiva (1568) to the construction of the planisphere and book IX to a universal horological instrument. Egnazio Danti, active in perspective matters (1583) was also responsible for placing an astronomical quadrant and aequinoctial armillary sphere on Santa Maria Novella (1580-90). In addition he wrote on the astrolabe in Italian (1569, 1578), on the use of the armillary sphere (1573), and furnished editions of the Sphaera Mundi of Sacro Bosco (1571) and of the Sphere of Proclus (1573). In the seventeenth century one merely needs to look at Aguilonius (1613), Accolti (1625), Kircher (1646) or Maignan (1646) to see that the interdependency of the two traditions became more obvious with time. 

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The visual evidence is equally striking, a woodcut by Wohlgemut shows painting and astronomy together (fig. 39). Artists show, moreover, a considerable interest in representing the armillary sphere with care; Taddeo Gaddi, Petrus Christus, Botticelli, Carpaccio, Joos v. Wasserhove and Honthorst may be mentioned as examples, or Leonardo who actually demonstrates how such instruments may be drawn. Whence it would seem that the question of Ptolemy’s influence is to be answered by attention to the tradition of astronomy rather than geography. 

If we return for a moment to the early days of astronomy, it becomes clear, however, that astronomy and geography were traditionally linked. From our school days we are all familiar with the ingenious method used by Erastothenes in determining the values of latitude (and longitude) on which were based his geographical researches: using the difference in the sun’s shadow between Siene and Alexandria and so on. The romance of the story tends to obscure its essential lesson: that geography in Antiquity was entirely dependent on astronomy and specifically that branch involving shadow projections. Such projections were discussed both in Euclid’s Elements and Optics and served, moreover as the starting point for the analemma (cf. Vitruvius IX, 7). 

The point that interests us is the interconnection of fields; shadow projection linked astronomy, geography, optics, and topography; a tradition continued by Ptolemy, through Alhazen, Witelo and Levi ben Gerson, inventor of the “Jacob’s staff” for surveying, which he combined, in turn, with the camera obscura for astronomical study. Gregor Reisch’s Margarita philosophica (1504) shows us that all these disciplines were sometimes classed under geometry. But then comparison with other illustrations of these disciplines (cf. Wohlgemut, fig. 39 or J. Amman in the Catalogus gloriae mundi) reminds us that their categories were much more fluid than our own. Whence we may conclude that the context of perspective is to be found amongst the many offspring of mother geometry and in particular her daughter, astronomy. 

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Vision and Geometry 

Basic to the Euclidean theory of vision was an angle axiom which denied a simple inverse relation between (apparent) size and distance. Linear perspective introduced an inverse size-distance axiom. Panofsky assumed that theories of vision and representation were necessarily linked and hence believed that the advent of linear perspective required a change in the theory of vision. He cited Pena’s 1557 edition of Euclid’s Optics as evidence. In fact the Euclidean theory of vision remained unaffected by perspective. Perspective, which had been linked with optics, became linked increasingly with geometry. In the sixteenth century, this occured particularly in Urbino (Commandino, Benedetti, Guidobaldo del Monte).

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In point of fact modern central perspective is a mathematically exact abstraction for a physiological image which is wholly deceptive. We see not with one fixed eye, but with two constantly moving eyes; the image which we receive on the retina is a spheroid world projected on a concave plane. Thus, in a perspective drawing, straight lines are presented as straight, but in our visual image they are actually curved.

A.G.M.Little, Perspective and Scene Painting, Art Bulletin, XIX (1936): 492.

Euclidean Theory of Vision

The nature of the Euclidean Theory of Vision mentioned in the above paragraph relates to the fact that the human eye projects a view of reality onto a spherical retina, which has significant consequences (or differences) relative to a perspective projection. The eye is a type of pin-hole camera with a curved image plane (i.e. the paraxial optics approximation); and objects located at wider field angles (and hence greater lateral distances) are projected (each one) to a different magnification, size or scale (on a spherical shaped retina). This contrasts with the rules of linear perspective, whereby objects located at different lateral object distances are all projected to an identical size on a flat picture plane. Unfortunately, reality is even more complex; and what happens will depend upon the nature of perspective methods/principles involved (i.e. the potential combination of one or more Natural, Visual, Graphical, Mathematical, Instrument Perspective types).

Alan Radley

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In the seventeenth century Paris became the world centre for mathematical perspective (e.g. Aleaume, Desargues, Niceron, Bosse). Thus Abraham Bosse, the first professor of perspective of the French Academy, could urge painters that they must draw what is there (Euclidean geometry) and not what is seen (Euclidean theory of vision). His colleagues, particularly Charles Le Brun, preferred to expel Bosse from the Academy rather than face the consequences of this clear statement of logic.

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From the 1660’s to the 1820’s artists either a) chose limited conditions where the effects of vision and perspective co-incided or b) spoke in general terms about (Euclidean) vision and (linear) perspective. The rise of descriptive geometry which claimed to offer universal principles for representation led to the first serious claims that representation and vision must be co-incident. Hence Panofsky was reading back into the sixteenth century a development that occured in the early nineteenth century.

The nineteenth century also brought other developments. Mathematicians explored various alternatives to (rectilinear) Euclidean space. These experiments eventually had consequences for modern art as Henderson and Corrada demonstrated. Meanwhile, physiologists became aware of serious problems with earlier analogies between the eye and camera obscura (Leonardo, Kepler, Scheiner). Helmholtz discovered that curved lines, when seen from nearby, appear straight and hence suggested a distinction between physical space which was Euclidean, and visual space which was non-Euclidean (and possibly Riemannian).

Mach explored this distinction in his Analysis of Sensations and devoted a chapter to it in his Knowledge and Error (1905). The emergent schools of psychology focussed on different aspects of this distinction. The Berlin school, later the Gestalt school, emphasized geometrical space. The Leipzig school (Wundt, Titchener) emphasized visual space. In the twentieth century this distinction has remained with physiologists such as Doesschate. Sometimes the names for the elements have changed. Sir Ernst Gombrich, for example, referred to visual space as the optical world or the mirror and to geometrical space as the physical world, the experienced world or the map.

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Image Analysis: The Correspondence or Equivalence Problem

A noteworthy outcome from any ‘snapshot’ or single-viewpoint imaging procedure relates to the fact that whenever we form a projection of a three-dimensional (3D) object existing in a 3D space; then irrespective of the shape of the picture plane (flat, curved, hemispherical, conic etc); the resulting object form will not be absolutely determined by a single image snapshot alone (unless a surrounding spherical picture plane is used).

In fact, there will be several different object forms that could have created an identical image pattern, leading to optical illusions, false perceptions, and other deceptive visual effects (ref. ambiguous images from human vision, photographic, microscopic, telescopic instruments, etc). Often the use of multiple and/or moving image frames can help to identify which of all the possible object forms is the true one.

Alan Radley

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The late J. J. Gibson called visual space the visual field and linked geometrical space with the visual world. There is more to this change of terms than is at first apparent. In the nineteenth century it was generally assumed that visual space was subjective and geometrical space was objective. In Gibson’s approach both are susceptible to measurement and hence in some sense objective. In Gibson’s formulation there is also no opposition between vision and geometry. Geometry applies to both physical space and to (psychological) visual space. The question remains whether the same branch of geometry applies to both kinds of space? Those who claim that (linear) perspective is dead often mean that artists have given up trying to record physical space and are focussing on visual space with its non-Euclidean surfaces. There is something to this but it is not the whole story.

Multiple Goals 

The history of art is often told as if art had only one goal. Arnason’s History of modern art can easily be read as if this were the case: for a long time the goal was imitation of nature and hence perspective was important. Then artists discovered the challenge of abstraction as a goal and abandoned perspective. One of the enduring contributions of Sir Ernst Gombrich has been to emphasize multiple functions or goals of art: e.g. magic (which used to be termed primitive art); pattern (Sense of Order), mimesis (Art and Illusion, Image and the Eye, Illusion in Nature and Art), expression and abstraction (Meditations on a Hobbyhorse) and symbolism (Symbolic Images). In this approach some functions of art do not need and sometimes even preclude perspective, while others encourage its use.

The twentieth century has added new functions or goals. One might might be termed exploring, which can be subdivided into the mental world, the perceptual world and chance. The third of these, of which Jackson Pollock is an excellent example, aims to remove any clear one-to-one correspondence between artist and art. This goal, which results in abstract painting, was for a time frequently identified with modern art. It is becoming ever more obvious, however, that the other two areas of exploring, namely, the mental and the perceptual world, have inspired a much richer repertoire of images. Exploration of the mental world has led to depiction of dreams, phantasies and other psychological dimensions. As a result realism has been applied to new realms and in the process it has been transformed into sur-realism, magic-realism, super-realism, hyper-realism.

Sometimes these new versions of realism involve a deliberate mixing of external elements (from the world of nature) and internal elements (from the world of the psyche). Linear perspective continues to play a significant role in these explorations. But it is often used in conjunction with other methods. Delvaux frequently uses curved perspective in his streets. Dali moves subtly between regular perspective and anamorphosis. Magritte uses what appears to be linear perspective but then deliberately plays with its underlying principles of occlusion and transparency. Hence walls, which traditionally block out objects are often transparent and windows which are traditionally transparent are often that which block a view in Magritte’s paintings. These experiments are spreading to other media. The recent film, Toys (1992), adapts Magritte’s painting Golconda (Menil Foundation, Houston) while playing on the paradox of the window as a mirror.

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Implicit in these experiments is an insistence that artists are not bound to a one-to-one correspondence between object and representation. Or to put it positively, artistic freedom is increasingly seen in terms of demonstrating alternatives. Hockney’s combinations of photography and painting are a further expression of these trends. In Hockney’s case there is also another explicit concern. Linear perspective, he claims, created a wall between the viewer and the object represented. Inverted perspective, according to Hockney, offers a way of removing that wall and integrating both viewer and representation within the same space. Artists such as Rauschenbach have offered yet another reason for inverted perspective: it corresponds, they claim, to visual experience. This idea emerged in the circle of Florenskij in the 1920’s and has became accessible to the West partly through the writings of Shegin.

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Alternative Picture Planes

Since the early nineteenth century there has been a growing fascination with recording visual space as opposed to geometrical space, and this has been an important stimulus for the exploration of alternative picture planes. The most obvious versions entail imitating the convex surface of the eye as in the work of Barre and Flocon that was made accessible through an excellent translation by Robert Hansen, who developed his own method of hyperbolic perspective and subsequently explored the history of the subject. According to Hansen this is not simply a fascination with subjective, psychological factors. The challenge lies in developing objective methods to record visual experience.

A recent exhibition organized by Marcia Clark confirms that this quest is about more than simply recording visual space from a single viewpoint as was traditionally the case. Artists are intent on capturing a series of viewpoints simultaneously. This is one of the incentives not only for Termes’ spheres, but also for his photographs and paintings on polyhedral surfaces. Other artists, Hockney among them, are explicitly searching for means to incorporate the dimension of time into their representations of space. This new interest in dynamic aspects of vision accounts for many of the recent experiments with alternative picture planes using spherical perspective, cylindrical perspective, hyperbolic, fisheye, tetraconic, polyconic and many another variants.

Seventeenth century artists developed a perspectival peep show (perpektyfkas) which invited viewers to use a “roving eye” to explore interiors. The work of Frisia, continues this tradition with a curious twist. His work shows us rooms and environments from an interior viewpoint, yet seen outside. Indeed, there is a curious way in which twentieth century experiments can be seen as exterior-interior or externalisations of spaces which are deliberate plays with the ambiguities of interior-exterior. If linear perspective traditionally captured a particular viewpoint, the new alternative projection methods are integrating a series of viewpoints.

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Transformations

The incentives for exploring alternative projection methods have not been simply perceptual. The rise of photogrammetry, particularly with the advent of satellite photography, has raised a series of practical challenges. How does one translate a view of the spherical earth from space onto the flat surface of a map? Even very serious cartographical publications have explored the potentially humourous consequences of projecting a human face onto a series of map projections.

Meanwhile artists such as Barre and Flocon have studied Renaissance projections (e.g. Postel) and Frisia (cf. the eighth contribution) has incorporated Mercator projections and planisphere projections into his work. The rise of holography has seen new uses of spherical, cylindrical, conical and other curved surfaces both for recording and viewing of these images.

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Ambiguities and illusions

The nineteenth century drew attention to a series of geometrical optical illusions. The twentieth century has continued to be fascinated by these and there have been some attempts to explain them in terms of perspectival experience. Some artists have recognized that these illusions lend themselves to artistic treatment. Reutersvärd (1934) was among the first to do so with the impossible figure that came to be known as the Penrose tribar, and as Ernst has shown, inspired a whole series of visual commentaries including Escher’s famous Waterfall (1961). Indeed a number of Escher’s works are deliberate juxtapositions of sections in linear perspective which, when combined, result in impossible spaces. This is yet another expression of explorations of non-correspondence between object and representation.

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Virtual Reality

In the Renaissance, artists used instruments such as the mirror and the perspectival window as a means of recording the visible world with linear perspective. In the latter twentieth century, images of the mirror and the looking glass continue to be used, now in the context of the computer screen, and with a new goal of seeing the hitherto invisible.

As Ivan Sutherland put it:

We lack corresponding familiarity with the forces on charged particles, forces in non-uniform fields, the effects of non-projective geometric transformations, and high energy, low friction motion. A display connected to a digital computer gives us a chance to gain familiarity with concepts not realizable in the physical world. It is a looking glass into a mathematical wonderland.

Ivan Sutherland, 1965

This visionary statement in 1965 was one of the starting points for Sutherland’s head mounted display (first published in 1968), experiments at NASA and led to the emerging fields of visualisation and virtual reality. Renaissance perspective sought to look through the window: virtual reality is attempting a new type of immersion “through the looking glass”.

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Virtual reality raises new problems of perspective. Even very simple systems such as the PC based Superscape Virtual Realities System entails a basic perspectival space for the general environment within which there are other viewpoints that can be manipulated simultaneously: from either inside or outside an automobile, from behind a person or off to the side of a helicopter. Renaissance perspective typically involved either interiors or exteriors. Virtual reality is introducing new, dynamic interplays between egocentric and exocentric viewpoints, including the ability to move through both walls and windows enabling seamless transitions between interiors and exteriors.

In the emerging field of cyberspace these principles are being extended as navigational tools among data cells in visual databases. In the eighteenth century Kant described verbally how we use perspectival space as a tool for orientation in our mind. Virtual reality and cyberspace are helping us to visualize this process. Spatial, perspectival co-ordinates are basic, not only in the virtual cockpit, but in all our activities.

Scale and Fractals

As mentioned earlier, one of the fundamental tenets of Renaissance perspective was the inverse size-distance law which stated that if one doubled the distance the represented size was one half, if one trebled the distance the size was one third and so on. Mandelbrot’s article about the size of the coast of Britain implicitly introduced a spanner into this assumption by showing that size was a function of scale as well as distance. In a sense we have been vaguely aware of this ever since the seventeenth century. The shape of an ordinary image is transformed entirely when we change the scale of its image radically in in a telescope or a microscope.

Fractals may have brought this problem into focus, but fractals assume a principle of iteration, i.e. their basic patterns are repeated and hence remain independent of scale. Hence they do not solve the problem which they have raised. What is needed is a new approach to perspective that takes into account scale as well as distance, whrereby any given shape only applies within a given range of scales. This is increasingly important in a world where we travel between scales with greater frequency.

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Attempts to Expand the Field of Vision

Over the centuries there have been many attempts to go beyond these limitations in representation. Artists, explored cylindrical and spherical surfaces on which to depict their paintings. In the 20th century, photography has contributed greatly to these developments. A whole range of lenses evolved from extreme wide angle (84°-179°), to wide angle (63°-83°), normal (34°-62°), medium telephoto (24°-33°), telephoto (8°-23°) and super-telephoto (4°-7°).

More dramatic developments happened on three fronts: fisheye, multi-cameras (polydioptric) and catadioptric cameras. The term fisheye was introduced by the American physicist Robert W. Wood in 1906 based on how a fish would see underwater: cf. Snell’s window whereby the 180-degree horizon is reduced to 97.2 degrees through refraction. In the open air, without refraction, there are 180 degrees, as in a classic fisheye, or through multiple cameras in the form of a dome.

In the 1920’s and 1930’s, fisheye lenses were used to study cloud formations through whole-sky lenses and later to study forest canopies with canopy cameras and so-called hemispherical photography. Some of the early fisheye cameras were simply heavily modified versions of regular cameras. But soon these cameras became highly professional devices expanding the range well beyond 180 to 220, 270 and even 280 degrees.

A second front lay in the development of multiple (or poly-dioptric) cameras. Simple versions typically entailed symmetrical numbers such 2, 4, 8 cameras leading to cylindrical panoramic images. In the past decades, these have been complemented by spherical configurations such as the Google street view, Panono and AMD Radeon Loom cameras leading to ever more dramatic omnidirectional images. The third front lay in catadioptric lenses, which typically involves mirrors: parabolic, hyperbolic, elliptical or planar. These principles were used in a Hamburg planetarium (1957) and are regularly used in omnidirectional images, which can subsequently be translated into cylindrical panoramic images.

To a certain extent these three methods competed with one another. At the same time, there were studies comparing catadioptric and fisheye images. There were also enormous efforts in translating between different formats, as for instance how three separate fisheye views could be merged as a single panorama. These translations become the more complex because terms such as spherical can mean a 3-dimensional sphere rendered on a 2-dimensional flat-surface, or it can represent a catadioptric image or again a physical refractive sphere. In addition to the original shape and the recorded shape, there is the further question of the projected surfaces, which can range from a shallow concave bowl, or deeper concave bowl to a full hemispherical or full spherical dome, which can in turn entail its interior or exterior surface.

In the beginning of the 20th century all cameras used film. A century later, almost all cameras are now digital whereby the challenges of translation have shifted from hardware to software. For instance, the software Lightroom allows users to edit photos in perspective mode for flat surfaces, cylindrical mode for cylindrical applications and spherical mode for spherical surfaces.

In the realm of curved and cylindrical projections, two areas have been particularly fruitful in the past half century, namely, flight simulators and gaming. High level flight simulators for space, the military and the airline industry are not considered here because they are designed for a tiny group of specialists. Readily available commercial systems include those of Microsoft, Roger Dodger Aviation’s 3 screen version and the large curved projections screen by SMAX.

Video arcades with Arcade Machine Cabinets emerged in the 1970s and 1980s but by the 1990s they “subsequently declined in the Western hemisphere as competing home video game consoles such as the Sony PlayStation and Microsoft Xbox increased in their graphics and game-play capability and decreased in cost.’’ Today, personal gaming now typically takes place on cell phones and portable gaming modules. Meanwhile, there have been very impressive gaming screens such as Norman Design Screen, Light 32 medium VCZ, Dome Production System.

Television has seen the ever more dramatic promises of 8K and now demos of 10K screens (e.g. Chinese company BOE). Much more dramatic versions are available, e.g. Samsung; LG OLED TV; 3 Screens (NdimensionZ). At the high end, there are 3D, cylindrical and spherical television Screens: LG Ultra HD 3DTV, VNS, Pufferfish. Typical retail stores for new televisions show mainly flat screens, only a few with minimum curvature and nothing more.

The area which has seen the most serious applications of these new technologies is cinema, which has, as predecessor, the field of panoramas, which have been documented in an excellent study by Oettermann (1979). This principle was adapted to the world of cinema by Disney in their Circle-Vision 360° (1955) and developed independently at a much higher resolution by the founders of IMAX for the World Fair in Montreal (1967).

At a more popular level, an important further step in this quest for 360 degree images was Apple’s Quick Time VR. The number of images required varied considerably. A minimal version required 12 images. Other versions required 16, 22, or 26 images. There were two basic versions: one created panoramic movies looking outward and the other produced object movies looking inward. Facebook has adapted this approach with newer technology. Now there are typically 14 cameras. In terms of emerging technologies, there are drones, which permit 360 degree views from the air or complex 360 degree cinematic cameras, which lead to full 360 degree views. Effectively, each of the cameras applied the principles of perspective. Having, 8, 12, 14, 16 or 22 simply multiplied the principle to cover the entire field of vision.

360-degree views have become increasingly popular, ranging from performances of Peter Pan’s Neverland  to 8-D cinemas with 360 degree screens. Subsets of a 360 degree view are found Paseo Lumenscapes and the Atmasfera 360 screen in Kiev. The best-known examples are IMAX theatres, some of which encompass a large portion of a sphere. Other examples such as Géode IMAX theatre at the Cité des Sciences in Paris and at the Epcot Center (near Orlando) effectively recreate a complete dome. All these examples continued to extend the late mediaeval quest for matching in ways that reflected a) copying the natural world; and b) conformed in differing degrees to human visual experience of the physical world.

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Optics as Extensions of the Eye

During the Middle Ages and the Renaissance there was considerable attention to camera obscuras in the context of analogies with the eye. Instruments to extend the capacities of the eye were much slower in evolving. One explanation has been that because lenses had the same name as lentils there was an ingrained bias against the reliability of anything other than that seen by the unaided eye and that this only changed in 1610 when Galileo had sighted the moons of Jupiter.

Our own account invites a very different reading of the evidence. During the fifteeenth century the emergence of linear perspective introduced new instrumental demonstrations of visual cones and pyramids which effectively destroyed all claims about the unaided eye’s reliability. Optics, which had once been a study of eyes in isolation, now included instruments which extended the scope of vision. Moreover, as the concerns of astronomy acquired an ever more central role in the latter fifteenth century, attention shifted from the problems of direct vision to those of reflection and refraction introduced by celestial observations.

It was no coincidence therefore that Leonardo da Vinci’s optical writings were intended specifically as an introduction to a great treatise on astronomy and cosmology, or that Kepler, a century later, should compose his greatest optical writing on the astronomical part of optics. In short the so called revolution in optics that occurred with Galileo’s telescopic observations in 1610, involved something much more than suddenly overcoming scruples about the reliability of existing lenses or acquiring better lenses. It arose also from a century old confidence that perspectival instruments were crucial in helping an otherwise inevitably deceived unaided eye. If perspective had made painting increaseingly a discipline of representational aids; perspective made optics a science of visual aids.

Once telescopes fell within the scope of optics, microscopes soon followed: the same problem of using instruments to enlarge the size of objects, except that they are small and close rather than large and far. It is perhaps significant in this context to note that Hooke, of microscope fame, also published perspectival instruments in the Philosophical Transactions of the Royal Society; that Brander wrote on perspective in the context of both telescopes and microscopes as well as surveying instruments; themes which were pursued by Lambert.

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Conclusions

If perspective is defined in a narrow sense as linear perspective then one of the major reasons for its continued popularity is a growing historical awareness which seeks both to understand methods developed in the Renaissance and apply new technologies in the analysis thereof. Some of the major themes of the earlier treatises such as regular solids remain significant to this day.

Yet there are significant contrasts between Renaissance methods and modern developments. The Renaissance paid lip service to equations between perspective and vision, while at the same time linking perspective increasingly with geometry and commiting themselves to recording geometrical space of the physical world. Some twentieth century artists have continued this tradition in their explorations of realism, hyper-realism, and sur-realism. Others have abandoned this commitment and focussed increasingly on the exploration of visual space, both exterior and interior. This has led to new goals of art in terms of exploring perceptual, mental, dream, psychological and even psycho-pathological states.

As a result, whereas Renaissance artists focussed attention on linear picture planes, twentieth century artists explored many alternative shapes of picture planes. They also contradicted the traditional transparency-occlusion principles of perspective in their quest for artistic freedom. Hence whereas Renaissance artists established a one to one correspondence between object and representation, twentieth century artists strive to demonstrate the contrary.

What emerges from this is that the chief significance of perspective which evolved in the fifteenth and sixteenth centuries probably lay neither in painting, where its strict rules were usually ignored; nor at first in geometry where its spatial implications were hidden by the two dimensional conventions of that tradition; but rather in focussing attention on the importance of instruments for vision as well as representation. Hence in introducing new standards for accurate drawing it also brought new criteria for optical truth. Seeing no longer involved the naked eye alone. The study of sight was now inseparable from its extensions: not just in terms of windows and rods, but also in terms of eyeglasses and combinations thereof, which led to both telescopes (or perspective glasses as they were called) and microscopes. Hence there was a good reason why Renaissance authors made no sharp distinctions between optics and perspective. The differences between vision and representation were outweighed by the perspectival instruments which remained common to both and on which both disciplines depended for their legitimation.

These connections between optics, perspective and instruments help explain why there is usually a section on optics in perspective treatises; why a perspectival window is found among the instruments at the French Academy of Sciences and why it should recur along with related perspectival instruments on the title page of a Lausanne edition of Newton’s Optics (1740). A list of these instances could easily become an article in itself, because we are only slowly becoming aware of the extent to which the history of perspective is as central to the history of science as it is history of art.

It may be no coincidence that we are becoming aware of the significance of perspective at a time when computers are being used to reproduce perspective and even to elucidate principles of Renaissance perspective. For the extension of criteria for optical veridity from the unaided eye to include both instruments of vision and representation marked an important step in relating individual experience to a framework that is common to many and in this sense more objective. Computers essentially take this process of objectify-cation one step further and thus mark a further implication of the principles of linear perspective. Perhaps that it why perspective is witnessing a renaissance of interest. Computers are helping us to see more clearly the monumentality of the shift that perspective brought to the Renaissance: not a new way of seeing but a new instrument for recording and verifying our many ways of seeing.

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Epilogue

If we look back over the developments of the past century it becomes clear that perspective did not simply die. As was suggested elsewhere (Sources of Perspective), the goals of art were greatly diversified and for a short time it seemed as if the new non-representational goals would replace the older traditions of realism and the so-called figurative arts. In retrospect, it has become clear that realism has not gone away. Certain cultures may insist that thought, art, or even creativity is primarily conceptual rather than perceptual, some schools may argue that nature and nurture need to be conflated, some may emphasize the role of constructivism, deconstructivism, or even destructivism, others may pose questions concerning the extent to which correspondence theories of knowledge apply in practice and yet realism has not managed to go away.

In trying to understand the larger context within which perspective evolved it was noted that the rise of systematic drawing systems entailed an increasing emphasis on contained spaces which led naturally, as it were, to artifice and ultimately artificial reality. It was shown that this shift had its own socal implications. Hence, what began as an event that brought together members of the public (of a certain class) in an inclusive sense, gradually became an exclusive performance for an ever-smaller circle of persons, which then in turn expanded once again.

It was suggested that developments in twentieth century perspective could fruitfully be seen in terms of a re-evaluation of the principle of correspondence, or rather as a challenge to some of its basic tenets by introducing reversals of the transparency-opacity laws, playing with the idea of windows as walls, and reversing usual assumptions concerning inner and outer. One basic consequence of these experiments was a widening of the categories of correspondence and non-correspondence to include a whole spectrum of alternatives: theoretical, assumed, possible, transposed and no direct correspondence. Experiments with alternative planes furthered this trend, as did the introduction of new media. Which led in turn to a much greater fascination with transformations from one kind of system to another, sometimes systematic, sometimes algorithmic, or sometimes purely accidental. In terms of goals of art this led to exploring.

In other contexts, this has led to the rise of simulation, virtual reality and virtual worlds whereby a new multiverse emerges in competition with the traditional universe: the real and the simulated; the physical and the imaginary which were once clearly separated are now being deliberately superimposed, mixed, confused, melded to the extent that the two worlds are integrated in our everyday experience.

This led to a re-consideration of Panofsky’s intriguing phrase concerning the objectivization of the subjective, which has come to mean something very different from that which he appears to have meant when he first gave his lecture at the Warburg over 90 years ago. Perspective, in its new manifestations, is transforming our concepts of space, time and the very concept of representation.

Renaissance perspective was originally a method for demonstrating clear relationships between representations and the objects upon which they were based. As such, Renaissance perspective revealed its sources, vaunted them almost. By contrast, perspective in the twentieth century has become a game of hiding its sources, veiling its connections with the original, playing with the principle of correspondence, shifting, transforming, denying it until it sometimes becomes impossible to determine the source. Some have used this ability as an expression of freedom, as proof that the spirit is not bounded, limited, hemmed in by any one-to-one link with the external world, nature, or reality.

Others have tried to avoid even the mention of all such words, as if one could make that which is outside us go away, only to find themselves haunted by notions such as difference, the other and the like. So, there is a new trend towards finding new ways of making relations with things beyond us, a renewed concern with the importance of context and re-contextualization. And in the process, we are finding that perspective is something much deeper than a handy convention or a temporary fad. Long ago, Kant recognized that perspective served as a key for both physical and mental orientation. We are learning that it is a tool for conceptual navigation: a way of finding not only where we are, but also for discovering who, what, how and even why we are. No wonder then that perspective, which introduced the notion of an open horizon, is such an open field.

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-- < ACKNOWLEDGMENTS > --

AUTHORS (PAGE / SECTION)

Kim Henry Veltman (1980 - 2017).
Alan Stuart Radley (2023).
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BIBLIOGRAPHY

Veltman, K.H. (1980) 'Ptolemy and the Origins of Linear Perspective' - Atti del convegno internazionale di studi: la prospettiva rinascimentale, Milan 1977, ed.
Marisa Dalai-Emiliani (Florence: Centro Di, 1980), pp. 403-407.

Veltman, K.H. (1992) 'Perspective and the Scope of Optics' - unpublished.

Veltman, K.H. (2017) 'Perspective from Antiquity to the Present' - unpublished.

Veltman, K.H. (1994) 'The Sources of Perspective' - published as an online book (no images). Later published with images as 'The Encyclopaedia of Perspective' - Volumes 1, 2 - (2020) by Alan Stuart Radley at the Perspective Research Centre.

Veltman, K.H. (1994) 'The Literature of Perspective' - published as an online book (no images). Later published with images as 'The Encyclopaedia of Perspective' - Volumes 3, 4 - (2020) by Alan Stuart Radley at the Perspective Research Centre.

Veltman, K.H. (1980s-2020) 'The Bibliography of Perspective' - began as a card index system in the 1980s; before being transferred to a dBASE-3 database system on an IBM PC (1990s). Later the bibliography was made available on the web on the SUMS system (2002-2020). In 2020 the Bibliography of Perspective was published as part of'The Encyclopaedia of Perspective' - Volumes 6, 7, 8 - by Alan Stuart Radley at the Perspective Research Centre.
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Copyright © 2020-23 Alan Stuart Radley.
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