Central Perspective

This section presents a detailed analysis of a particular class of Visual Perspective (first type), named Central Perspective. This category of Graphical Perspective includes well-known forms such as Linear and Curvilinear Perspective.

Linear Perspective is often employed for transcription purposes in technical and engineering drawing, or artistic purposes. Whereas Curvilinear Perspective has been used to represent/map/study wide-field views of dimensional reality.

Perspective Illusions

In a sense, perspective is all about illusion—creating the appearance of three-dimensional space (3D Object Space) on a two-dimensional surface (2D Image Space), or producing a ‘convincing’ dimensional reality. Of course, we can have perspective forms that are inherently 3D such as stereoscopy and Virtual Reality, etc. But in this section, we are concerned with Graphical Perspective or the artistic creation of an optical illusion of depth by utilising flat images.

Perspective is the production of the illusion of space and especially depth. It is the art of depicting 3D objects upon a plane surface so that the picture may affect the eye of the observer in the same way as the natural objects themselves!

However, a flat plane cannot be equal to a 3D natural object in all respects, and it is the viewer’s mind and perceptive system that imbues a perspective painting with an impression of depth. Hence, certain forms of Graphical Perspective can produce the illusion of three-dimensional space on a flat picture surface—and by employing flat images that conform to special Perspective Phenomena, including:

  • The diminution of the scale (of remote objects)
  • Spatial foreshortening
  • Use of ‘natural looking’ vanishing points (apparent congruence of parallel lines)

It is important to note that not all forms of Graphical Perspective employ every single one of the different Perspective Phenomena. Parallel Perspective forms (for example) use only spatial foreshortening but not diminution of scale, or vanishing points.

Central Perspective

Central Perspective is a form that makes good use of Perspective Phenomena to

Examples of Graphical Perspective include Parallel, Reverse, and Axial Perspective types, etc. But in this section, we shall explore only those forms that conform to what has become known as Central Perspective (also named Central Projection). Ergo, we present an inquiry into the geometrical and optical sources of Central Perspective, examining associated visual, representation, and modelling forms and their various application areas.

Before looking in detail at the different types of Central Perspective, it is crucial to delineate the basic features of Graphical Perspective (technical forms only).

Graphical Perspective

In general terms, perspective refers to the procedure (or results) of forming an image or view—a representational pattern—of a dimensional scene. A perspective ‘pattern’ exists either as a ‘live’ or real-time view, or as a ‘captured’ image of a dimensional scene, whereby said view/image can be formed by: 

  1. A shadow producing light source plus occulting object (Natural Perspective).
  2. The eye of a living being (Natural Perspective). 
  3. A mathematical procedure (Mathematical Perspective [types 2,3]).
  4. A graphical procedure to produce an image of a dimensional reality (Graphical Perspective – e.g. drawing, painting, photograph etc) .
  5. An optical instrument (Instrument Perspective). 

Graphical Perspective (type 4) is a form of Technical Perspective; which refers to any systematic process that produces a detailed visual image, measurement, representation, model or view, of a dimensional object or scene. Technical Perspective is formed using optically, mathematically, geometrically, or logically correct/known/consistent principles.

Several forms of Technical Perspective are identifiable, including Natural, Mathematical, Graphical, Instrument, and Media Perspectives. However, Graphical Perspective is the only one whose primary purpose is to create a valid illusion of spatial reality. Ergo, Graphical Perspective is used to create the appearance of a spatial reality or dimensional scene/object.

What about Virtual Reality (VR) you may ask – or Photography/Cinema/Television – do these Instrument Perspectives not create the illusion of a spatial reality? Yes, they do provide the illusion of dimensionality, but usually only by employing the methods of Graphical Perspective (at the very least).

In conclusion, Graphical Perspective is a systematic way to probe spatial reality (in visual terms), or to analyse certain observable truths about a physical or imagined space. Said visual truth is embodied in a visual image. Understanding how this image or pattern is formed is essential to explaining how/why we can have multiple different types of perspective, and to be able to know how to correctly apply/map each type appropriately to a specific scenario.

The Artistic Method

Perspective is all about accurately capturing, and representing plus displaying; dimensional reality. But how, precisely, do the methods of perspective achieve such goals? To answer this question, let us begin by considering how an artist typically produces a perspective drawing/painting of a dimensional scene. Patently, the artist must first select a subject to paint (choose an object/scene), which is to be looked at from a fixed location, being the so-called Station Point or View Point.

One might assume that the artist can then just begin sketching what he/she sees whilst looking at the scene in an ad-hoc manner. But normally it is necessary to fix the angle and field of view in some way; and because there are a large number of possible viewing angles and/or scene aspects that can be observed from a particular View Point. In this respect, normally an explicit frame for the observed scene is employed, named the Windowpane (which may be imaginary).

This Windowpane is normally rectangular in shape, and planar (or flat in) in form. serves the purpose of fixing the artists location, angle and field of view, relative to the scene in question.

Ergo, the artist begins by looking at a scene through an imaginary Windowpane (glass window), and drawing/painting what is seen directly onto the flat Windowpane. Whereby for a representation created in such a manner, each painted object in the scene is thus a flat, scaled down version of the object on the other side of the window. Ergo, the technique of Graphical Perspective works by representing the light that passes from a scene through an imaginary rectangle (represented as the plane of the drawing/painting), to the viewer’s eye.

What happens when another person looks at a painting that has been created in this way? Well it as if the viewer were (him or herself) looking through the same imaginary a window and viewing a representation that is identical to the original scene (at least in some senses – and not including binocular perceptive effects). This happens because each portion of the painted object lies on the straight line from the viewer’s eye to the equivalent portion of the real object it represents, whereby the viewer then sees no difference (sans binocular effects) between the painted scene on the Windowpane and the view of the real scene. In other words having an identical Station Point has the desirable effects of canceling out what would appear to be distortions in the image when viewed from a different point.

For a painting constructed as described, the Picture Plane can be thought of as the glass of the notional Windowpane through which the viewer looks into the representation of reality that lies beyond. In practice the Picture Plane is the same as the actual physical surface of the painting produced in this manner.

Unfortunately, there exists a problem with the above description of Graphical Perspective. 

When the resulting representation is viewed from the same spot as the Windowpane was created (or identical Station Point located in-front of the Windowpane /Picture Plane), then the depicted image is identical to what was seen through the unpainted window. However it is often the case that the viewer is not located precisely at the same Station Point relative to the Picture Plane (as the original artist); either being located at the wrong distance, or off to one side of the painting central line (or at an angle relative to the Plane of Painting).

Luckily, many times the visual effects of such a different location is small, and human perception is able to readily account for and ignore said effects. Ergo, in practice, unless the viewer chooses an extreme viewing angle, the perspective normally looks more or less correct. This is referred to as “Zeeman’s Paradox”.It has been claimed that a drawing in perspective still seems to be in perspective at other spots because we perceive it as a drawing, and notably because it lacks other so-called ‘depth cues’ (see discussion below). Nevertheless, the field of view is often narrow enough (typically approx. 60 degrees) that the distortions are minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted.

Graphical Perspective (Parallel Form(s) with Non-Planar Picture Plane)

Short Decription ..

Graphical Perspective – Summary

How Graphical Perspective Works

type appropriately to a specific scenario.

How Graphical Perspective Works

Perspective is all about accurat

Before getting into the technical details of Central Perspective proper, it is useful to summarise the primary features of Graphical Perspective.

To begin, we can state that Graphical Perspective is a purely human made representation of a spatial reality, being a form of Artificial Perspective. We can have three main types of Graphical Perspective, namely Central Perspective (flat/plane or non-planar Picture Plane), Perpendicular Perspective (or Descriptive geometry), and Spherical Perspective (sphere of vision). Other hybrid forms are possible but we shall skip these here (see Taxonomic Tree).

Typically, in Central and Perpendicular Perspective we have a flat or Planar shaped Picture Plane, being an image plane which is located between the ‘eye point’ (or oculus) and the object being viewed. Depending upon the geometrical details of how this Picture Plane is located relative to the scene, and  how the scene structural elements are projected into this plane, we are able to define a particular form of Graphical Perspective.

When we insert a vertical plane perpendicular to the sight line (which extends out to the scene or object of interest) – and project what is optically seen from that eye point, then we have a form of Central Perspective (1-2-3 point Perspective, etc). Whereby the orientation of the picture plane is normally perpendicular to the axis that comes straight out of your eyes. If your eyesight is entirely horizontal then the picture plane is perpendicular to the ground and to the axis of your sight. As an alternative, when we insert a plane at a specific angle relative to the scene, and project scene structural elements according in a parallel fashion and independent of distance, then we have Parallel Perspective.

In summary, Graphical Perspective (Technical Form), refers to drawing/painting or prescribing the Visual Features of a dimensional scene, and by placing an imaginary Windowpane between the projection plane (or eye-point) and scene structural elements, and then drawing/prescribing said visual elements directly onto the same plane.

Horizon

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Vanishing Points (Central / Spherical Perspective)

In the case of Central Perspective (i.e. 1-2-3 Point Perspective, Curvilinear Perspective, etc), then we have a form of Graphical Perspective that employs Vanishing Points to aid in representing reality as it is viewed/captured. A question arises – what exactly is a Vanishing Point? How do they arise – and in what sense can they be said to be objective or real?

A Vanishing Point is a point within a view/image space (normally represented on a two-dimensional (2D) image plane) towards which mutually parallel lines in three-dimensional (3D) space appear to converge (or disappear). In terms of a perspective projection, a vanishing point is a point on the image plane of a perspective representation (e.g. drawing/painting, photograph etc) existing on the image plane where the 2D perspective projections of mutually parallel lines in 3D or Natural/Physical Space appear to converge.

Patently, within a dimensional space (Natural/Physical Space); we can have mutually parallel lines (straight or rectilinear form) located anywhere, pointing in any direction, of any length, and in multiple numbers. Now natural scenes do sometimes exhibit straight lines, and even mutually parallel lines (for example within woodland branches etc). But mainly mutually parallel and straight lines are normally seen in human-made environments. Most often these are seen in architecture, for example houses, churches, landscapes, and long lines for example on roads, rail tracks, or other objects aligned in long lines etc.

It is a fact that wherever and whenever we have two mutually parallel lines (existing in spatial reality) then we will have potential for formation of a vanishing point (by means of a perspective view). A typical feature of human manufactured environments is the design of rectinilear forms – with horizontal ground plans including  

Central Projection

Central Perspective is a direct embodiment of a form of Graphical Projection named Central Projection; a type of representation that mimics what the eye sees when looking at a dimensional scene or three-dimensional (3D) object.

Central Perspective is a form of three-dimensional (3D) Graphical Projection in which the Perspective Window or Picture Plane is

In a strict mathematical sense, the term Central Perspective refers to all forms of Graphical Projection in which an object/scene is ‘viewed’ wholly from a frontal direction or along the central line of the projection (or directly along a single line-of-sight); whereby objects in the scene are depicted and scaled according to one or more (related or systematic) 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 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 (one or more of) the orthogonal, lateral and vertical directions (in object space) appearing to recede towards one or more related 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, producing a vista which is viewed directly along a single line-of-sight, 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 dimensional scene using logically consistent projective principles.

A well-known form (or category) of Central Perspective is Renaissance or Linear Perspective, which is explained in the sections below.

Renaissance or Linear 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). Note that Renaissance Perspective may also sometimes refer to the Two or Three-Point Perspective types (see explanation of Linear Perspective below).

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). 

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.

Unfortunately, confusion exists about the nature of perspective and Linear Perspective in particular; whereby people sometimes fail to understand precisely how perspective arises, the circumstances in which a specific form is valid, and situations in which it may correctly, or alternatively fail to, represent reality to a helpful degree.

Herein you will discover the authentic sources of Linear Perspective, and discern its relationship to a broader set of vision/representation based objectives.

Graphical Perspective and Dimensional Reality

An evident goal of perspective is to obtain an accurate understanding of a dimensional reality, or (typically) to prescribe/index/represent/model a region of natural space consisting of three spatial dimensions (3D); whereby for example, we wish to explore the dimensional form of a dimensional scene and/or the 3D object(s) it contains. Oftentimes, we might also wish to explore/record/represent the time dimension that is intimately associated with a region of natural space (using moving images).

We are concerned here with the nature, viewing and representation of the third dimension or 3D, and in particular (in the case of Graphical Perspective) with those issues which are relevant to the representation of perspective scenes on two dimensional picture planes (including paper, computer screens etc). Graphical Perspective concerns the copying of reality to produce accurate drawings, paintings, models, maps and projections of dimensional space. 

The graphical representation of 3D has a long history going back many thousands of years to the ancient Greek, Egyptian and even Chinese civilizations. During the intervening centuries, a number of different systems have been used to aid in the realistic representation of 3D, but here we shall focus mainly on the method that has become commonly known as Linear Perspective.

Perspective Category Theory

Patently, Linear Perspective, as commonly understood, is a form of Graphical Perspective. However, the projection principles that are embodied in this form of perspective sometimes apply also to Instrument Perspective(s), including image capture and projection systems; plus photography, cinema, Computer Generated Imagery (CGI), and Virtual Reality, etc.

Notice that all of aforementioned forms of perspective are human-inventions of one kind or another. Ergo, we can state that Linear Perspective, in general, is a human development, and is concerned with accurate analysis of spatial scenes/forms. That said, much debate has surrounded the precise nature of perspective, and of Linear Perspective in particular; and as to wether or not perspective is a (human applied) convention of not. In other words, many experts contest any statement that affords objective reality to any form of perspective (and Linear Perspective in particular).

At this stage, we shall avoid getting sidetracked with the issue of the status of perspective; but rather simply throw the question into the long grass, so to speak. In terms of our Perspective Category Theory, we can place Linear Perspective on the taxonomic tree as follows: Visual, Optical, Technical, Graphical, 3D Graphical Projection (Standard) and Central Perspective.

It is noteworthy that Linear Perspective involves mathematical constructive principles; but we have avoided placing it (primarily) under the Mathematical Perspective category, and because we have reserved that solely for pure mathematical techniques of modelling spatial reality; and not for methods that are primarily concerned with the construction/representation of a spatial reality.

In any case Mathematical Perspective often contributes to the other kinds of Technical Perspective; including Visual Perspective (second type), plus Graphical, Instrument, Forced and Media Perspective(s). Such mathematical contributions happen, either due to the natural visual effects resulting from the laws of physics, inherent instrument perspective, or the application of human-designed algorithms to a visual scene.

Products, Goals and Functions

Perspective refers to any systematic process that produces: a detailed visual image, measurement, representation, model or view, of a dimensional object or scene. These are the Products of Perspective.

Consequently, three goals of perspective can be identified. They are first viewing reality: observing spatial form; second, matching reality: surveying or modelling spatial form; and finally making representations of reality: copying / constructing images of spatial forms. These are the goals of perspective (in human terms), and it is clear that visual knowledge and images are fundamental to each type.

Perspective, in general, works to enable viewing, prescribing, matching, modelling, exploring, representing, and making images, of the physical world. These are the Functions of Perspective.

In terms of products/goals, Linear Perspective is most often associated with with the graphical representation of spatial reality; while sometimes aiding in viewing and matching reality at the same time. It is this representation feature of Linear Perspective that we shall focus on here, and consider problems related to making faithful representation(s) of the physical world.

Perspective Components (of visual transformation)

It is helpful to consider perspective as a visual process, or a set of visual transformations, that affect changes to the appearance of a dimensional scene.

Ergo, perspective is a process that can be broken down into a set of Perspective Components of visual transformation. A particular Category of Perspective reflects a corresponding group of visual changes that occur while viewing/surveying/representing a dimensional scene. The Perspective Components correspond to all of the optical adjustments happening to a scene’s Visual Features; including the appearance of points, lines, plane figures, solid shapes, shades, shadows, reflections, translucency, colour, texture, etc.

A typical goal of perspective generally is to capture, view or record certain visual features of a three-dimensional (3D) object space or scene. Another way of describing a typical perspective imaging procedure is to state that Graphical Perspective is concerned with how 3D reality looks from a particular point of view, and as projected onto a (typically) fact picture plane space. The net result of this procedure is that the image – or spatial features of the object/scene – appears visually distorted or altered, in certain ways, and because this is the inevitable geometrical result of projecting a 3D world onto a 2D plane (at least for this type of perspective which employs vanishing points to scale Visual Features according to apparent distance from the picture plane).

Accordingly, standard Perspective Components (of visual transformation) delineate the specific visual effects happening to Visual Features whenever a perspective view/image/measurement/model/representation is made. Whereby often, said visual-transformations can be linked to one of the so-called ‘retinal variables’ that are detectable by the human eye, or linked to other Visual Features detectable by another image forming process/instrument (see below).

Human Vision and the Depth Cues

Perspective is the science of spatial form; or more specifically the appearance of form. Accordingly perspective has close connections to the third dimension or depth. But before we can explore how perspective is able to probe spatial reality; we must understand certain aspects of, and the fundamental connections between; space, human vision and representation.

People often say to me that 3D as depicted in drawings, paintings, or on a television or computer screen is not “true 3D”. But they are then unable to explain this statement any further, and sometimes they also add that flat screen images cannot be true 3D because they do not show stereoscopic views. Along the way people occasionally mention that stereoscopic glasses are needed for 3D, one being red and the other blue for the left and right eyes. In one sense what these people are saying is correct, in that a flat picture plane does not show stereoscopic images. However they are entirely incorrect when they assume that only stereoscopic images are “true” 3D.

Vision experts have long known about the different aspects of depth perception, and that these can be grouped into two categories: monocular cues (cues available from the input of one eye) and binocular or stereoscopic cues (cues available from both eyes). Each of these different “cues” is used by our brains, either independently, or else together, in order for us to perceive the third dimension.

It is important to note that not all of the cues are required to be present simultaneously in order to give us an accurate or realistic impression of 3D. It has been demonstrated, for example, that we can get a realistic impression of 3D when just one or two cues are present, as in a perspective drawing for example.

I think it it is worth reminding ourselves of the cues in a list at this point. Monocular cues include Perspective, Motion Parallax, Color Vision, Distance Fog, Focus, Occlusion, and Peripheral Vision. Binocular cues include stereopsis – (or binocular disparity sometimes also called binocular parallax) which is the difference in shapes and positions of images due to the different vantage points from which the two eyes see the world. The other binocular cue is convergence, or range-finding stereopsis which is the human ability to judge the distances to objects due to the angle of convergence between the eyes.

Note that some vision experts would argue for the inclusion of other yet more subtle (monocular) optical cues, including occluding edges, horizons and other affects due to the “direct perception of surface layout”, but to simplify an obviously complex topic we shall ignore these additional factors here.

The greater number of items on the monocular list, gives a first clue that perhaps stereoscopic vision effects are not the primary way in which we as humans perceive depth or the third dimension. You can easily test this yourself by closing one eye, and immediately you notice that the world still appears to be spread out before you in all of its three dimensional glory! With one eye closed you may have difficulty with the finer points of depth perception such as picking up a pin off the floor. However largely for ordinary tasks if you lost one eye, then you would still be able to rely on the other eye for 3D vision, in fact exclusively by relying on the monocular visual cues.

On passing I would like to note here that those who suggest that stereoscopic 3D (aka red- blue parallax films) is the only true 3D, and further that its mechanisms are well known, are in fact claiming that they have more than a head start on some of the greatest experts in human vision who ever lived. World renowned scientists agree that science has yet to even begin to understand the mechanism by which human beings combine or overlay two different parallax views in real time into a single correlated image sensation.

Some even proclaim this image combination feat to be a miracle of the human perceptive system – and so it may be – because the source images are thoroughly misshapen and also distorted one relative to the other.

Perspective Defined (Generally)

Let us pause at this point and take a stock of where we are.

I hope I have been able to convince you of the fact that stereoscopics are not required to give an impression of 3D. If they were we would not be able to make much sense at all of television, films, photographs or even the vast majority of drawings and paintings. These methods, one and all, solely rely on the monocular cues for depth depiction, yet we have no difficulty understanding the 3D worlds depicted in which objects lie at different apparent distances from the viewer.

One of the most important of the monocular cues for depth perception is perspective. Let us now agree on a very simple definition of perspective. Perspective (from Latin perspicere, to see clearly), is an approximate projected representation of a scene as seen from a particular viewing location.

The two most characteristic features of perspective (or Renaissance/Linear Perspective) are that objects are represented with a smaller scale as their distance from the observer increases, and also that the scene experiences so called spatial foreshortening, which is the distortion of items when viewed at an angle.

A rudimentary knowledge of the different types of perspective is essential if we wish to understand how are able to see in 3D.

Types of Perspective

At this point I would like to make a distinction between two different types of perspective. Firstly, there is the type that arises from the perception of depth in human vision (sometimes called Visual Perspective [2nd Type] or True Perspective), and secondly there is the type that is created to facilitate the perception of depth in graphical images (Graphical Perspective).

Regardless of the features of the specific definition adopted, experts are in agreement that perspective is a very powerful depth cue in both the graphical and vision forms. It stands to reason therefore that in order to maximize the effectiveness of this cue in any representative method, it is important to mimic the overall optical affects of Visual Perspective as closely as possible. However, once again there are complications and disagreements over which is the most natural and realistic form of Graphical Perspective. 

It turns out that there are many different forms of Graphical Perspective, including Linear, Curvilinear, Spherical, Parallel, and Axial etc. Arguments continue to rage over which is the more natural. Linear perspective, which was first developed during the period of the Italian Renaissance, is perhaps the most familiar form of perspective to the Western eye. Nevertheless, vision and optical experts have noted that linear perspective is not a good approximation to so-called natural or real visual perspective. 

Curvilinear Perspective

In particular, at the outer extremes of the human visual field, parallel lines become curved, as in a photo taken through a fish-eye lens. It may surprise you to learn that the human visual field has a natural curvilinear shape! However painters, building designers and scientists have been aware of this fact for hundreds and possibly even thousands of years.

It has been claimed for example that the Ancient Greeks made the Parthenon columns bow outwards to account for – and correct – the curvilinear shape of the human visual field. Also painters like Leonardo Da Vinci and Turner added curvilinear effects into their depictions to more closely mimic reality as seen by the human eye. 

It has also long been known that it is possible to graphically re-create scenes in which the geometry conforms to an overall curvilinear shape similar in form to the views projected by a fish-eye lens. This form of perspective has sometimes been called Curvilinear Perspective, and it is a form of perspective which has an undeniable origin in the natural optics of scenes. Curvilinear perspective was ably explored in “Curvilinear Perspective, From Visual Space to the Constructed Image” by Albert Flocon and Andre Barre in their 1986 book. Artist Dick Termes has also produced many works based on curvilinear and 6-point perspective.

Curvilinear Perspective has a geometry which is closely related to the human visual field. In particular the rules of optics cause objects located at large distances from the central visual plane to be contracted in size, a true to life effect that is not depicted by Linear Perspective. Also others have noted that the human eye projects images onto a spherical retina, causing images to curve outwards in the same way as images in a wide field lens. 

In figure 2 below, we see two images from Flocon and Barre’s detailed mathematical study of Curvilinear Perspective, being drawings which ably represent the basic features of the natural curvilinear shape of human visual field. Especially noteworthy here is the curvilinear shape of wide-angle scenes, and the “realistic” (if exaggerated) foreshortening of scale in the lateral dimension. 

Curvilinear Realities

The facts of human vision presented here will come as a complete surprise to many. The question arises as to why it is that the facts of human vision should surprise us? Perhaps we are all too close to our own sense of vision to notice the natural curvilinear shape of every wide-field scene we ever look at, and likewise we do not generally take any notice of the miracle of 3D perception because it is ever present. Or perhaps we are all to-familiar with concepts such as Linear Perspective and/or the narrow field-of-view of photographs. In fact narrow-field photographic images do work rather well – in terms of 3D impression. 

Next time you are looking at 2D television or at a photograph notice how strong the affect of depth or the third dimension really is. You have no trouble here forming a good conception of the different depths of the objects that are depicted, and can form an accurate overall impression of scene geometries. No 3D glasses glasses are used here, and in each case we use “monocular” cues to form an accurate internal mental “model” of these scenes, which aids and supports our comprehension of 3D. 

At this point you may be asking yourself why it is that photographic, film and also television images do not exhibit scene curvatures. The answer is that they would if they covered a wide enough field of view – say around 180 degrees, and in any case optical designers have worked hard to ensure that the camera lenses involved eliminate such “distortions”. Note here that the so-called “Fish-Eye” wide-field lenses do show extreme curvilinear distortions similar to those depicted in Flocon and Barre’s detailed mathematical study.

Also when you have a moment, get a 30 cm ruler (longer is better), and whilst looking forward bring it close to the bottom of your nose, and notice how its shape at the outer edges curves upwards and forwards. It may take you a few minutes to be able to see this effect, because you are so accustomed to not noticing it ! But once you do you will be amazed to see your curved field of view as it really is for the first time. 

It is an established fact that wide-field optical perspective views are naturally curvilinear in form. Fish-eye lens views are not curved because of any effect introduced by the lens itself, but rather because that is how reality looks when you decide to project a specific scene over a very wide field of view! Eagles and birds see the world like this, that is in the ultra-wide field aspect. This fact leads me to conclude that curvilinear perspective has a strong foundation in reality. 

I am not claiming here that curvilinear perspective is necessarily a more real depiction technique than the linear ones that we are used to seeing, but only that all things considered it is an equally valid form of representation! Perhaps the main reason why Curvilinear Perspective seems so strange to us is that we have become so used to seeing everything in terms of straight lines and right angles. We may be missing out as a result on some quite spectacular images as a result. 

Therefore, although curvilinear scenes may at first seem like a distortion of reality, our discussion has shown that this shape is in fact rooted in the natural optics of scenes and also at the same time in the human visual field which is inherently curved in shape. 

Can the 3D World ever be “Truly” Represented?

Overall, many experts are in agreement that the human visual field is in fact curvilinear in shape. It is important to note here that Curvilinear Perspective is related to one of the monocular depth cues experienced when viewing real scenes, that of peripheral distortion experienced when viewing wide-angled scenes. Nevertheless, and despite the arguments in favour of Curvilinear Perspective being a good approximation to human vision when looking at – or observing – extremely wide-field scenes; it is nevertheless true that for ordinary vision, and over a normal field angle (say less than 90 degrees) the rules of Linear Perspective (one-point perspective) do in fact serve as a good approximation to human vision as it is employed in everyday circumstances.

Thus graphical images projected according to the rules of Linear Perspective do enable the viewer to adequately ‘perceive’ the changes in appearance of objects in relation to the Components of Visual transformation that happen as a result of depth.

The most realistic 3D would be one which employed all of the depth cues, however no method to date has been devised which has been able to employ them all. In fact it may not even be an achievable goal to construct a system so realistic that it employs all of these cues. Such a system would be indistinguishable from reality, and may in fact be an impossibility because it is known that human vision uses other yet more subtle scene based optical cues to form an impression of 3D. 

The fact that no single method employs all of the different depth cues (perhaps) leads to the conclusion that no one method of depth representation can be claimed to be more “real” than any another. What about holograms you may ask – don’t they employ all of the cues, both mono and stereo? I am afraid not. It is true that holograms do employ both monocular and binocular cues, but they do not usually employ moving images and so miss out on the moving cues. Other cues are often missed here including changes such as color, shadow, occlusion and also peripheral vision due to the relatively narrow field of view of most holograms. Overall I would conclude that no representative method currently employs all of the depth cues, and so none is true 3D in the strictest sense of the word. 

As an aside the author has invented a new type of mirror, named the “Hologram Mirror”, which produces an image of the self which “floats” in space in-front of the mirror’s surface. [14] Here, unlike with holograms, image occlusion affects are created, and the viewer obtains a strong and realistic impression of 3D. Similar optical devices may be used for producing improved types of 3D displays, specifically for interfacing (naturally) with future computing systems. A short explanation of the “Hologram Mirror” principle is salient. In Figure 3 (left drawing) we can see that the mirrors labelled 2 and 3 form an upside-down image of a subject (1) at 4, whereupon a (partially transparent) mirror labelled as 5 re- images this intermediate image into an upright, life-sized reflection of a person (7) that is observable “floating” in space at a short distance in-front of the same person (1). 

Conclusions

n conclusion, I hope that I have been able to convince you of the fact that you don’t need stereoscopic 3D glasses to see things in “true” 3D, and also not to dismiss out of hand the “reality” of curvilinear perspective scenes and/or multiple and distorted perspective views.