Our journey into the multiple dimensions of perspective begins here.

We start by defining the scope of perspective topics to be examined. Next, we introduce perspective concepts and related terms to accurately delineate, explore and meticulously analyse, the field of perspective (as a whole). Do not worry if said principles appear highly technical or are a little daunting to commit to memory—because we shall return to these ideas—at a slower pace and in greater detail—many times on this site.

We wish to establish perspective as a primary subject discipline, and accordingly, a fundamental Perspective Category Theory is developed. 

Nature of Perspective

We begin by asking: what is perspective?

Unfortunately, the task of answering this question is fraught with difficulties. Thousands of articles, books, theses, and treatises have been written to tackle the puzzling complications that arise in this respect. Perspective turns out to be one of the most fundamental, theoretically challenging, perplexing, yet elusive topics.

But why is perspective such a challenging topic? This is partly because many different types of perspective are known. We have visual, mathematical, and graphical kinds of perspective, etc. Secondly, multiple types of perspective tend to operate simultaneously, so it is difficult to identify the root causes of specific visual effects. For example, we cannot easily separate environmental optics (or Environmental Perspective) from the human visual system (Visual Perspective).

As a result, perspective is an inherently complex topic. Anyone who seeks a deeper understanding of perspective has to engage the old noggin, as my father used to say. But how can we deal with such complexity? Perhaps only by defining all classes of perspective, identifying the salient features of each type, before characterising the particular circumstances that bring each form into play.

We must open our eyes to all of the varied dimensions of perspective!

The Categories of Perspective

In the Oxford English Dictionary (2nd Ed.), perspective is variously defined as the science of optics, an optical instrument, the art of representing objects in a three-dimensional (3D) space, a drawing or picture, a visible scene, and the act of looking, etc. Noteworthy is that perspective also refers to a mental view or prospect, etc.

Accordingly, we can define perspective, in general terms, as the formation of an Image—or a representational pattern—of a state of affairs present in a spatial reality (e.g. natural, illusive, imagined space, etc). Perspective may refer to the process or procedure that forms said Image (e.g. Graphical Perspective), or it may refer to the end result, the Image itself (e.g. a perspective drawing).

Said Image may be either of the visual type to reflect visual features of an object/scene, or be of the non-visual type and reflect non-visual characteristics (ostensibly). Accordingly, there are two main classes of perspective: Visual Perspective and Symbolic Perspective (non-visual perspective), which correspond to the two basic kinds of images: Visual and Non-Visual Images.

As the name suggests, Visual Perspective (of the first type, or not overtly related to the human visual system) refers to when a Visual Image is used to view, match or represent the visual appearance of a spatial object/scene. Alternatively, Symbolic or Non-Visual Perspective deals with Non-Visual Images (e.g., literal, logical, ontological, or algebraic images). Noteworthy is that for the non-visual class of perspective, the ‘images’ formed may relate to spatial/visual features or other non-visual features of the object in question.

Figure 1: The Categories of Perspective

(Enlarge figure by opening in a separate window)

Figure 1 shows the Categories of Perspective and details processes that give rise to one class of Visual Perspective, namely Optical Perspective. Note that missing from this diagram are several other kinds of Visual Perspective, including Sonar (sound ‘imagery’), Magnetic, Mechanical, Gravitational Perspective(s), etc.

Perhaps without realising, we are building a class hierarchy or taxonomic tree of perspective. It is noteworthy that classes are often overloaded on said tree, whereby one class of perspective is the daughter of a higher level class. For example, we have Technical Perspective, a type of Optical Perspective, which is, in turn, a type of Visual Perspective (ref. one branch of the class tree structure).

As our discussion continues, we shall build our knowledge of the lower-level classes, particularly those under the Optical Perspective branch.

Visual and Non-Visual Images

The reader may find the inclusion of the symbolic or non-visual class of images—and associated perspective forms—a little surprising. At least when defined alongside perspective proper (the visual, graphical perspective types etc). Still, non-visual images are fundamental throughout literature, science, culture, etc.

Word Image can build up a picture of an object/scene using purely language-based descriptions, whereby the result is sometimes identical to a Visual Image. This can happen by metaphor, descriptive methods, or mathematical formula. For example, the ‘visual image’ of a three-dimensional cube of a specific size and colour can be produced by either image class.

Nevertheless, it is usually the case that Visual and Word Images of the same object/scene, will differ significantly. Shape, fine details, surface contours, spatial relations, optical effects etc, are often more accurately captured with Visual Images. Alternatively, Non-visual Images can convey hidden or difficult-to-see facets missed by visual methods. Said features include chemical, atomic and sub-atomic makeup, invisible forces at play, or other invisible facets, such as micro-processes, etc.

We can make an interesting distinction between permanently and temporarily invisible features. For example, it is sometimes the case that previously invisible features of an object/scene become visible when viewed using the correct perspective or using an appropriate viewpoint, scale, timescale, etc.

Contact Perspective

Contact Perspective refers to the direct mapping of spatial form, namely a spatial object/scene, and by utilising physical contact using one, two or three dimensional examination procedures (1D/2D/3D space). Contact Perspective is perhaps the purest—or most ‘real’ class of perspective—and because it deals with the physical world in its actuality. 

We can identify two different kinds of Contact Perspective:

  • Handling Perspective – scales, rulers, callipers, sensors/actuators, etc.
  • Moulding Perspective – casting processes, projection moulding machines, etc.

It would seem difficult to argue with (or contradict) data that arises from direct contact with the object of attention, at least in scientific or materialist terms. However, it is essential to realise that many of the same data-gathering tasks arise with both Contact and Optical Perspective. For example, the problem of determining true object shape versus apparent and/or partial object shape.

Note that we have included specific perspective-related handling and calculating methods within our Perspective Category Theory, namely scales, rules, callipers, slide rules, etc., as falling under Instrument Perspective. Ergo, all measuring/calculating devices are classified under the Technical, Optical Perspective branch. We have done so to simplify the classification scheme. Such devices often involve using human vision to take readings or otherwise to apply said methods to physical reality.

In sum, the various kinds of Contact Perspective have strong links to the other types of Visual Perspective; and they have been widely used to develop, validate and certify Mathematical, Graphical, Instrument, and Media types of Perspective. 

Optical Perspective

Optical Perspective is concerned with capturing, measuring, or representing realistic views/images of dimensional space. Optical Perspective encompasses all forms of perspective that use, or purport to use, light, or electromagnetic radiation, to analyse/represent a spatial scene; including all wavelength ranges from gamma rays, x-rays, visual spectrum, microwaves, radio, etc. .

Some forms of Optical Perspective employ artificial/simulated ‘light-rays’ that lie beyond the bounds of ordinary physics. Said techniques can sometimes operate on an impossibly vast or tiny visual scale; or produce images that pass straight through solid objects, etc. However, the use of such techniques does not necessarily make the resultant visual images any less real or accurate (ref. heavenly, sub-atomic and medical optics, Virtual Reality, etc).

Noteworthy is that we have adopted the term Optical or Technical Perspective for a broader range of meanings than purely ‘optical’ phenomena such as photography (for example). Included in that term is any systematic optical related process that produces Visual Images (of any type) that concern spatial reality.

As we shall learn, there are many different categories of Optical Perspective. Still, all involve spatial imaging principles that reflect scene geometry aspects with varying degrees of visual realism or accuracy.

Optical Model of Spatial Reality

Another way of characterising a Visual Image produced by Optical Perspective, is as an Optical Model of spatial reality (existing or imaginary). How so?

Well, a perspective image is typically a two-dimensional representation (or projection) of structure in a three-dimensional world, a simplified or idealised conception of that world. The same image faithfully reflects, generally on a smaller scale, aspects of the existing structure to function as a compass (for a particular extent of prescribed space). In this manner, the perspective image becomes a map (or model) of the external spatial reality. Often, the model is put forward as a basis for perceptions, calculations, predictions, or further investigation.

But if the perspective image is a model, it stands to reason that it results from modelling procedures. In this respect, we consider an optical instrument that produces sufficiently accurate perspective images in form, detail, etc., as a modelling process for the projected spatial reality.

In light of this discussion, Optical Perspective is the production of an Optical Model (or Visual Image) that is the outcome of a systemic Optical Process comprised of a set of imaging and/or modelling procedures.

We have identified five Optical Processes that produce perspective images:

  • Optical Imaging – environmental optics, eyes/vision, optical instruments, etc.
  • Mathematical Modelling – Analytical, Descriptive, and Projective Geometry.
  • Graphical Modelling – artistic drawing, painting, CGI, etc.
  • Optical Modelling – geometrical optics, ray-tracing, image processing, etc.
  • Computer Modelling – GIS, GPS, Virtual Reality, Hypermedia, etc.

We have defined Optical Perspective to operate wherever a real, simulated, or modelled optical view/image of a spatial reality is produced (by whatever method).

It is worth noting that the aforementioned Optical Processes are by no means independent of one another. It is often the case that several different modelling processes are involved in producing a particular perspective image. For example, movie Computer Generated Imagery (CGI) typically involves Mathematical, Graphical, Optical, and Computer Modelling simultaneously.

Now that we have adequately delimited our primary subject matter, we can explore Perspective Category Theory in detail.

Perspective Category Theory

We adopt a scientific, logical approach to the field of perspective.

Some people may object to applying scientific principles since perspective is evidently both an art and science. However, doing so has many advantages, most notably because we can establish concepts and principles upon which the entire field of perspective is built.

Patently, perspective sometimes deals with unreal, imaginative, and illusionistic three-dimensional spaces. However, these same spaces are often visualised, or represented using realistic ‘looking’ visual images (or perspective proper). Henceforth, visual realism is a key component of all perspective forms.

Desired is a new perspective theory, which hopefully can be used to understand, model, and apply complex perspective phenomena in the broadest range of practical circumstances. A first step in building a theory is to identify a group of concepts—or set of matching universal forms—that purports to represent all of the different facets of perspective.

The sections below introduce a set of key principles upon which Perspective Category Theory is built. Wherein most of these concepts can be explicitly/theoretically proven—but if not, then said principles can hopefully be demonstrated empirically.

The Optical World

Optical Perspective seeks to view, match, or represent aspects of the Optical World—or the changing appearance of objects as conveyed to the camera, instrument, or the eye/detector by emitted or reflected light (ref. using real, modelled, represented or imaginary light-rays). Ergo, a key goal of perspective is to explore the Physical World by using the Optical World to probe spatial reality accurately or with sufficient realism.

The natural products of human vision are named as the Visual World (unaided eyesight)—or the transformation of the Optical World according to the rules/processes of human vision (ref. physiological and psychological optics). In addition, the Represented World refers to human attempts to depict aspects of the Physical, Optical or Visual World(s).

Finally, we have Perspective Illusions or the Illusive Optical World, which make(s) false impressions of size, location, depth, position, or transparency concerning a scene/object. These are produced by natural and human-made methods and by utilising optical phenomena plus special optical arrangements, such as mirrors, holograms, etc., or computer-generated Virtual or Augmented Reality systems.

The Physical, Optical, Visual, Represented and Illusive Worlds possess fundamental mapping relations. However (as we shall learn), the precise nature and validity of such correlations are by no means straightforward or guaranteed. Ergo, a secondary goal of Optical Perspective (biggest sense) is to understand how the different facets of each World fit together to produce visual reality.

Optical Perspective can be separated into the Technical and Non-Technical categories of Perspective—as explained below.

Technical Perspective (definition)

Technical Perspective refers to any systematic process that produces a detailed visual image, measurement, representation, model or view, of a three-dimensional object or scene. Technical Perspective is formed using optically, mathematically, geometrically, or logically correct/known/consistent principles.

Technical Perspective can be separated into distinct categories and sub-categories. Still, importantly all types of Technical Perspective are recognised by having a direct connection to human vision, environmental optics, and/or related visual processes/methods or associated instruments/machines.

Technical Perspective includes all naturally occurring optical effects that can be classified under optics of the environment, including, for example, optics of the heavens (sun, moon, and stars etc), shadow projection, panoramic views from mountain tops, underwater optics, etc. Also classified under the heading of Technical Perspective is the vision of all animals, including human vision and birds, frogs, fish, insect eyes etc (named as Visual Perspective of the second type). 

It is essential to realise that most types of representation (including visual illusions) are included under the technical class. Specifically, this is so—wherever said effect relates to the ordinary laws of environmental optics or the expected results of human vision (ref. physical and psychological optics).

Non-Technical Perspective (definition)

Non-Technical Perspective is defined as any non-systematic process that produces a detailed visual image, representation, model or view, of a three-dimensional object or scene. Non-Technical Perspective does not involve the application of a comprehensive visual theory/process/method of image formation or optical projection. Non-Technical Perspective is formed using optically, mathematically, geometrically, or logically incorrect/unknown/inconsistent principles (including nonphysical processes).

Examples of subjects that are normally classified under Non-Technical Perspective include Aperspective, Axial, Inverted, Negative, Time (random combinations of images taken at different epochs), freehand drawing, birds’ eye, etc. Whereby a Non-Technical Perspective category is recognised by evident non-connection to the ordinary laws of optics, or because it fails to match the expected results of human vision. Non-Technical Perspective includes graphical methods that today are classified as Ancient Perspective (e.g. Aspective or Egyptian Perspective) or Byzantine Perspective (Reverse/Inverse/Inverted Perspective).

The first systematic or Technical Perspective ever developed is usually agreed to be Renaissance Perspective (i.e. Central or Linear Perspective). Whereas today other forms of Technical Perspective are known; including Parallel, Oblique, 6-point, and many different kinds of optical projection including mapping projections of the earth and heavens, etc.

Natural Perspective (definition)

Natural Perspective is defined as any naturally occurring instrument/process that produces a detailed visual image, view or shadow/outline of a three-dimensional object or scene. Natural Perspective includes Visual Perspective (second type) or human and animal vision (eyes).

Environmental Perspective is a type of Natural Perspective that refers to naturally occurring optical effects such as projection of shadows/outlines, line-of-sight problems, translucency/reflection/colour effects, etc., that happen in the natural environment (and without human interference). For example, astronomical, atmospheric, underwater optics,  optics of crystals, rainbows, etc. Noteworthy is that all forms of Architectural buildings are herein classed as a form of Environmental Perspective.

Categories of Technical Perspective (primary categories)

The primary categories of Technical Perspective are listed below and encompass the vast majority of perspective methods employed in modern times.

Technical Perspective has six primary categories:

  • Natural Perspective (views of natural and built worlds) – including Visual Perspective (second type) or direct looking at reality using human or animal vision (view of a three-dimensional form / scene). Includes also Environmental Perspective or naturally occurring optical effects such as projection of shadows/outlines, line-of-sight problems, translucency/reflection/colour effects, astronomical, atmospheric, underwater optics, etc; Environmental Perspective Includes human designed optical vistas (e.g. architecture).
  • Mathematical Perspective (views of natural and built worlds) – modelling reality / shaping appearance(s); includes the second mathematical type or Geometrical Perspective, and also the third type dealing with optical projections (projection perspective);
  • Graphical Perspective (views of natural and built worlds) – copying reality / creating appearance(s) or representations of reality;
  • Instrument Perspective (views of natural and built worlds) – looking at, capturing and measuring reality; and projecting appearance(s);
  • Forced Perspective – aka ‘false’ or ‘trick’ perspective (designed views of natural built worlds) – visual illusion by the construction of a false reality, or by the representation of a false reality (distorted/transposed scene geometry);
  • Media Perspective (views of natural and built worlds) – connecting/linking, ordering, constructing (mimesis), matching, mixing, exploring, and cross-matching: multiple perspective view(s).

We shall go on to explore the six-categories of Technical Perspective in great detail throughout this site (beginning under the Definitions section). Literally hundreds of perspective sub-categories (and forms) fall under these six headings.

Note that the form of Visual Perspective listed under the Natural Perspective category is identified as the second type of Visual Perspective; which relates directly to human vision. This form is contrasted with the first form of Visual Perspective, existing at the top of the Perspective Taxonomic Tree, under which all classes/categories of Optical Perspective are contained.

The Systemisation of Space (and Time)

Technical Perspective fosters the systematic analysis of space (and time). 

By accurately prescribingindexing, and modelling reality; perspective enables accurate systemised worlds to be developed on media. Through perspective views, new kinds of visual solutions can be created in the human mind or model and subsequently be recorded on media. 

The explicit three-dimensional worlds made possible using perspective machines/methods enable new ideas, concepts, forms, and conglomerations of objects to be explored. Some objects/processes are really existent ones, and some only imaginary (at a particular time). 

Perspective gifted humans with a god-like power to create the world in our image; because it ensures that objects/processes of all kinds are accurately conceived, perceived, manufactured, and employed! 

Goals / Products of Perspective

Three goals of Technical Perspective can be identified (in human terms). 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.

Expressed in quantitative terms, the primary goals of Technical Perspective are certification of sightmeasurement, and representation, whereby we make accurate images, views, maps, and models, etc.

Noteworthy is that the different categories of Technical Perspective reflect varying degrees of abstraction and levels of accuracy in achieving a close correspondence with spatial reality. And usually, the goal is optical veracity—or comprehensive visual truth. Even Forced Perspective and Perspective Illusions are concerned with fooling the eye/brain into believing in a ‘false’ Physical World. Such illusions are produced by using natural looking perspective methods.

The objectives of Non-Technical Perspective are less distinct. They reflect similar goals but involve arbitrary visual techniques that are less systematic and not grounded in strict scientific principles. Ergo Non-Technical Perspective is not primarily concerned with achieving a high degree of correspondence with Physical Reality, but rather with improving the clarity or perceptibility of certain visual features (for example).

As stated, both the Technical and Non-Technical Perspective types fall under the heading of Optical Perspective, and refer to any process that forms a detailed visual image, measurement, representation, model, or view, of a three-dimensional object or scene. These outcomes are named the Products of Perspective. Noteworthy is that all of these products are normally related to Optical Processes of one kind or another (apart from certain direct-contact instrumental measurements, etc).

In summary, all categories of Optical Perspective seek to capture, copy or display aspects of the Optical World; and they only differ in terms of the degree of realism produced (reality correspondence) and the particular visual features that provide said realism.

Transformation of Visual Features

It is helpful to consider perspective as a set of visual processes, or a set (or chain) of visual transformations, that affect changes to the appearance of a three-dimensional scene (for a particular scenario).

Perspective is a process that can be broken down into a set of visual processes that occur while viewing/surveying/representing a spatial scene. These effects 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.

Accordingly, standard visual transformation(s) can be identified that delineate changes 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.

The types of visual transformation(s) are numerous and varied in degree of effect – and may include (for example) – varying shape changes to objects according to position within a perspective image. It may be very difficult (if not impossible for a specific scenario) to identify/quantify/measure all of the existing visual effects.

A better approach, therefore is to identify and map standard types of Perspective Phenomena that lie behind changes to Visual Features (for a particular category of perspective), and hence to comprehensively analyse how a particular perspective image or view comes to be (in general).

Perspective Phenomena

An individual category of perspective demonstrates, and is recognised by, certain Perspective Phenomena. The Perspective Phenomena refer to all of the intrinsic, or general, changes to Visual Features that occur according to a particular Category of Perspective and its inherent processes. 

The Perspective Phenomena comprise, in each case, all of the defining features of a particular Category of Perspective; including types such as Visual, Mathematical, Graphical, Instrument Perspective, etc. An example might be a specific form of Graphical Perspective, for example Linear Perspective, whereby some of the (apparent) Perspective Phenomena include:

  • The diminution of remote objects;
  • The foreshortening of dimensions not perpendicular to the direction of vision;
  • The convergence of parallel lines to a point on the horizon;
  • ++ several other Perspective Phenomena that affect Visual Features.

Patently each class of perspective will exhibit a unique set of Perspective Phenomena, whereby certain phenomena are shared (or embodied) by a number of different categories of perspective. Embodiment of the Perspective Phenomena may happen by certain naturally occurring visual processes, or else by human design, but irrespective of cause we refer to the instantiation of said phenomena as the Methods of Perspective.

Methods of Perspective

A Perspective Method is any natural, practical or theoretical technique that aids in forming a detailed visual image, measurement, representation, model or view, of a three-dimensional object or scene. Typically one or more such methods are combined to render a view/image/measurement/model according to a particular instance of a category of Technical Perspective (for example).

Formerly, a Perspective Method is often encapsulated in one or more mathematical formulae/technique(s) that is/are applied systematically to a scene—to produce a view, image, model or representaion. This methodical procedure can happen naturally due to (for example) image formation in a human eye, or else it can be applied using a graphical technique (for example, the method of linear perspective).

Typically one or more Perspective Methods are combined to render an image according to a particular Category of Technical Perspective. Practical examples include Perspective Instruments such as a lens, camera, the perspective window, planisphere, projector, computer, etc. Perspective Methods include all instrument design features plus operating methods. Examples of theoretical methods include a particular projection algorithm, the deliberate creation of ‘designed’ vanishing point(s), field-of-view, projected scale, etc.

The visual results of perspective can appear in various ways, with varying results. Ergo there are many different Categories, Phenomena, Methods, and Forms of Perspective which operate to produce a vast range of visual effects (image forms). Hence it is by no means clear which particular perspective technique is the most real. This problem is central to the ongoing arguments over whether or not perspective is innate (real/natural) or simply a convention (human invention).

Perspective Forms

Each type of Optical Perspective can be realised in a specific Perspective Form; whereby there may be a whole variety of different kinds of Perspective Forms possible for each Perspective Category or Sub-Category.

For example, Linear Perspective is a Sub-Category of Graphical Perspective. Still, it can manifest itself in a whole variety of different Perspective Forms, including in forms such as in a drawing, painting, computer-generated image, virtual reality, etc. Each instantiated Perspective Form may be significantly, or subtly, different from the other Forms; even for a notionally identical image/view of the same three-dimensional scene/object.

Patently each Perspective Form will be instantiated by means of specific set of Perspective Methods.

Functions of Perspective

Perspective, in general, works to enable viewingprescribing, matching, modelling, exploring, representing, and mixing images, of the physical world. These are the Functions of Perspective, which are supported by each perspective category, to a greater or lesser degree.

Perspective Classification

How is it possible to classify a subject with so many different facets?

We have classified these facets into distinct cubby-holes as best we can. Nevertheless, specific categories of perspective retain a degree of non-precision—or are highly subjective—and such is the case with Visual and Graphical Perspective(s).

As stated, all types of Technical Perspective have strong links to Mathematical Perspective, and most are subject to the vagaries of human perception. Indeed perspective raises fundamental questions about the nature and limits of human perception/understanding of reality, whereby related problems have been puzzled over for 2000 years—and are still debated today.

Perspective is a foundational topic, having much in common with ‘theoretical’ subjects like algebra, geometry, and number theory. Still, it also has close links to physical disciplines like art, representation, scientific imaging/measuring instruments, etc. Perspective is a set of natural, analytical and experimental methods that enable humans to better perceive, measure, model, and create views/images of spatial reality according to our particular needs.

We aim to bring a rigorous understanding to the central topic of perspective. Accordingly, we have developed a new categorical theory of perspective that elucidates links between the different types of perspective, outlines the big picture, and encompasses all aspects in a single model. Our primary objective is to identify all of the Goals, Classes, Categories, Phenomena, Products, Methods, Forms, and Functions of Perspective; without worrying so much about the underlying philosophical questions (at least initially).

Visual Perspective (first type) is concerned with forming views of three-dimensional reality. Ergo understanding the different categories of space is of central importance if we wish to understand perspective.

Categories of Space (Mapping Relations)

Perspective seeks to reflect visual aspects of the Physical World (typically).

Perspective is concerned with obtaining views of dimensional space, and it deals with transformation of one kind of space to another kind of space. Whereby perspective happens when a three-dimensional Object located in three-dimensional Object Space (real, virtual, or imaginary) is transformed into a two or three-dimensional Image located in dimensional Image Space (real or virtual).

Perspective remains a complex topic because it involves the comparison of fundamentally different categories of space. For example, reconciliation of the 3D space of the Physical World (Physical Space), with the 2D space of Graphical Perspective (Graphical Space). Often, and despite a strong desire to attempt equality, this reconciliation is impossible to achieve completely; because one deals with dimensions that must map without any physically-based 1:1 correspondence (e.g. transform 3D space to a 2D space).

It is essential to realise that one kind of perspective will typically involve multiple types of space. Let us take the example of a person looking at a painting of distant mountains. The viewer creates a Visual Space using pictorial elements perceived from a Graphical Space (the painting), including an inherent Mathematical Space expressed by the artist as a Linear Perspective projection. The overall effect is an apparent (or illusionistic) perspective view of a Natural or Physical Space (ref. Environmental Perspective).

The upshot is that anyone who wishes to apply perspective to a visual problem has to decide first which categories of space are involved; before identifying which Perspective Categories, Phenomena, Methods, and Forms are operating, and so to understand the generated Perspective Product(s). Ergo understanding the different categories of space, and operating processes of perspective, is of central importance if we wish to explain how and why the Optical, Visual and Represented Worlds appear as they do.

The Manifold Nature of Perspective

Technical Perspective is key to vision, spatial modelling, and creating appearances.

We have defined Technical Perspective as any systematic process that produces a detailed visual image, measurement, representation, model or view, of a three-dimensional object or scene. Whereby said Perspective Products are comprised of a specific set (or pattern) of Visual Features including points, lines, plane figures, solid shapes, shades, shadows, reflections, translucency, colour, texture, scale, etc.

Perspective prescribes how an object’s Visual Features appear from a particular location and angle-of-view, distance, etc; and according to a specific observation scenario. But it is often challenging to discover the root causes of even a simple perspective effect. This is because it is usually the case that we have several optical, visual, modelling, or representation-based Perspective Methods happening simultaneously. Our only hope is to identify and classify perspective changes to Visual Features, and then analyse how each piece of the perspective ‘puzzle’ fits together to produce the visual world as we know it.

In any practical situation, we may have several different Categories of Perspective working together to produce the Products of Perspective. Accordingly, many different types of space are involved simultaneously. In a perspective process, we see a flow of Visual Features translated between different spaces that accordingly experience a series of visual transformations.

Hopefully, the reader begins to see how the different facets of perspective work together to produce perspective’s (frequently) manifold nature.

Perspective Instruments

Our discussions thus far have side-stepped an essential aspect of perspective: the use of instruments and machines to view, image, measure, model and represent visual aspects of the Physical World. An Instrument Perspective; is generated whenever an instrument, of one type or another, is used to view, capture, measure, or project an image of a scene. 

For image viewing and capture, we have cameras, telescopes, microscopes, etc. Other instruments have been developed for image projection and display. In this respect, many innovations have occurred, ranging from the earliest such as the Magic Lantern and Kinetoscope to modern cinema techniques, such as wide-screen and 3D cinema, plus immersive systems, including IMAX theatres. 

Today we have new types of media and networked computing devices, including smartphones and tablets, which can display and manipulate a variety of perspective views. However, it is essential to realise that perspective instruments and machines are conceptualised, built, and operated, according to mathematically, geometrically, and logically consistent principles (hopefully). In other words, all such devices are enabled by valid perspective theory/methods.

Patently, perspective instruments and machines have the potential to add further complexity to the already complex perspective topic. But we should not be worried because often perspective instruments can simplify, expedite and clarify matters. This happens because perspective devices give us new choices regarding what we see, how, when and where. In the process, we are magnifying our natural visual capacities and boosting the efficiency, number, and usefulness of perspective images and techniques, measurements, models, etc.

Limitations of Category Theory 

We have defined Technical Perspective as any systematic process that produces a visual image, measurement, representation, model, or view, of a three-dimensional object or scene. Using Technical Perspective, we can approach objects from different viewpoints and scales, leading to distinct visual/geometrical functions that can be isolated, catalogued, and explored.

Six distinct categories of Technical Perspective have been identified: Natural, Mathematical, Graphical, Instrument, Forced, and Media Perspective(s). These categories are assigned according to the nature of the creative process(es) employed to produce the attendant perspective view(s)/image(s). By clearly identifying perspective categories, we can analyse and (hopefully) explain the precise origins of perspective phenomena and better understand/manage attendant physical processes.

Let us take the example of a person looking at a painting of distant mountains. The viewer uses eyesight (Visual Perspective) to perceive pictorial elements from a Graphical Space (the painting), including an inherent Mathematical Perspective expressed by the artist as a Linear Perspective projection. The overall effect is an apparent (or illusionistic) perspective view of a three-dimensional natural scene (ref. Natural or Environmental Perspective). 

Notice how Perspective Category Theory helps us to name which types of perspective phenomena are operating in a practical situation. However, notice that Linear Perspective is both a form of Mathematical and Graphical Perspective simultaneously. Hence a particular perspective image/view may fall under one or more categories. Most types of perspective involve mathematical principles, or Mathematical Perspective, to one degree or another. Similarly, all kinds of Visual Perspective (2nd type) involve Environmental Perspective, etc.

Another vital factor to consider, which complicates matters further, are the psychological dimensions of Visual Perspective, which refers to the operations of the human mind on the perception of perspective images/views. The implications of the latter topic in our discussion are important and complex and must be adequately considered.

In any case, and especially considering the multi-variant nature of perspective, the aforementioned categories help us break down what would otherwise be a highly complex (and often obscured) group of optical/geometrical/perceptual processes. Ergo, Perspective Category Theory helps us to identify and separate Technical Perspective into an eminently manageable set of more straightforward and readily identifiable phenomena.


We have concluded our brief introduction to perspective methodology; previewing perspective concepts employed throughout this site. Further, we have developed the founding principles of Perspective Category Theory.

What have we learned? Only that perspective is by no means an old, abandoned, or dead subject. On the contrary, today, we see more types and applications of perspective than ever before. Perspective instruments such as telescopes, microscopes, cameras, televisions, computers, etc., take perspective methods to a new level of usefulness. The primary functions of perspective are significantly improved in power, scope, accuracy, and the sheer number of eminently practical application areas.

The primary perspective goals are certification of sightmeasurement, and representation, whereby we make accurate images, views, maps, and models. In this manner, we can develop and test real-world theories. Concordant with such developments has been using perspective to aid in the accurate mapping of environments, and creating new environments.

Overall, perspective is the preeminent way for humans to image, view, match, and represent a whole universe of visual complexity. Perspective has a fascinating past, exciting present, and promising future. By utilising advanced perspective methods/instruments, we vastly expand the field/range/depth of human vision. And given the amazing triumphs provided by the remarkable history of perspective, who can say what incredible heights perspective can take humanity to?

Prepare yourself for an invigorating ride as we embark on a fascinating journey into the marvellous and varied world(s) of perspective!



Alan Stuart Radley, 15 February 2023.
Kim Henry Veltman, from: 'The Sources and Literature of Perspective', 2004.


Radley, A.S. (2023) 'Perspective Category Theory'. Published on the Perspective Research Centre (PRC) website 2020 - 2023.

Radley, A.S. (2023) 'Dimensions of Perspective', book in preparation.

Radley, A.S. (2023) Perspective Monograph: 'The Art and Science of Optical Perspective', book(s) in preparation.

Radley, A.S. (2023) 'The Dictionary of Perspective', book in preparation. The dictionary 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 dictionary was made available on the web on the SUMS system (2002-2020).

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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