In this section, we explore the etymology, definitions and categories of perspective.
Most people have observed that objects diminish in size as they recede farther from the eye; that parallel lines appear to converge; that in general, the appearance of objects differs from reality. The rules governing this difference are determined by the principles of perspective. Yet this simple statement hides vast complexity.
As we shall learn, there are many types, definitions, categories, and applications of Optical Perspective. Still, all deal with how the three-dimensional world appears from a particular viewpoint (or multiple viewpoints). Perspective is concerned with observing, capturing, recording, modelling, shaping, representing, and projecting visual appearances.
More recently, perspective techniques have been employed to link, order, and explore vast numbers of digital images on new media systems.
Let us now examine the fascinating topic of perspective.
How can we adequately define perspective?
One definition in the context of pictorial and scenic art is: “The term ‘perspective’ may refer to any graphic method, geometrical or otherwise, that is concerned with conveying an impression of spatial extension into depth, whether on a flat surface or with form shallower than that represented. Perspective composition results when the artist adopts a visual approach to drawing and consequently portrays perspective phenomena such as diminution of the size of objects at a distance and the convergence of parallel lines in recession from the eye.”
However, this simple definition fails to encompass the multiple and diverse ways perspective has been employed in the past, today, and in topics such as vision, mathematics and computing.
The Oxford Dictionary (2nd ed.), defines perspective thusly:
- The science of sight; optics.
- An optical instrument for looking through or viewing objects with; a spy-glass, magnifying-glass, telescope, etc.
- The art of delineating solid objects upon a plane surface so that the drawing produces the same impression of apparent relative positions and magnitudes, or of distance, as do the actual objects when viewed from a particular point.
- The appearance presented by visible objects, in regard to relative position, apparent distance, etc.
- The relation or proportion in which the parts of a subject are viewed by the mind; the aspect of a matter or object of thought, as perceived from a particular mental ‘point of view’. Hence the point of view itself; a way of regarding (something).
- A drawing or picture in perspective; a ‘view’; spec. a picture so contrived as seemingly to enlarge or extend the actual space, as in a stage scene, or to give the effect of distance.
- A picture or figure constructed so as to produce some fantastic effect; e.g. appearing distorted or confused except from one particular point of view, or presenting totally different aspects from different points.
- A visible scene; a (real) view or prospect; esp. one extending in length away from the spectator and thus showing distance, a vista.
- A mental view, outlook, or prospect, esp. through an imagined extent of time, past or (usually) future; hence sometimes = expectation, ‘look-out’.
- Drawn or viewed in accordance with the rules or principles of perspective.
- The action of looking into something, close inspection; the faculty of seeing into a thing, insight, penetrativeness.
Patently, according to these various facets, perspective is a term that can portray multiple meanings! How then is it possible to better grasp the concept of perspective—and develop a complete understanding of the different ways it can be applied to practical situations?
Well, to begin, we ask: what is the etymological root of the term ‘perspective’?
The English word perspective comes from the Latin perspectiva, a complex term with a long history of different interpretations of meaning. Indeed the confusion continues in the present day because it is often apparent that perspective means different things to different people.
‘Perspectiva’, says Durer, ‘is a Latin word and means “durcheshung”: a view through something’. But the word perspectiva also sometimes refers to perspicere in looking directly at something or ‘seeing a form clearly‘ with your eyes.
While the etymology of the term prospective is interesting; these various interpretations of perspective can be resolved (ostensibly) by acknowledging that there are different types of perspective; and fundamentally (at a minimum), we have Visual Perspective (2nd type)—or direct looking at reality; and Graphical Perspective—or representation of realty using drawing/painting, etc.
Whatever the steps involved in the establishment of its modern meanings, by the 1480’s Brunelleschi had developed what painters today call ‘perspective’—a series of methods of graphic representation of a three-dimensional scene (typically 2D view of a 3D scene); as opposed to the overlaid processes of purely ‘visual perspective‘ employed when looking directly at a scene.
As artistic techniques evolved over the next 4-500 years, perspective was used to create many illusory three-dimensional spaces. Examples include trompe-l’oeil, view d’optique, capriccios, intarsia, grotesques, anamorphic views, quadrature (illusionistic ceiling painting), the illusion of expansive mountain-top vistas, and large-scale panoramas etc.
Today perspective, in one form or another, is key to vision, spatial modelling, and creating appearances. And the disparate applications of perspective theory within fields like art, mathematics, and drawing are too numerous to list. Indeed, much of modern scientific progress, plus a considerable fraction of productive efforts within art, culture, and technology, is built upon perspective methods.
In sum, our discussion teaches that perspective has not been in the past, and is not today, a single topic. We have multiple meanings for the term. Ergo, it is vital to establish basic definitions.
Categories of Perspective
There are two main classes of perspective: Visual and Symbolic Perspective.
As the name suggests, Visual Perspective (first type) refers to when a Visual Image is used to view, match or represent the visual appearance of a three-dimensional object/scene. At the PRC we study (chiefly) one particular form of Visual Perspective, named as Optical Perspective.
Optical Perspective seeks to capture, copy or display aspects of the Optical World. Optical Perspective can be separated into the Technical and Non-Technical classes of perspective. Whereby in this section we are explore the different facets of Technical Perspective.
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 (ref. Products of Perspective). Technical Perspective has distinct categories and sub-categories, as explained herein. 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/instruments/machines.
Technical Perspective includes all naturally occurring optical effects or—Environmental Perspectives—that can be classified under optics of the environment, including, for example, optics of the heavens (sun, moon, stars etc), shadow projection, line-of-sight problems, 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 (for example) birds, frogs, fish and insect eyes etc. Noteworthy is that all forms of Architectural buildings are herein classed as a form of Environmental Perspective.
It is essential to realise that most types of representation (including visual illusions) are included under the technical class, and 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).
According to our new theory of Technical Perspective, it has six sub-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;
- Graphical Perspective (views of natural and built worlds) – copying reality / creating appearance(s);
- 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 (views of built world) – 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).
To some extent, the different categories of Technical Perspective tend to overlap in terms of geometrical origins, and similarity of visual effects. Nevertheless, it is helpful to identify the sources of Technical Perspective by utilising categories.
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, which are supported by each perspective category, to a greater or lesser degree.
With perspective, we can approach objects from different viewpoints and scales, leading to distinct functions that can be isolated, catalogued and explored. Hence perspective allows accurate systemised worlds to be developed on media, and in the human mind.
Let us now analyse each category of Technical Perspective in turn.
Visual Perspective (second type)
Firstly, we have a form of Natural Perspective named as Visual Perspective—sometimes called ‘true’ perspective—that applies when a human views a scene in the real world (unaided eyesight). We shall skip over other forms of natural perspective, including subjects such as shadow projection, environmental optics, animal vision, etc. (temporarily)
Visual Perspective refers to looking at a scene from a specific viewpoint; whereby produced is a set of image transformation effects correlated with vanishing points (for example). Said effects are dependant on scene geometry relative to the observer viewpoint, or the particular distances and viewing-angles of scene structural elements.
It is important to note that visual perspective produces images that are enabled/limited by human vision. Several related visual processes accrue, including scene projection onto a curved retina, binocular perception, narrow field of distinct vision, shallow and curved plane of distinct vision [horopter], and scene perspective changes due to a moving head/eyeball.
Human vision is inherently complex and involves many interrelated physiological and psychological mechanisms, wherein certain key visual processes, such as binocular perception, are not fully understood.
Secondly, we have Mathematical Perspective, which refers to applying algorithmic rules to transform the appearance of scene/object geometry.
One example, is transforming an object’s form/scale and/or outline/shadow (visual appearance) according to a particular mathematical law/rule. Ergo, mathematical perspective is comprised of geometric transformations such as projection, translation, reflection, and rotation etc, plus related image distortion effects.
Mathematical Perspective often involves applying a valid theory of spatial geometry to a real-world problem (sometimes on a grand scale). Examples include modelling sun position relative to an earth-bound observer, Global Positioning Satellite (GPS) calculations, calculation of latitude/longitude position, mapping of spherical earth coordinates onto 2D maps, and astronomical calculations such as prediction of planetary orbits.
Mathematical Perspective often contributes to the other kinds of Technical Perspective. 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.
We can identify three different kinds of Mathematical Perspective:
- Algebraic Perspective [non-visual class] – involves algebraic formulae.
- Geometrical Perspective [analytical visual class] – graphical calculation.
- Projection Perspective [projective visual class] – spatial projection.
Algebraic Perspective (Type 1) refers to mathematical images composed of letters and symbols. Accordingly, we have categorised this as a form or sub-class of Symbolic Perspective.
Geometrical Perspective (Type 2) employs Analytic Geometry, also called coordinate geometry, or use of algebraic symbolism and methods to represent and solve spatial problems. Analytic Geometry refers to spatial modelling using graphical points, lines, and multi-dimensional objects and shapes, surfaces, and solids. Ergo, due to the inherent links with Analytical Geometry, we have categorised Geometrical Perspective as a form of Visual Perspective.
Another form of Visual Perspective is named simply as Projection Perspective (Type 3), which refers to applying projective principles to the creation of an image or view of a spatial scene or object. This latter type of Projection Perspective employs Descriptive and/or Projective Geometry to produce images of a spatial reality.
At the PRC, we study the visual classes of Mathematical Perspective; namely Geometrical Perspective (images formed by graphical calculation) and Projection Perspective (images formed by visual projection).
Another form of Technical Perspective is—Graphical Perspective—which attempts to create accurate representations of reality. For example, Graphical Perspective may simulate an illusory three-dimensional (3D) view—produced by notional and ‘natural looking’ vanishing points, etc.
Often Graphical Perspective is employed for transcription purposes in technical drawing, or else for artistic purposes. Graphical perspective is also extensively employed for computer-generated imagery, including applications such as Computer Aided Design (CAD), animation, and movie special effects (SFX, SPFX, FX) etc. Examples of graphical perspective are linear, parallel, axial.
Linear perspective is a well-known form of Graphical Perspective, that developed slowly over several hundred years. Below we see a sampling of images involving the early application of linear perspective, which is employed to create various artistic effects (but most importantly, illusion of spatial depth).
Noteworthy is that in a remarkable painting by Fra Angelico from the year 1435, he employs several of what we would today call ‘depth cues’ to provide an impression of depth, but without overtly using perspective techniques (such as linear perspective). Nevertheless, the result is a strong impression or illusion of a third dimension, produced by scale and focus changes, occlusion, light, shading changes, colour intensity, and geometric plus contrast effects.
Nonetheless, the mathematical rules of linear perspective did not emerge until 1450, with the legitimate construction by Alberti.
Noteworthy is that a debate, and long-running disagreement, exists between various artists, art-movement theorists, scientists, and visual designers; as to which kind of graphical perspective is the most natural or ‘real’.
Some experts claim that linear perspective is the most realistic, while others opt for curvilinear perspective, the distorted views of cubism etc.
The debate often crystallises around the number of vanishing points, whereby the use of 1, 2, 3, 6 (or more) vanishing points are employed within a single representation. The upshot is that various forms of Graphical Perspective are applied, producing many diverse (and sometimes surprising) visual effects.
Next, we have Instrument Perspective; which is generated whenever an instrument, of one type or another, is used to capture, measure or project an image of a scene (i.e., an aspect of reality).
Capturing, is when an instrument forms an image of a scene, being an image that has undergone a particular set of transformations relative to the scene geometry. One example is when a camera lens creates a two-dimensional (2D) image of a 3D scene—and introduced are image transformations which are unique to the lens itself (or another lens with identical focal length, plus similar optics).
Other kinds of Instrument Perspectives are commonly employed for image capture, including those provided by movie and television cameras.
Here moving images capture also the dimension of time, and changing/roaming perspectives are produced. Associated camera techniques include zoom and pan, ‘dolly’ or tracking shots, first-person tracking, plus use of multiple lenses on the same camera etc.
It is essential to realise that the Instrument Perspectives of today, provide far more informative views of reality because modern views are more detailed, sensitive, and all-encompassing.
For example, we have enormous and high-tech telescopes (on the ground and in space); that can form close-up views of incredibly distant objects (up to 13 billion light-years distant—the same objects being unbelievably faint), enabling the universe to be seen on distant scales. Plus, we have electron microscopes that can magnify up to 1-50 million times!
Sometimes a special type of device, named here as a Perspective Instrument, is explicitly employed to make accurate measurements in the real world. Whereby, a perspective instrument—is defined as an instrument whose operating principle is based on a particular class of perspective effect, enabling accurate dimensional measurements to be taken in relation to a particular aspect of reality.
Types of perspective instruments used for specific purposes include: for general tasks: the ruler, callipers and compass. For navigation: the quadrant, cosmolabe, the proportional compass, and the sextant; for astronomy: astrolabes, sundials, and planispheres; and for cartography: the theodolite, etc.
Of course, Instrument Perspectives are not limited to image capture/measurement but also relate to 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.
Sometimes an optical instrument is used for image capture and display simultaneously. One example is when a plane mirror forms a virtual image of an object. The reflected image appears at the same distance behind the mirror as the object is in front, only laterally inverted. Ergo the illusion of a doppelganger mirror world is created. Mirror Perspective is a unique instrument perspective that can form amazing optical effects commonly employed in magic tricks, stage illusions, etc. For example, clever mirror arrangements can produce unusual perspective-related: image fictions, ambiguities, distortions, and paradoxes
Noteworthy is that the visual, mathematical, and instrument perspective types; often tend to merge, blend or combine, on a particular viewing occasion (and often without the observer overtly noticing).
For example, when looking at a wildlife movie on your television at home, you see a combination of instrument perspective from the camera that captured the images; plus instrument perspective from the television that displays the images; and finally, you see everything using your visual perspective while looking at the television from a specific point-of-view.
Plus, remember that various kinds of mathematical perspective, will inherently be involved in, and contribute to, all of these other forms of perspective.
We can conclude that everyday perspective effects are complex indeed!
Perspective techniques are sometimes used to create illusions.
Typically a perspective illusion makes false impressions of size, depth, position, or transparency for objects. One example is when dimensionality is adjusted within a scene, making an object appear farther away, closer, larger, or smaller than it is.
The four types of perspective illusions are:
- Visual Perspective illusion: illusion by adjusted appearance (false direct view of reality);
- Graphical Perspective illusion: illusion by constructed appearance (false constructed view of actual or imaginary scene);
- Instrument Perspective illusion: illusion by secondary visual, or projected appearance (false formed view of actual or imaginary scene);
- Forced Perspective illusion: illusion by construction of a false reality, or by the representation of a false reality (distorted/transposed scene geometry)
Perhaps a fuller explanation of each illusive technique is required.
Firstly, we have Visual Perspective illusion, forming a false view of reality. Here the illusion is caused by the visual system and related ‘false’ perceptions. For example, a classic optical illusion is the apparent bending of a stick that is half-submerged in water (scene appearance does not match the actual geometry).
Secondly, we have Graphical Perspective illusion, where the goal is to copy reality or create the appearance of a reality (actual or imaginary scene). When we wish to copy reality, a representation of ‘true’ reality is created, such as a perspective drawing of a real-world scene (graphic reflects actual geometry of scene). As an alternative, we can use graphical perspective to represent an imaginary world; for example, a perspective drawing of a scene from the ‘Lord of The Rings’ book (graphic reflects actual geometry of imaginary scene).
Thirdly, we have Instrument Perspective illusion, where the goal is to form a false view of, or project a false appearance of, an actual or imaginary scene (view distorts/changes certain aspects of depicted scene geometry, but accurately reflects certain other aspects of depicted scene geometry). Noteworthy is that the illusion is caused by the instrument itself. An example is Virtual Reality, where a 3D virtual scene is projected into a person’s field-of-view.
Another well-known example of an Instrument Perspective illusion, is a hologram, which produces a three-dimensional image by reconstruction of light wavefront(s).
Next, we have Forced Perspective. The goal is illusion by constructing a false reality or representing a false reality. The scene’s geometry is distorted/transposed relative to its expected appearance (ref. unnatural scaling or unnatural position of objects), forming an optical illusion. Noteworthy is that the illusion is caused by the nature of the scene geometry itself. For example, said illusion makes an object appear farther away, closer, larger, or smaller than it is (the scene appearance is wholly at odds with the actual geometry).
An example of Forced Perspective, is when stage scenery employs false horizons, unnatural object scaling, false vanishing points, painted panoramic backdrops, mirror illusions etc., and in order to create the illusion of a distant background or the presence of ghostly figures etc. It is noteworthy that forced perspective illusions may come in various forms, for example, unique arrangements of 2D and 3D scenography, unusual photographic, graphic elements, and even large-scale holograms, etc.
Another form of Instrument Perspective illusion is projection mapping, similar to ‘mixed reality’ techniques such as video mapping and spatial augmented reality. In this technique, real-world objects, often irregularly shaped, are employed as a display surface for video projection (usually rear or back-screen projection).
Another interesting form of instrument-generated illusion is a Transparent Perspective Illusion that forms transparent, see-through, or multi-layer views/images of a three-dimensional object/scene. Whereby there are two basic types of transparent perspective. Firstly, we have projection of transparent views/images of three-dimensional objects/scenes onto a computer display. A second type is similar to ‘mixed reality’ techniques such as spatial augmented reality. In this latter technique, 3D transparent and multi-layer views of an object are projected into the physical environment stereoscopically through a Virtual Reality or Augmented Reality headset (for example).
Often both types of transparent perspective employ data from CAD models, or data from 3D computer scanning methods such as tomography, x-ray, or else from magnetic resonance imaging. It is also sometimes possible to project transparent and multi-layer views of the object’s interior structures onto a real-world object, enabling a person to see inside the object in question.
Importantly the different categories of illusion may blend. For example, use of a large photographic background depicting a mountain range could be used as a stage background (graphic illusion), whereby this illusive element contributes also to the visual and forced types of perspective illusion.
Many other types of illusion are possible, being types that are not related to perspective but rather have optical, physiological, and cognitive sources.
Finally, we have Media Perspective, which refers to the particular imaging capabilities, and visual transformations provided by a new media system.
Said Media Perspective, typically consists of multiple networked instances of other instrument/media perspectives. Examples of technologies that fit under the umbrella of media perspectives include: Internet and Web-based information systems, networked digital television systems, smartphones, and networked devices, satellite imaging systems, GPS, plus Virtual Reality (VR), Augmented Reality (AR), Mixed Reality (MR), and Extended Reality (XR) systems, etc.
Whereby, in the case of a highly developed media perspective system, views from one or more of the other categories of perspective (listed above), may be connected/linked together and then ordered, constructed, matched, mixed, explored, and cross-matched, etc. The result is the formation of a new class of Multi-view Perspective. Henceforth, a new era is emerging, of multiple and all-encompassing virtual ‘worlds’!
To produce a true Multi-view Perspective, it is a requirement that all of the contained perspective views must be adequately linked, indexed, and hence integrated together, and in such a manner that the resultant digital world is readily explorable. Specific examples of media perspective systems that work this way include Google Maps, Earth, NeRF-W, and Microsoft Photosynth. Unfortunately, the latter approach has now been abandoned/discontinued, but it is nevertheless an excellent example of an early-stage multi-perspectival media system.
Such systems create full-blown virtual worlds from a combination of thousands of images, often being photographs of the same area but taken from slightly different viewpoints or a variety of locations in space/time.
Through advanced mathematical techniques, it is then possible to merge all of these images into a single virtual space. Ergo such a virtual world, enables efficient navigation and rapid viewing, of a huge number and a great variety of different perspective views; including images taken from ‘mathematically’ created and/or normally ‘impossible’ viewpoints!
In any case, perceptions change in different media—and since our spatial perception is altered by every new medium we introduce—we need to manage, measure, and understand such effects.
The Kaleidoscopic Nature of Perspective
How can we summarise the fascinating topic of perspective?
As we have seen, perspective has exceptionally rich sources, combined with near countless influences/applications.
Moreover, it is a topic with multiple interlacing stories, a myriad of relations, and a deep fractal-like structure that opens up like a Russian doll into new levels of understanding almost without limit. However, we should not be put off by this complexity. Instead, marvel at how the single topic of perspective provides new vistas of knowledge wherever we happen to look!
The modern advances that are enabled by or which relate to perspective, are numerous and mind-blowing. New advances are coming fast, furious, and everywhere. A sampling of innovations includes super-high-resolution digital cameras (4k – 16k detector pixels in widest aspect); super-high-resolution digital monitors/televisions (4k, 8k, and 16k display pixels in widest aspect); plus high-bandwidth media formats and associated networking capabilities.
Ergo, today images are more detailed, colourful, distinct, wide-field, and realistic. Plus, images—of both still and moving kinds—are far more numerous, relevant and readily available, and hence impactful, than ever before. Implicit is that perspective is a term that encompasses a diverse set of visual, modelling, and also representation methods. Hence we must consider its multi-variant forms as employed in both Western and Oriental cultures.
Perspective is such a vast, broad, and profoundly influential topic that it should be taught as a distinct subject in schools, colleges, and universities. Whereby perspective is evidently the very definition of an interdisciplinary subject, with countless sources, links, and fundamental relations that penetrate to the very core of key phenomena in the arts, sciences, and technology. In any case, here at the PRC, we aim to fill this evident knowledge gap, providing comprehensive perspective-related theories, papers, books, images, courses, lectures, and documentary films—whereby all of these materials shall be free to access/download.
In conclusion, perspective has profoundly influenced many critical theories, important discoveries, and practical methods within art, science, and technology. Perspective is a keystone to progress, and it is truly a foundation of human civilisation.
-- < ACKNOWLEDGMENTS > -- AUTHORS (PAGE / SECTION) Alan Stuart Radley, 21st January 2023. --- BIBLIOGRAPHY 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) '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. --- Copyright © 2020-23 Alan Stuart Radley. All rights are reserved.