Perspective is fundamentally concerned with our relationship to three-dimensional space. It describes how spatial reality is viewed, imaged, projected, measured, represented, transformed and experienced, and how the apparent properties of objects and scenes change according to viewpoint, direction, distance, orientation, projection geometry, scale, resolution and the visual or imaging system involved.
Perspective is therefore much more than a technique for drawing the illusion of depth on a flat surface. It provides a framework for understanding how 3-D object space becomes a visual, optical, graphical, photographic, mathematical, instrument-generated or computational image space.
At its broadest, perspective connects spatial reality → perspective process or system → image or perspective space → visual interpretation.
What Is 3-D Space?
Three-dimensional space is a spatial continuum in which three values or coordinates are normally required to establish the position of a point.
In ordinary geometry these dimensions are commonly described as length, width and height or depth. A Cartesian coordinate system represents them using three perpendicular axes — x, y and z — meeting at an origin.
Physical objects occupy three-dimensional spatial reality. A cube, building, landscape, human body or planet possesses spatial extent rather than merely the two dimensions of a flat image.
However, within perspective the term 3-D can refer to several very different things. It can mean a physical three-dimensional object or scene, the visual experience of spatial depth, a three-dimensional model, or an image that merely produces a convincing impression of depth.
Three Different Kinds of Space
A useful starting point is to distinguish three broad kinds of space involved in perspective:
- Physical or Object Space — the three-dimensional spatial reality containing the objects or scene being viewed, imaged, measured or represented.
- Natural Optical or Visual Space — the image or appearance produced through optical processes and, in human vision, through Visual Perspective Type 2.
- Artificial or Represented Image Space — the graphical, photographic, instrument, mathematical or digital space in which a representation of spatial reality is formed.
These spaces should not be assumed to be identical. Perspective is concerned precisely with the relationships, correspondences and transformations between them.
Object Space and Image Space
Object Space, also called Target Space, contains the physical, artificial, simulated or imagined object, scene or spatial information addressed by the perspective process.
Image Space, or Perspective Space, is the space in which the resulting view, image, measurement, model or representation is formed, located, displayed or perceived.
The relationship can be summarised as:
Object / Target Space → Perspective Process or System → Image / Perspective Space → Visual Interpretation
The resulting image may preserve some properties of the original spatial reality while transforming, reducing, concealing or distorting others. Perspective therefore involves both correspondence and transformation.
Perspective Is the Bridge between 3-D Space and Its Images
Visual and optical perspective concerns the changing appearance of things as they extend through the third dimension or depth.
The central problem is therefore not merely how to draw objects, but how information contained in spatial reality is transformed when it is:
- seen from a particular viewpoint;
- formed optically by an eye, lens or camera;
- projected onto a flat or curved image surface;
- converted into graphical geometry;
- measured or mapped;
- captured photographically;
- modelled mathematically;
- reconstructed computationally; or
- displayed within Virtual, Augmented or Mixed Reality.
Perspective provides the conceptual and technical framework for analysing all of these transformations.
Physical 3-D Space Is Not the Same as a 3-D Image
A physical three-dimensional object occupies real spatial volume. A perspective image of that object does not necessarily do so.
A conventional photograph or Linear Perspective drawing can be completely flat while nevertheless providing a powerful impression of spatial depth. Conversely, a stereoscopic or Virtual Reality image can provide binocular depth information without reproducing the original physical scene itself.
The phrase 3-D image must therefore be used carefully. It may refer to:
- a flat image containing monocular depth information;
- a stereoscopic image supplying different views to the two eyes;
- a multi-view or holographic image;
- a volumetric display;
- a navigable three-dimensional computer model; or
- a represented spatial environment such as Virtual Reality.
These are different Perspective Types and should not be treated as though the term “3-D” always means the same thing.
1-D, 2-D, 2.5-D and 3-D Perspective
Perspective representations can also be considered according to dimensionality.
- 1-D Perspective concerns a point or one-dimensional spatial relationship represented or measured within a larger object or image space.
- 2-D Perspective concerns lines, planes and flat Forms represented within two- or three-dimensional spaces.
- 2.5-D Perspective describes techniques that create an impression of three-dimensional space while movement or geometry remains partly restricted to a two-dimensional organisation. Examples include some parallel-projection graphics, parallax scrolling and pseudo-3-D computer-game environments.
- 3-D Perspective concerns one-, two- or three-dimensional spatial Forms situated in three-dimensional Object Space and represented through two- or three-dimensional Image Space.
This dimensional distinction helps separate the actual dimensionality of a physical object or model from the dimensionality of the image or display used to represent it.
Five Broad Forms of Artificial 3-D Representation
The Dictionary distinguishes at least five broad ways in which artificial perspective can represent three-dimensional spatial reality:
- Uni-angular Monocular Representation — a spatial scene viewed or represented from one fixed viewpoint or viewing direction. Ordinary Linear Perspective drawings and photographs are examples.
- Uni-angular Binocular Representation — a fixed central view presented stereoscopically using two slightly different viewpoints corresponding approximately to the two eyes.
- Multi-angular Representation — several viewing directions or viewpoints contribute to the representation. Holographic and some multi-view systems provide examples.
- Unlimited-Angle Representation — a model or environment can be explored from potentially unlimited positions and viewing directions, as in CAD models or Virtual Reality.
- Omnidirectional Representation — the represented image or object is designed to be encountered from directions around the observer or object.
This classification shows that the difference between a flat perspective drawing, stereoscopic cinema, a hologram and Virtual Reality is not simply that one is “2-D” and another “3-D”. They provide fundamentally different degrees of viewpoint freedom, angular information, binocular information and immersion.
How a Flat Image Can Appear Three-Dimensional
One of the most remarkable functions of perspective is its ability to produce an impression of three-dimensional space upon a two-dimensional surface.
A drawing, painting, photograph or screen image can remain physically flat while containing visual information that enables the viewer to infer depth and spatial structure.
Important Perspective Phenomena and depth information include:
- Diminution of Size — more distant Forms occupy smaller projected dimensions;
- Foreshortening — Forms change apparent dimensions according to orientation and depth;
- Diminution of Form — detail becomes progressively less distinguishable;
- Degradation of Form — apparent shape or structure changes;
- Diminution of Colour and Contrast — important components of Aerial Perspective;
- Texture Gradients — repeated surface structures become progressively compressed;
- Occlusion — nearer Forms hide parts of more distant Forms;
- Light and Shade — tonal structure contributes information about surface shape, volume and depth;
- Vanishing Points and Vanishing Traces — spatial directions and planes acquire organised limiting structures; and
- Horizon relationships — help organise spatial direction and recession.
Not every Perspective Type uses every phenomenon. Parallel Perspective, for example, can preserve parallelism and omit ordinary diminution with depth while still displaying Aspect Foreshortening and three-dimensional Form.
Monocular and Binocular 3-D
The perception of three-dimensional space should not be reduced to stereoscopic vision alone.
Monocular Perspective can provide powerful information about depth through size, form, overlap, texture, motion, focus, colour, contrast, convergence and many other visual relationships.
Binocular Perspective adds information derived from the slightly different views obtained by the two eyes. Binocular disparity and vergence can contribute particularly strongly to the perception of nearby depth and dimensional relief.
Natural human Visual Perspective Type 2 therefore involves a combination of monocular, binocular, optical, physiological and perceptual information rather than a single mechanism for “seeing in 3-D”.
The Geometry of Perspective Space
Perspective space is not restricted to one geometry.
Depending upon the method or system, Object Space or Image Space may be organised using:
- Cartesian coordinate space;
- Euclidean space;
- non-Euclidean space;
- flat or planar space;
- curved space;
- cylindrical space;
- spherical space;
- polar or radial systems; or
- other mathematical or computational spatial frameworks.
A conventional Linear Perspective image commonly uses a flat picture plane. Curvilinear, cylindrical and spherical perspective employ different mappings. Visual Perspective Type 2 involves the curved retina together with the optics and perceptual processes of the human visual system.
The geometry of the image surface is therefore an active part of the Perspective System rather than an incidental background.
Primary and Secondary Geometry
A useful distinction can also be made between Primary Geometry and Secondary Geometry.
Primary Geometry concerns the three-dimensional arrangement of objects, spatial directions and projection relationships in Object Space.
Secondary Geometry concerns the resulting points, lines, shapes and spatial relationships represented within Image Space.
A major function of perspective is therefore to establish understandable correspondences between primary spatial geometry and its secondary image geometry.
Perspective of Form
The geometrical appearance of three-dimensional objects changes when they are viewed or projected from different positions.
Perspective of Form concerns changes in apparent geometry, including:
- size;
- shape;
- outline;
- position;
- orientation;
- angle;
- foreshortening; and
- convergence.
The physical Form may remain unchanged while its projected Form changes substantially. Perspective is therefore inherently relational: the appearance depends upon the relationship between object, viewpoint, direction, picture or image surface and projection system.
Perspective Is More than Geometry
Three-dimensional appearance is not produced by geometrical Form alone.
The Dictionary distinguishes at least four principal divisions of Perspective Form:
- Perspective of Form — geometrical changes of apparent size, shape, position and orientation;
- Gradient of Colour Perspective — changes in colour, hue or saturation;
- Gradient of Acuity Perspective — changes in clarity, sharpness, resolution and visible detail; and
- Gradient of Chiaroscuro Perspective — changes in light, shade and contrast contributing to depth and spatial recession.
A convincing appearance of 3-D space can therefore emerge through the interaction of geometry, colour, clarity, illumination, contrast, texture and other visual information.
Space Itself Is Invisible
An important concept is that space itself is invisible. What we actually see are objects, surfaces, boundaries, light, colour, shadows, textures and other structures occupying or revealing relationships within space.
Human beings therefore infer and organise three-dimensional spatial reality by recognising Forms and relationships within perspective views and images.
Known structures such as:
- ground planes;
- parallel and perpendicular lines;
- grids;
- horizons;
- verticals;
- familiar objects;
- repeated intervals; and
- known geometrical Forms
help us segment, order, index, measure and comprehend otherwise invisible spatial relationships.
The Correspondence Problem of 3-D Space
One of the deepest problems in perspective is the relationship between a three-dimensional object and its projected image.
A single two-dimensional monocular image does not normally contain enough information to determine one unique three-dimensional object or scene.
Different 3-D Forms, positions and spatial arrangements can potentially produce the same or very similar two-dimensional projected image.
This is the Correspondence or Equivalence Problem of Perspective.
To interpret an image, the visual system, surveyor, artist or computer may need additional information such as:
- known viewpoint;
- known picture-plane geometry;
- a metric grid;
- recognisable Forms;
- known dimensions;
- multiple views;
- context;
- projection scale;
- resolution; or
- other geometrical constraints.
Perspective therefore concerns not only the encoding of three-dimensional space into images but also its subsequent decoding.
The Viewpoint Problem
Every perspective view represents spatial reality from a particular viewpoint or set of viewpoints.
Move the observer and the projected appearance changes. Different surfaces become visible, proportions change, Foreshortening changes, relative positions alter and Vanishing Points can move.
Each viewpoint therefore supplies unique but incomplete information about the original three-dimensional Form.
A more comprehensive understanding of 3-D space often requires Multi-View Perspective: combining images from different viewpoints and directions.
The Scale–Shape–Size Problem
Perspective also reveals that size, shape and scale cannot always be separated.
The familiar Size–Distance relationship explains why, under defined central-projection conditions, an object’s projected dimensions decrease as its distance increases. But this relationship alone does not determine apparent or measured Form.
The result also depends upon:
- viewpoint;
- orientation;
- Foreshortening;
- visible shape;
- occlusion;
- projection geometry;
- projection scale;
- resolution; and
- the method of measurement.
Increasing scale or resolution may reveal previously invisible structural details and thereby alter both apparent shape and measured size. This is the Scale–Shape–Size Problem.
3-D Space Can Be Transformed
Perspective does not merely copy space. It can also systematically transform spatial appearance.
Perspective spaces can include:
- Convergent Space — apparent sizes and directions converge with recession;
- Parallel Space — principal parallel directions remain parallel;
- Curvilinear or Spherical Space — spatial geometry is mapped through curved projection systems;
- Accelerated or Forced Space — apparent depth or recession is exaggerated;
- Decelerated Space — apparent depth is reduced;
- Anamorphic Space — spatial Form is deliberately distorted for a particular viewpoint or optical condition;
- Blended or Composite Space — several spatial or image geometries operate together;
- Mirror or Virtual Space — apparent space exists through reflection or other virtual-image processes;
- Multi-View Space — several viewpoints or directions are represented together; and
- Simulated or Virtual Space — computational or physically constructed environments reproduce or alter spatial appearance.
This ability to transform space is fundamental to art, architecture, cinema, stage design, optical illusion, photography, computer graphics and immersive media.
From Linear Perspective to 3-D Models
Traditional Linear Perspective converts aspects of three-dimensional Object Space into a two-dimensional geometrical Image Space.
Contemporary New Media Perspective can go further by building an actual three-dimensional mathematical model of the scene.
CAD, CGI, GIS and Computer Vision systems can represent objects through three-dimensional coordinates and surface geometry. A virtual camera can then generate many different perspective views from the same model.
The representation is no longer limited to one completed fixed image. The viewer or camera can move through the model and generate changing views of the same spatial structure.
Multi-View, Multi-Scale and Multi-Time 3-D Space
A comprehensive Perspective Model can potentially organise spatial information across three major variables:
- Multi-View — linking different viewpoints and viewing directions;
- Multi-Scale — connecting views obtained at different spatial scales and resolutions; and
- Multi-Time — connecting changing states of spatial reality across different times and rates of change.
This moves perspective beyond the traditional idea of a single image taken from a single viewpoint at a single moment.
The longer-term goal is the construction of increasingly comprehensive visual models in which the observer can explore spatial reality across position, direction, scale and time.
Perspective Category Theory and 3-D Space
Perspective Category Theory helps identify which processes are responsible for a particular three-dimensional appearance or representation.
- Natural Space relates principally to Natural Perspective.
- Visual Space relates to Visual Perspective Type 1 and Visual Perspective Type 2.
- Optical Space relates to Optical Perspective.
- Mathematical Space relates to Mathematical Perspective.
- Graphical or Represented Space relates to Graphical Perspective.
- Instrument Space relates to Instrument Perspective.
- Simulated or Illusory Space relates to Simulated Perspective.
- Digital or New Media Space relates to New Media Perspective.
These spaces frequently overlap. A physical scene can be captured optically by a camera, processed digitally, displayed on a screen and finally experienced through human vision.
The complete sequence may therefore involve several transformations of the original 3-D space through a Perspective Image Chain.
Perspective = Method or Process + Outcome
The relationship between perspective and 3-D space also demonstrates why perspective must be understood as both process and outcome.
A Perspective Method or System performs an operation upon spatial information. The result is a view, image, measurement, representation, model or spatial appearance.
Accordingly:
Perspective = Method or Process + Resulting Image, View or Spatial Appearance.
The same three-dimensional scene can therefore produce many different Perspective Forms depending upon the process used to view, project, capture, calculate, represent or display it.
The Main Goals of Perspective in 3-D Space
Perspective systems ultimately allow us to perform several fundamental operations upon three-dimensional spatial reality:
- View — observe, image or capture a spatial scene;
- Match — measure, compare, survey, classify or cross-match spatial information;
- Represent — copy, model, map, index, link and explore three-dimensional reality;
- Create Illusion — alter apparent size, shape, depth, position or spatial structure; and
- Create Immersion — produce the experience of being located within a real or represented spatial environment.
Perspective therefore underlies a remarkable range of activities extending from direct eyesight and drawing to photography, surveying, cinema, architecture, scientific imaging, CAD, Computer Vision and Virtual Reality.
Why Perspective and 3-D Space Matter
The relationship between perspective and three-dimensional space lies at the centre of how humans understand and represent the physical world.
We live within three-dimensional spatial reality, but much of our knowledge of that reality reaches us through views and images: the retinal images of eyesight, photographs, drawings, maps, diagrams, films, screens, scientific instruments and digital models.
Every such image is selective. Viewpoint determines what can be seen. Projection changes apparent Form. Scale and resolution determine which structures remain visible. Optical systems modify colour, contrast and clarity. Representation may preserve some spatial properties while sacrificing others.
Perspective is the field that studies and organises these relationships.
It therefore provides a bridge between physical 3-D space, optical and visual appearance, geometrical projection and represented image space.
Understanding that bridge is fundamental to understanding depth, viewpoint, spatial Form, projection, images, visual perception, Linear Perspective, Parallel Perspective, Curvilinear Perspective, Spherical Perspective, stereoscopy, photography, Computer Perspective, 3-D modelling, Virtual Reality and the wider visual organisation of spatial reality.