A Perspective Principle is a theoretical mechanism or relationship through which a spatial object, scene, image or model produces a particular perspective view, image, representation or spatial appearance.
A Perspective Principle can involve the relationship between real or modelled spatial structures, viewpoint, picture plane or image surface, projection geometry, light, mathematics, graphical construction and computational processes.
Perspective Principles therefore explain the underlying mechanisms of perspective. They describe why and according to what relationships particular perspective effects arise, while Perspective Methods describe how those principles are applied in practice.
Definition of Perspective Principle
The Dictionary defines a Perspective Principle as a theoretical imaging mechanism comprising real or modelled scene or object structures, picture-plane or image-surface geometry and one or more optical, mathematical, graphical or computer principles operating together to produce a representation of spatial reality.
Volume 1 gives a closely corresponding definition: a Perspective Principle is a theoretical mechanism involving scene/object structure, picture-plane or image-surface geometry and optical, mathematical, graphical or computer processes.
The concept is therefore deliberately broader than the principles of conventional Linear Perspective alone.
Principles Explain How Perspective Works
A Perspective Principle identifies an underlying relationship or mechanism that helps explain how a perspective image, view or spatial appearance arises.
Examples can involve relationships between:
- object or scene structure;
- viewpoint or station point;
- Direction of Vision;
- distance;
- object orientation;
- picture-plane or image-surface geometry;
- projection direction;
- light and optical behaviour;
- mathematical transformation;
- graphical construction; and
- computer calculation.
Different combinations of these relationships generate different perspective processes and Forms.
Perspective Principle and Perspective Method
A Perspective Principle should be distinguished from a Perspective Method.
- Perspective Principle: the underlying theoretical mechanism or relationship.
- Perspective Method: a repeatable procedure that applies one or more principles to form, calculate, record, project, measure, transfer, check or represent a perspective view or image.
The principle explains why or according to what relationship the perspective process works. The method describes how it is carried out.
One Perspective Principle can therefore support several different Perspective Methods, and one method can apply several principles simultaneously.
Perspective Principle and Perspective Process
A Perspective Process is the operation through which principles and methods form, capture, calculate, project, represent, simulate, display or interpret a view, image or spatial relationship.
The principle therefore describes the underlying mechanism, while the process is the operation in which that mechanism functions.
Perspective processes may be natural or artificial and can involve optical, visual, mathematical, graphical, instrument, simulated or New Media conditions.
Perspective Principle and Perspective System
A Perspective System is broader than an individual Perspective Principle.
Volume 1 defines a Perspective System as a scheme of visual or optical perspective operating as a unit for image making, image projection, image analysis or matching.
A complete Perspective System can contain:
- one or more Perspective Categories;
- Perspective Types;
- Perspective Principles;
- Perspective Methods;
- physical or virtual structures;
- instruments or computational processes; and
- resulting Perspective Forms.
A Perspective Principle is therefore one theoretical component within a potentially much larger system.
Perspective Principle and Perspective Type
A Perspective Type identifies a particular method, system, process or subdivision of perspective.
A Perspective Principle instead identifies an underlying theoretical relationship that may operate within one or several types.
The same principle can consequently occur within several related Perspective Types, while a single Perspective Type may depend upon several interacting principles.
Perspective Principle and Perspective Form
A Perspective Principle should also be distinguished from the resulting Perspective Form.
Perspective Form describes the visible, optical, geometrical or spatial appearance produced by a perspective process.
Forms can involve:
- changes in apparent size;
- changes in shape or aspect;
- convergence or divergence;
- changes in position or orientation;
- gradients of colour;
- gradients of clarity;
- light-and-shade effects;
- curvature or distortion; and
- apparent depth, volume or immersion.
The Perspective Principle helps explain the mechanism responsible for producing such a Form.
Perspective Principle and Perspective Phenomenon
A Perspective Phenomenon is an apparent and generalised change in the visual features of an object or scene arising through a particular perspective process.
Examples include:
- Diminution of Size;
- Foreshortening;
- Diminution of Form;
- Degradation of Form;
- Diminution of Colour;
- Diminution of Contrast;
- vanishing points;
- horizon relationships; and
- gradients of light, shade or texture.
The phenomenon is the apparent effect; the principle concerns the mechanism or relationship through which the effect arises.
The Principle of Central Perspective
The Dictionary gives a specific Principle of Perspective for central projection.
Selected object points are connected to a fixed centre of projection or station point. The projectors intersect a picture plane, and those intersections establish the corresponding image points.
The visual cone or Pyramid of Vision can therefore be considered as intersected by a picture plane to produce the perspective image.
This is the fundamental projective principle underlying conventional Central and Linear Perspective.
Object Point, Visual Ray and Image Point
In central projection, an object point, centre of projection and corresponding image point are related by a common projector or visual ray.
The perspective image of the object point occurs where that ray intersects the selected picture plane.
The complete image can therefore be understood as a large set of corresponding object-point, projection-ray and image-point relationships.
The Picture Plane as Part of the Principle
Picture-plane or image-surface geometry forms an explicit part of the Dictionary’s definition of a Perspective Principle.
The appearance produced by a perspective system therefore depends not only upon spatial objects and viewpoint but also upon the geometry of the surface onto which, or within which, the image is formed or represented.
A flat picture plane, curved image surface, spherical surface or other image geometry can participate in different perspective principles and generate different Perspective Forms.
Perspective Principles Are Not All Linear-Perspective Principles
The broad concept of Perspective Principle should not be restricted to conventional Linear Perspective.
Linear Perspective possesses particular principles concerning central projection, vanishing points, picture planes and diminution. Other Perspective Types operate according to different geometrical, optical or computational relationships.
Parallel Perspective, for example, preserves parallel projectors rather than projecting from one finite centre and therefore does not possess the same finite vanishing-point structure as Central Perspective.
Curvilinear and Spherical Perspective employ different image-surface or projection relationships again.
Fundamental Principles and Axioms of Linear Perspective
The Dictionary separately records a group of eight Fundamental Principles or Axioms associated specifically with Central and Linear Perspective.
These concern the geometrical relationships between:
- parallel spatial lines;
- parallel spatial planes;
- vanishing points;
- vanishing traces or lines;
- the picture plane; and
- the station point or eye.
They should be understood as principles of this particular projective arrangement rather than universal rules for every Perspective Type.
Axiom 1 — Convergence of Lines to a Vanishing Point
A family of mutually parallel spatial lines extending in a direction that recedes relative to the picture plane appears to converge towards a corresponding Vanishing Point.
The Vanishing Point represents the projected image or limit of that common spatial direction.
Different spatial directions normally possess different vanishing points.
Axiom 2 — Convergence of Planes to a Vanishing Trace
The Dictionary states that parallel planes receding from the observer appear to approach one another and possess a corresponding Vanishing Trace.
This extends the concept of vanishing beyond individual lines and Vanishing Points to the directional structure of planes.
Axiom 3 — A Line in a Plane
If a spatial line lies within a particular plane, its Vanishing Point must lie upon the Vanishing Trace of that plane.
The directional structure of the line is therefore contained within the larger directional structure of its plane.
Axiom 4 — A Plane’s Vanishing Trace Contains Its Vanishing Points
Conversely, the Vanishing Trace of a plane contains the Vanishing Points corresponding to all spatial line-directions lying within that plane.
The familiar geometrical horizon provides an important example: it is the vanishing line of the horizontal plane and contains the Vanishing Points of horizontal spatial directions.
Axiom 5 — The Intersection of Two Planes
A line forming the intersection of two planes belongs to both planes.
Its Vanishing Point must therefore occur at the corresponding intersection of the Vanishing Traces of those planes.
This demonstrates the close geometrical relationship between line direction, plane direction, Vanishing Point and Vanishing Trace.
Axiom 6 — Lines Parallel to the Picture Plane
Spatial lines parallel to the picture plane remain mutually parallel in an ideal flat Central Perspective image.
They therefore possess no finite Vanishing Point on that picture plane.
This is a fundamental distinction between directions that recede relative to the picture plane and directions that remain parallel to it.
Axiom 7 — Picture-Plane Parallel Lines and Vanishing Traces
The Dictionary states that a spatial line parallel to the picture plane has its perspective represented parallel to the Vanishing Trace of every plane in which that line lies.
This rule connects line orientation with the larger vanishing structure of the planes containing it.
Axiom 8 — Intersecting Planes Parallel to the Picture Plane
If the line along which two spatial planes intersect is parallel to the picture plane, the corresponding Vanishing Traces of the two planes are parallel.
The Dictionary specifically notes that these later axioms relate to conditions involving a relatively narrow Field of View and a flat picture plane.
Principles of Receding Parallel Lines
The Dictionary also summarises the same basic geometry as practical Perspective Principles.
Parallel lines that recede relative to the picture plane appear to converge towards a common point. That Vanishing Point can be found by passing a line through the eye or station point parallel to the spatial direction being represented.
Where this parallel line intersects the picture plane establishes the corresponding Vanishing Point.
Principle of Non-Converging Parallels
Parallel lines that are themselves parallel to the picture plane do not possess a finite Vanishing Point upon that plane.
A line passed through the eye parallel to such a spatial direction remains parallel to the picture plane and therefore does not intersect it at a finite point.
The resulting projected lines consequently remain parallel.
Principle of Receding Parallel Planes
Parallel planes that recede relative to the picture plane possess a common Vanishing Trace.
The Dictionary states that this line can be determined by passing a plane through the observer’s eye parallel to the receding planes and finding its intersection with the picture plane.
This is the plane equivalent of finding the Vanishing Point of a family of parallel lines.
Planes Parallel to the Picture Plane
A spatial plane parallel to the picture plane possesses no finite Vanishing Trace on that picture plane.
A corresponding plane passing through the eye and parallel to it will itself remain parallel to the picture plane rather than intersect it along a finite line.
The principle parallels the behaviour of individual spatial lines that are parallel to the picture plane.
Rules for Finding a Vanishing Point
The Dictionary expresses the Vanishing-Point principle in several equivalent construction rules.
To establish the Vanishing Point of a family of mutually parallel lines:
- Identify the common spatial direction of the parallel system.
- Pass an equivalent parallel line through the observer’s eye or station point.
- Extend that line until it intersects the picture plane.
- The intersection establishes the Vanishing Point of that directional system.
This is one of the fundamental geometrical principles underlying Central and Linear Perspective.
The Perspective of an Object Point
The perspective image of an individual point in object space occurs where the visual ray from that point intersects the picture plane.
This simple relationship is fundamental because more complicated lines, planes and Forms can be represented through corresponding collections of projected points.
The Size–Distance Principle
Another major Perspective Principle is the relationship between projected size and distance.
Under comparable projection conditions, equal Forms positioned progressively farther from a finite centre of projection occupy progressively smaller projected sizes.
This relationship underlies Diminution of Size Perspective and contributes to Perspectival or Optical Foreshortening.
For comparable Forms at the same orientation, projected size decreases approximately in inverse proportion to depth under the relevant central-projection conditions.
The Size–Distance Law Is Not the Whole of Perspective
Volume 1 emphasises that the Size–Distance Law is fundamental but does not by itself explain all perspective phenomena.
Apparent or measured size and shape also depend upon other conditions, including:
- viewpoint;
- orientation;
- projection geometry;
- scale;
- resolution;
- shape sufficiency; and
- the method of measurement.
Perspective Principles therefore normally operate as interacting relationships rather than as isolated universal formulas.
Aspect as a Perspective Principle
The apparent shape of a spatial Form also depends upon its orientation and projection angle.
A Form viewed or projected obliquely can appear compressed or geometrically transformed even where distance remains unchanged.
The Dictionary distinguishes this Aspect Foreshortening from the distance-dependent Perspectival or Optical Foreshortening produced through Diminution of Size.
Perspective therefore involves both distance-dependent and orientation-dependent principles.
Two Components of Foreshortening
The Dictionary distinguishes two principal contributors to Foreshortening:
- Aspect Foreshortening: contraction or transformation associated primarily with object orientation, viewing angle or projection angle.
- Perspectival or Optical Foreshortening: distance-dependent contraction resulting from Diminution of Size.
Both principles can operate simultaneously within the same perspective view.
The Window Principle
The Window Principle concerns the use of an actual or imaginary picture plane positioned in relation to a fixed viewpoint, viewing direction and Field of View.
In graphical perspective, the artist can imagine the picture as a transparent window through which the spatial scene is viewed.
The represented image corresponds to the projected relationships where the directions from the scene towards the eye intersect that picture surface.
The principle provides an important conceptual connection between graphical construction, optical projection and Camera Perspective.
The Transparency Principle
The Dictionary identifies the Transparency Principle as a basic operating principle associated with Linear and Optical Perspective.
The medium between visible object points and the projection or imaging system must allow the relevant light or projection rays to travel through it.
In ordinary vision this enables light from visible spatial surfaces to reach the observer and form the optical or visual image.
The principle is therefore more fundamental than the literal presence of a transparent glass Perspective Window.
The Reversibility Principle
The Dictionary also identifies a Reversibility Principle associated with Linear Perspective.
Under appropriate geometrical conditions, information contained within a perspective image can be worked backwards in order to reconstruct aspects of the original projection arrangement or viewpoint.
The Dictionary gives the example of working backwards from the perspective image of a regular square or cube in order to reconstruct the viewpoint responsible for its particular foreshortened appearance.
This principle has practical importance for processes such as Perspective Matching and Photogrammetry.
Reversibility Has Limits
The Dictionary does not present Reversibility as an unrestricted principle applying perfectly to every image.
Its example depends upon regular geometrical Forms and appropriate relationships to the picture plane.
The amount of recoverable information also depends upon what the perspective image preserves and upon the assumptions available about the original object or scene.
The Opening Principle
The Opening Principle describes the capacity of Linear and related technical perspective to organise and visually open represented space into extensive depth.
Converging parallels, Vanishing Points, horizon relationships and diminution can organise spatial structures such as roads, railways, colonnades, buildings and tunnels so that represented space appears to extend far beyond the immediate foreground.
The principle concerns the way perspective can organise, integrate and extend the visible structure of represented depth.
The Closing or Occlusion Principle
The Dictionary treats the Closing Principle as synonymous with the Occlusion Principle of Perspective.
Objects, surfaces and other spatial structures can block the view of what lies behind them.
This occlusion helps establish figure-ground relationships and provides information about relative spatial position and depth.
The Opening and Closing Principles are therefore complementary: perspective can reveal deep spatial recession while visible Forms simultaneously conceal other regions of space.
Operating Principles of Perspective
The Dictionary identifies several principles associated particularly with the historical and theoretical development of Linear Perspective:
- Transparency Principle;
- Reversibility Principle;
- Opening Principle; and
- Closing or Occlusion Principle.
These principles describe different aspects of how a perspective view is formed, structured, interpreted or reconstructed.
Optical Perspective Principles
Perspective Principles can be optical as well as geometrical.
Optical Perspective involves the formation, transmission, reflection, refraction, focusing or projection of light and other electromagnetic radiation.
The principles governing an optical perspective system can therefore include the geometrical direction of rays together with the physical behaviour of the imaging medium, lens, mirror, aperture, detector or other instrument.
Mathematical Perspective Principles
Mathematical Perspective employs geometrical relationships, coordinates, transformations, measurements and calculations to describe or generate spatial projections.
A Perspective Principle can therefore be represented mathematically and then used to calculate a corresponding view or image.
Central projection, parallel projection, projective diminution and geometrical transformations provide examples of perspective relationships that can be expressed mathematically.
Graphical Perspective Principles
Graphical Perspective applies perspective relationships through drawing, construction, diagramming and other representational methods.
A graphical method can embody principles of:
- central projection;
- parallel projection;
- vanishing-point construction;
- vanishing-line construction;
- measurement and subdivision;
- Diminution of Size;
- Aspect Perspective; and
- projection onto flat or other picture surfaces.
The graphical procedure is the Perspective Method; the geometry underlying that procedure consists of Perspective Principles.
Computer Perspective Principles
Perspective Principles also operate within Computer and Digital Perspective.
A computer system can encode spatial geometry mathematically and calculate a perspective view according to variables such as:
- virtual camera position;
- viewing direction;
- Field of View;
- projection type;
- object geometry;
- image-surface geometry; and
- other optical or rendering conditions.
The fact that these relationships are calculated automatically does not remove the Perspective Principles. It implements them computationally.
One Principle Can Operate in Several Categories
A Perspective Principle is not necessarily confined to one Perspective Category.
The geometrical relationship of central projection, for example, can be analysed mathematically, constructed graphically, produced through an optical instrument or calculated digitally.
The same underlying relationship can therefore appear through several different processes and media.
Several Principles Can Operate Together
A complete perspective image rarely depends upon only one isolated principle.
A photographed spatial scene can simultaneously involve:
- spatial geometry;
- Size–Distance relationships;
- Aspect Perspective;
- optical projection;
- focus and image acuity;
- Field of View;
- light and shade;
- colour and contrast;
- instrument imaging; and
- human visual perception.
The final perspective appearance can therefore arise from several interoperating principles rather than one universal perspective law.
Perspective Principles and Category Chaining
Perspective Principles can also operate successively as an image passes through several Perspective Categories.
A physical scene can undergo Natural Perspective, optical and Instrument Perspective during camera capture, New Media Perspective during digital processing and display, and finally Visual Perspective Type 2 when seen by a human observer.
The principles relevant at one stage need not be identical to those operating at the next.
This sequential interaction is part of what Perspective Category Theory describes as Category Chaining.
Perspective Principles and Composite Perspective
A Composite Perspective can contain several Perspective Categories, methods and principles operating together or in sequence.
A cinematic image, for example, may involve physical spatial arrangements, optical camera principles, graphical or simulated perspective, digital processing and computer-generated imagery before the final image is displayed and viewed.
Understanding the resulting perspective therefore requires identification of the different principles operating throughout the complete system.
A Principle Is Not Necessarily Universal
One of the most important cautions in perspective theory is that a valid principle within one Perspective Type or configuration should not automatically be extended to every other type.
For example:
- finite Vanishing Points are fundamental to many Central Perspective arrangements but not ordinary Parallel Projection;
- lines parallel to a flat picture plane behave differently from receding directions;
- flat rectilinear projection differs from curved, cylindrical and spherical projection;
- Diminution of Size does not operate in the same manner within Parallel Perspective; and
- wide Fields of View can introduce relationships not adequately described by assumptions developed for narrow flat perspective constructions.
Perspective Principles should therefore always be understood in relation to the particular system and conditions to which they apply.
Principles, Rules and Axioms
The Dictionary uses several related terms — principle, rule and axiom — in its treatment of Perspective.
These terms should not necessarily be assumed to have exactly the same scope.
A broad Perspective Principle describes an underlying theoretical mechanism. A construction rule can describe a practical geometrical consequence or procedure. An axiom identifies a foundational relationship assumed or established within a particular perspective system.
The context therefore determines the intended level of application.
Why Perspective Principles Must Be Stated Precisely
Perspective becomes confusing when a principle valid for one situation is presented as though it applied universally.
Precision requires identifying the relevant:
- Perspective Category;
- Perspective Type;
- viewpoint;
- projection direction;
- object-space geometry;
- picture-plane or image-surface geometry;
- Field of View;
- optical conditions;
- method; and
- intended Perspective Form.
Only then can the appropriate principles and their limits be identified accurately.
Perspective Principles in Art
Artists can employ Perspective Principles directly or indirectly.
A geometrically constructed drawing can deliberately apply central-projection, vanishing-point and measurement principles, while a freehand drawing may estimate the same apparent relationships through observation.
Artists can additionally represent optical perspective phenomena such as Diminution of Colour, loss of contrast, Acuity Perspective, Chiaroscuro Perspective and other depth information.
Perspective in art therefore extends considerably beyond the principles of line convergence alone.
Perspective Principles in Photography and Imaging
Photography demonstrates how geometrical and optical Perspective Principles can operate together.
The camera establishes a viewpoint, Direction of Vision and Field of View while the optical system forms an image according to its physical and geometrical properties.
The resulting photograph can contain Diminution of Size, Aspect Perspective, convergence, foreshortening, light-and-shade effects, changes of acuity and many other Perspective Phenomena.
The photograph is therefore not the consequence of one isolated principle but of a complete Perspective System.
Perspective Principles in Computer Graphics
Computer graphics demonstrate that Perspective Principles remain fundamental even when image production is automated.
A digital spatial model and virtual camera encode the geometrical conditions of the perspective system, while software calculates the resulting image according to the selected projection and imaging parameters.
Computer systems can therefore implement Central, Parallel, Curvilinear, Spherical and other Perspective Principles according to the required representation.
Common Misconceptions about Perspective Principles
- Perspective Principle does not mean only the rules of Linear Perspective. The Dictionary defines the concept broadly enough to include optical, mathematical, graphical and computer mechanisms.
- A Perspective Principle is not the same as a Perspective Method. A principle explains the underlying relationship; a method specifies a repeatable procedure applying it.
- A Perspective Principle is not the same as a Perspective Form. The principle concerns mechanism; the Form is the resulting visible or spatial appearance.
- A Perspective Principle is not the same as a Perspective System. A system can contain several categories, types, principles and methods operating together.
- One-, Two- and Three-Point Perspective are not themselves universal Perspective Principles. They are geometrical configurations or types of Central Perspective.
- Not every family of parallel lines has a finite Vanishing Point. Lines parallel to the picture plane have no finite Vanishing Point on that plane.
- Not every Vanishing Point lies upon the ordinary horizontal horizon. Its position depends upon the spatial direction being projected.
- Vanishing Points and Vanishing Traces describe related but different directional relationships. A Vanishing Point corresponds to a line direction; a Vanishing Trace or line corresponds to the directional structure of a plane.
- Perspective is not explained solely by Diminution of Size. Aspect, projection geometry, scale, resolution, image-surface geometry and other factors can also change appearance.
- Foreshortening does not have only one cause. Aspect Foreshortening and Perspectival or Optical Foreshortening have different underlying relationships.
- Principles valid for flat, narrow-field Central Perspective should not automatically be applied to every projection system.
- Computer-generated Perspective does not eliminate Perspective Principles. The computer calculates and implements them.
- One perspective image can embody many principles simultaneously. Geometrical, optical, graphical and computational mechanisms frequently operate together.
Why Perspective Principle Matters
Perspective Principles matter because they provide the theoretical foundation for understanding why perspective images and spatial appearances take the Forms that they do.
Without the underlying principles, perspective can easily become a collection of disconnected drawing rules, optical effects and technical procedures. Identifying the principle makes it possible to connect the observable phenomenon to its spatial, geometrical, optical, graphical or computational cause.
This distinction is particularly important because the same visible effect may arise through different processes, while the same principle may be implemented through several different methods and technologies.
Perspective Principle therefore provides a fundamental bridge between theory and practice: principles explain the mechanisms, methods apply them, processes perform the operations, systems combine them, and Perspective Forms are the resulting visible or spatial appearances.
Understanding Perspective Principle provides a foundation for understanding Perspective Method, Perspective Process, Perspective System, Perspective Type, Perspective Form, Perspective Phenomena, Central Projection, Parallel Projection, Picture Plane, Station Point, Vanishing Point, Vanishing Trace, Horizon Line, Diminution of Size, Size–Distance Law, Aspect Perspective, Foreshortening, Window Principle, Transparency Principle, Reversibility Principle, Opening Principle, Occlusion Principle, Linear Perspective, Optical Perspective, Mathematical Perspective, Graphical Perspective, Computer Perspective and Perspective Category Theory.