Mathematical Perspective

Mathematical Perspective is perspective based on mathematical relations, geometry, measurements, coordinates, transformations or calculations used to analyse or represent spatial objects, scenes, views or images.

It is much broader than the familiar construction of linear perspective. It includes symbolic formulae, geometrical models, spatial measurements, technical projections, transformations between coordinate systems, projective constructions, map projections, computer graphics and the reconstruction of spatial information from images.

Mathematical Perspective provides methods for answering fundamental spatial questions:

  • Where is an object located?
  • How large or distant is it?
  • In which direction is it viewed?
  • How does its appearance change with viewpoint?
  • How can a three-dimensional object be projected into two dimensions?
  • How can spatial information be recovered from an image?
  • Which properties are preserved or altered by a particular projection?

It connects geometry, measurement and calculation with the wider field of visual and optical perspective.

Mathematical relationships between viewpoint, direction, distance and projection explain the recession of a spatial plane towards a vanishing point.

What is Mathematical Perspective?

Mathematical Perspective describes the use of mathematical structures to model, measure, transform, project or reconstruct spatial objects, scenes and images.

These structures may include:

  • numbers and dimensions;
  • points, lines, curves and planes;
  • angles and directions;
  • coordinates and reference systems;
  • distance, scale and proportion;
  • surfaces, volumes and solids;
  • visual rays or projectors;
  • transformations;
  • projection centres and projection surfaces;
  • vanishing points, lines, planes and spheres.

Some Mathematical Perspective is entirely symbolic. Other forms produce visible geometrical diagrams, technical drawings, maps, models or projected images.

The Dictionary of Perspective therefore treats Mathematical Perspective as one of the principal categories of perspective rather than merely as another name for linear-perspective drawing. Within Perspective Category Theory, the category is defined by the mathematical source or mode of its process: spatial reality is calculated, measured, modelled, transformed or projected through mathematical relationships. 

Mathematical Perspective is broader than Linear Perspective

Linear Perspective is an important form of Mathematical Perspective, but it represents only one part of the category.

Conventional linear perspective generally employs:

  • a fixed viewpoint or station point;
  • a picture plane;
  • projection lines or visual rays;
  • parallel spatial directions;
  • vanishing points;
  • a horizon or vanishing line;
  • diminution with distance.

Mathematical Perspective also includes forms that do not use a single viewpoint or a flat picture plane. These include:

  • orthographic and parallel projection;
  • axonometric and isometric systems;
  • oblique projection;
  • cylindrical and spherical projection;
  • cartographic projection;
  • analytic and coordinate geometry;
  • projective transformations;
  • three-dimensional computer modelling;
  • computational image reconstruction.

It may also operate without producing a conventional picture. A formula, coordinate system or numerical model may describe spatial relationships mathematically without presenting them as a realistic view.

Three principal kinds of Mathematical Perspective

The Dictionary of Perspective distinguishes three principal kinds:

  1. Algebraic Perspective
  2. Geometrical Perspective
  3. Projection Perspective

These are related but perform different functions. 

Algebraic Perspective

Algebraic Perspective uses mathematical letters, symbols, equations and numerical relationships to describe spatial structures, views or transformations.

It belongs primarily to a symbolic or non-visual class. Its result may consist of formulae and calculations rather than a directly visible perspective image.

Algebraic Perspective may describe:

  • coordinates;
  • distances and angles;
  • scale and magnification;
  • transformations;
  • projection equations;
  • spatial constraints;
  • curves and surfaces;
  • camera and viewing parameters;
  • relationships between object space and image space.

For example, the position of a point may be represented through coordinates, while its projected image position may be calculated through an equation or transformation matrix.

The algebraic description is not itself necessarily a visible representation of the object. It supplies the symbolic rules through which a geometrical model or image can be produced, analysed or transformed.

Geometrical Perspective

Geometrical Perspective uses points, lines, curves, planes, surfaces and solids to analyse and model spatial relationships.

It includes analytic geometry, also called coordinate geometry, in which algebraic methods are used to define and solve geometrical problems.

Geometrical Perspective can show:

  • spatial direction;
  • relative position;
  • intersection;
  • parallelism and perpendicularity;
  • angle;
  • scale;
  • distance;
  • shape;
  • surface orientation;
  • projection;
  • geometrical limits and vanishing structures.

Unlike purely algebraic perspective, Geometrical Perspective commonly produces a visible diagram or model. It is therefore part of Visual Perspective Type 1, although the process generating it remains mathematical.

Projection Perspective

Projection Perspective uses projective principles to form an image or view of a spatial object or scene.

It establishes a defined relationship between:

  • an object or spatial model;
  • a projection centre or projection direction;
  • projection lines or rays;
  • a plane, surface or other receiving structure;
  • the resulting image or projected form.

Projection Perspective includes both:

Descriptive geometry

Descriptive geometry uses organised projections to describe and measure three-dimensional objects through two-dimensional views.

These may include:

  • plan;
  • front elevation;
  • side elevation;
  • sectional views;
  • auxiliary views;
  • first-angle projection;
  • third-angle projection.

Projective geometry

Projective geometry studies properties and relationships produced through projection. It provides the mathematical basis for central perspective, vanishing points, perspective transformations and many forms of spatial representation.

Projection Perspective may therefore produce either a measurable technical representation or a view intended to resemble spatial appearance.

The fundamental structure of mathematical projection

A mathematical projection usually involves several basic components.

Object or model space

The object, scene or mathematical model before it is projected.

Viewpoint or projection centre

The point from which projection lines originate in central projection.

Projection direction

The direction followed by the projectors. In parallel projection, the projectors remain mutually parallel rather than originating from one finite point.

Projection lines or visual rays

Mathematical lines connecting spatial points to corresponding projected points.

Projection surface

The plane, cylinder, sphere or other surface receiving the projection.

Image or projection space

The resulting two-dimensional or transformed spatial structure.

A simplified sequence is:

Spatial object or model
↓
Mathematical projection or transformation
↓
Image, diagram, map or represented structure

These components make it possible to distinguish the original spatial object from the process used to transform it and from the resulting image.

Forward projection and backward analysis

Mathematical Perspective can operate in two directions.

Forward projection

In forward projection, a known spatial object or model is transformed into an image or another represented structure:

3-D object or model → projection system → 2-D image

This direction is used in:

  • perspective drawing;
  • technical projection;
  • cartography;
  • computer graphics;
  • animation;
  • architectural visualisation;
  • virtual environments.

Backward analysis

In backward analysis, an existing image is examined to infer information about the spatial object, scene or viewpoint that produced it:

2-D image → mathematical analysis → possible 3-D structure

This direction is used in:

  • photogrammetry;
  • surveying;
  • computer vision;
  • image measurement;
  • camera calibration;
  • three-dimensional reconstruction;
  • forensic and scientific imaging.

Computer graphics commonly projects from model and world coordinates towards screen coordinates, while computer vision attempts to infer or reconstruct spatial information from image data. 

Coordinates, reference frames and transformations

Mathematical Perspective requires a means of locating and relating objects.

A coordinate system supplies a reference structure within which points, lines, surfaces and objects can be described.

Different stages of a perspective process may use different spaces:

  • object coordinates;
  • model coordinates;
  • world coordinates;
  • camera or viewpoint coordinates;
  • projection coordinates;
  • image coordinates;
  • screen or display coordinates.

A spatial object can then be transformed between these systems.

Typical transformations include:

  • translation — changing position;
  • rotation — changing orientation;
  • scaling — changing size;
  • reflection — reversing orientation;
  • shear — displacing one part relative to another;
  • projection — converting or mapping spatial dimensions;
  • perspective transformation — altering apparent spatial relationships according to viewpoint or projection.

These transformations form a major foundation of computer graphics, computer-aided design, mapping and digital imaging.

Measurement, scale and matching

Mathematical Perspective is not only concerned with producing pictures. It also provides systems for measuring and matching spatial reality.

Perspective measurement may involve:

  • length;
  • width;
  • height;
  • distance;
  • angle;
  • area;
  • volume;
  • proportion;
  • scale;
  • position;
  • orientation;
  • spatial correspondence.

Within your wider theory, Match is one of the fundamental functions of perspective. Mathematical methods can compare an image or model with known measurements and determine how accurately it corresponds to an object, scene or intended spatial structure.

This is important in:

  • architecture;
  • engineering;
  • surveying;
  • cartography;
  • photogrammetry;
  • manufacturing;
  • scientific imaging;
  • computer vision.

The Scale–Shape–Size Problem

Measurement is not independent of viewing or projection scale.

The Scale–Shape–Size Problem recognises that measuring visible size may also involve measuring shape, because the detectable form of an object can change according to:

  • projection scale;
  • magnification;
  • sensor resolution;
  • viewing distance;
  • image resolution;
  • level of detail.

At a coarse scale, only a general outline may be visible. At a finer scale, additional edges, structures and surface variations become detectable.

The measured object is therefore not always a completely fixed visual entity under every condition. The shape available for measurement may depend upon the scale and resolution at which it is viewed or represented.

The Viewpoint–Correspondence Problem

A single two-dimensional projection does not ordinarily determine one unique three-dimensional reality.

Different spatial objects or arrangements can sometimes produce:

  • the same outline;
  • the same projected position;
  • similar visible angles;
  • the same or closely corresponding image.

This is the Viewpoint–Correspondence Problem.

A flat image may not fully specify:

  • depth;
  • physical size;
  • distance;
  • hidden surfaces;
  • the precise shape of the original object;
  • the position of the viewpoint.

Additional constraints or information may therefore be required, such as:

  • known dimensions;
  • parallel or perpendicular relationships;
  • multiple views;
  • camera information;
  • lighting and shadow;
  • movement;
  • stereoscopic information;
  • prior knowledge of likely objects.

Mathematical Perspective can reduce the range of possible solutions, but it does not automatically remove every ambiguity.

Central projection

In central projection, all projectors pass through one projection centre.

This creates many of the familiar effects of linear perspective:

  • distant objects project smaller;
  • receding parallel directions converge;
  • sets of parallels possess vanishing points;
  • planes possess vanishing lines;
  • the position of the viewpoint affects the complete image.

One-point, two-point and three-point perspective are familiar graphical applications of central projection.

The projected image depends upon:

  • the position of the projection centre;
  • the position and orientation of the image plane;
  • the direction of the spatial objects;
  • the distance between object, viewpoint and plane;
  • the chosen field of view.

Parallel projection

In parallel projection, the projectors remain parallel.

Parallel projection does not produce the same diminution and convergence as central projection. Parallel spatial lines normally remain parallel in the resulting image.

Principal forms include:

  • orthographic projection;
  • oblique projection;
  • axonometric projection;
  • isometric projection;
  • dimetric projection;
  • trimetric projection.

Orthographic projection uses projectors perpendicular to the receiving plane and is widely used in technical and engineering drawing because it can preserve measurable scale and shape in appropriate views. 

Descriptive geometry and technical representation

Descriptive geometry provides methods for representing and solving three-dimensional spatial problems through organised two-dimensional projections.

A solid object may require several distinct views to specify its structure:

  • top or plan view;
  • front elevation;
  • side elevation;
  • sectional or auxiliary view.

These views are not intended primarily to reproduce how the object appears from an ordinary human viewpoint. They are intended to reveal measurable shape, orientation and construction.

Mathematical Perspective therefore includes both:

  • appearance-based projection, which may resemble a view;
  • measurement-based projection, which provides technical spatial information.

This distinction is important in architecture, engineering, manufacturing and design.

Projective geometry and vanishing structures

Projective geometry formalises relationships that remain meaningful under projection.

In perspective, these include:

  • incidence — whether points and lines meet;
  • collinearity — whether points lie on one line;
  • convergence;
  • projection centres;
  • corresponding points;
  • vanishing points;
  • vanishing lines;
  • projective transformations.

A set of parallel spatial lines possesses a common projected direction. Under central projection, this direction is represented by a vanishing point.

A plane contains many possible sets of parallel directions. Its corresponding vanishing points form a vanishing line.

Geometrical vanishing and optical vanishing

Mathematical or geometrical vanishing should be distinguished from optical disappearance.

Geometrical vanishing

Geometrical vanishing identifies the limiting projected direction of a set of spatial parallels.

It concerns where a direction converges in projection.

Optical vanishing

Optical vanishing occurs when an object or detail becomes too small, faint or low in contrast to remain detectable.

It concerns how far an object remains visible.

A geometrical vanishing point does not mean that a physical object has literally reached or disappeared at that point. It represents a mathematical direction or limit.

This distinction connects Mathematical Perspective with Optical Perspective while preventing two different meanings of vanishing from being confused.

Vanishing points in all directions

Vanishing geometry is not confined to three points on a frontal picture plane.

Every spatial direction can possess a corresponding geometrical vanishing direction. Around an observer or projection centre, these directions can be modelled as points distributed across an observer-centred sphere.

Opposite directions correspond to opposite points.

This all-direction model extends vanishing geometry beyond conventional one-, two- and three-point systems and provides a mathematical connection to:

  • spherical projection;
  • panoramic representation;
  • wide-field viewing;
  • virtual reality;
  • all-direction imaging;
  • the Sphere of Vision.

Planar and non-planar projection

Not every projection uses a flat image plane.

Spatial information may be projected onto or mapped through:

  • planes;
  • cylinders;
  • cones;
  • spheres;
  • hemispheres;
  • disks;
  • curved display surfaces;
  • irregular geometrical structures.

Non-planar projection is important in:

  • panoramic imaging;
  • cartography;
  • spherical perspective;
  • fisheye photography;
  • planetariums;
  • dome projection;
  • virtual reality;
  • environmental displays.

A straight spatial line need not remain straight under every mapping. Its represented form depends upon the geometry of the projection system and receiving surface.

Mathematical Perspective and Graphical Perspective

Mathematical Perspective and Graphical Perspective frequently overlap, but they are not identical.

Mathematical Perspective is defined by mathematical relations, measurements, geometrical principles, transformations or calculations.

Graphical Perspective is defined by construction or representation through drawing, painting, diagramming, drafting or another graphic method.

A perspective image may belong to both categories.

For example, a linear-perspective drawing may be:

  • mathematical because it follows projective geometrical relations;
  • graphical because it is drawn on a surface;
  • visual because it produces a visible image.

However:

  • an algebraic projection formula may be mathematical without being graphical;
  • a freehand spatial sketch may be graphical without following a rigorous mathematical system.

The distinction concerns the principal process or source, not simply the appearance of the finished image. Perspective Category Theory also allows one method or image to belong legitimately to more than one category. 

Mathematical, Optical and Visual Perspective

Mathematical Perspective should also be distinguished from Optical and Visual Perspective.

Mathematical Perspective

Calculates, measures, models, transforms or projects spatial relationships.

Optical Perspective

Forms or transmits views and images through light, reflection, refraction, focusing or another optical process.

Visual Perspective Type 1

Includes the resulting visible image, diagram, projection or representation.

Visual Perspective Type 2

Concerns the retinal and perceptual experience produced when a human observer views the result.

A computer-generated image may therefore involve:

Mathematical model and coordinates
↓
Mathematical transformations and projection
↓
Graphical rendering
↓
Digital display
↓
Optical transmission from display to eye
↓
Visual Perspective Type 2

Several perspective categories can operate together in one image chain.

Historical development

The mathematical history of perspective extends far beyond one Renaissance invention.

Antiquity and medieval optics

Ancient geometry and theories of visual rays provided early methods for relating sight, direction, angle and apparent size.

During the medieval period, Greek geometrical and optical traditions were preserved, translated and extended through Arabic and European optics.

Renaissance perspective

During the fifteenth and sixteenth centuries, relationships between geometry, optics, architecture, surveying and pictorial representation led to the systematic development of central perspective.

Seventeenth century

Connections between perspective, conic sections and geometry became increasingly formal, and perspective emerged as one of the mathematical sciences.

Eighteenth and nineteenth centuries

Perspective and stereotomy were incorporated into descriptive geometry. Parallel projection, axonometry, isometric drawing, conic sections and technical drawing became increasingly systematic.

Twentieth and twenty-first centuries

Analytic geometry, photography, photogrammetry, computing and digital modelling expanded Mathematical Perspective into dynamic and computational processes.

It now includes computer graphics, computer vision, artificial intelligence, three-dimensional scanning, GIS, scientific imaging, VR, AR and real-time spatial systems. 

Mathematical Perspective in computer graphics

Computer graphics is founded upon sequences of mathematical spatial transformations.

A typical graphics process moves from:

Model coordinates
↓
World coordinates
↓
View or camera coordinates
↓
Projection coordinates
↓
Image or screen coordinates

The system calculates:

  • object position;
  • orientation;
  • scale;
  • camera location;
  • field of view;
  • clipping;
  • projection;
  • depth;
  • screen placement.

Perspective projection can then simulate the diminution and convergence associated with a finite viewpoint, while orthographic projection can present objects without perspective diminution.

Mathematical Perspective in computer vision

Computer vision often works in the reverse direction.

It analyses images to estimate:

  • object position;
  • depth;
  • scale;
  • motion;
  • surface orientation;
  • camera position;
  • spatial correspondence;
  • three-dimensional structure.

A single image may be insufficient to determine all of these properties uniquely. Multiple images, known dimensions, motion, lighting, depth sensors or statistical models may be required.

This makes the Viewpoint–Correspondence Problem central not only to traditional perspective but also to robotics, autonomous systems and artificial intelligence.

Applications of Mathematical Perspective

Mathematical Perspective operates across many disciplines.

Art and architecture

Perspective construction, proportion, architectural projection and spatial design.

Engineering and manufacturing

Technical drawing, CAD, machine design, measurement and fabrication.

Surveying and photogrammetry

Recovering dimensions, positions and terrain information from observations and images.

Cartography and GIS

Transforming the curved Earth or another spatial model into maps and digital geographic structures.

Photography and cinema

Camera calibration, field of view, lens modelling, compositing and visual effects.

Computer graphics and games

Three-dimensional modelling, animation, rendering, virtual cameras and image projection.

Computer vision and robotics

Image analysis, depth estimation, navigation, object recognition and spatial reconstruction.

Scientific and medical imaging

Measuring, modelling and representing structures that may be too small, distant, internal or otherwise inaccessible to direct vision.

Virtual and augmented reality

Generating mathematically consistent views that respond to changing head, eye or camera position.

Why Mathematical Perspective matters

Mathematical Perspective provides a bridge between physical space and abstract understanding.

It makes it possible to:

  • measure spatial reality;
  • compare objects and images;
  • model structures that cannot be viewed directly;
  • transform between dimensions and coordinate systems;
  • produce accurate technical representations;
  • generate projected and simulated worlds;
  • reconstruct space from images;
  • identify the limits and ambiguities of representation.

It also demonstrates that perspective is not merely an artistic convention.

Perspective is involved whenever mathematical relationships are used to organise, calculate, transform, project or understand spatial reality.

Mathematical Perspective in The Art and Science of Perspective

Detailed geometrical constructions, diagrams and worked examples are developed in The Art and Science of Perspectivebook series.

Volume 1, The Past, Present and Future of Visual and Optical Perspective, introduces the historical, theoretical and technological development of mathematical and graphical perspective.

Volume 2, Dictionary of Perspective, defines Mathematical Perspective and its many related terms, methods, geometries and projections.

A later volume in the series, Graphical and Mathematical Perspective, will provide a more detailed examination of mathematical modelling, geometrical systems, projection methods, transformations, graphical construction and their applications.

Primary Categories

Explore the principal categories through which perspective can be studied and classified.

Natural Perspective →
Perspective arising through natural viewing and the appearance of spatial reality.

Mathematical Perspective →
Perspective based on mathematical, geometrical and projective principles.

Graphical Perspective →
Perspective constructed or represented through drawing and other graphical methods.

Instrument Perspective →
Perspective produced or mediated through cameras, lenses and other imaging instruments.

Simulated Perspective →
Perspective produced through artificial or simulated representations of spatial appearance.

New Media Perspective →
Perspective associated with digital imaging, computer graphics, virtual environments and emerging visual technologies.