Spherical Perspective is a family of visual, optical, graphical and mathematical systems in which spatial directions are organised through a sphere, spherical surface or complete all-direction field.

It can represent a hemisphere or a full 360-by-180-degree environment, including directions:

in front of and behind the observer;
to the left and right;
above and below;
and along every intermediate direction.

Spherical Perspective is therefore not simply a wide version of Linear Perspective. It replaces the restricted frontal picture plane with a system capable of organising space around an observer, around an object or across a spherical display.

It includes:

Sphere of Vision Perspective;
Sphere of Revolution Perspective;
Sphere of Rotation Perspective;
spherical and digital globes;
five- and six-point graphical systems;
spherical panoramas;
glass-sphere views;
mirror- and metal-ball reflections;
spherical cameras and image mappings;
dome, globe and spherical displays;
virtual-reality and all-direction environments.

Five- and six-point perspective drawings are important forms of Spherical Perspective, but they are not requirements of every spherical system. Spherical Perspective may be optical, graphical, mathematical, digital, reflected, projected, displayed or experienced through movement.

INSERT IMAGE 21 — SPHERICAL PERSPECTIVE

Suggested caption: Spherical Perspective organises spatial appearance through an all-direction field rather than one limited frontal picture plane.

What is Spherical Perspective?

Spherical Perspective maps spatial directions onto:

the surface of a sphere;
the interior of a sphere;
a spherical field surrounding a viewpoint;
a flat image derived from a sphere;
or a sequence of views obtained by looking around or moving around an object.

The spherical system may be used to:

view a surrounding environment;
record a 360-degree scene;
represent all directions within one image;
examine an object from many viewpoints;
display an image inside or outside a sphere;
navigate a digital globe;
create an immersive environment;
transform spherical information into a flat map or image.

A sphere can therefore operate as:

a model of viewing directions;
an image-capturing structure;
a projection or mapping surface;
a display surface;
a path of movement around an object;
a graphical framework for multi-directional vanishing.

The word spherical refers to the geometry or spatial organisation of the perspective system. A conventional drawing of a ball made in Linear Perspective is not automatically Spherical Perspective.

Spherical Perspective is not one single method

Spherical Perspective is a broad family rather than one fixed construction.

The Dictionary includes systems based upon:

equirectangular mapping;
fisheye projection;
stereographic projection;
hemispherical projection;
six-point graphical construction;
spherical reflection;
globe-like representation;
panoramic photography;
spherical image capture;
internal and external spherical displays.

Different systems preserve and transform different spatial properties.

One spherical mapping may preserve local angles. Another may preserve area, distance from a selected point, direction or some other relationship. None can preserve every spatial property when the spherical field is transferred onto a flat surface.

The correct interpretation therefore depends upon:

what is being viewed;
where the observer is located;
whether the viewer looks inward or outward;
whether the sphere moves;
whether the image is optical or constructed;
whether it remains spherical or is flattened;
whether the result is static, moving or interactive.

Two fundamental directions of Spherical Perspective

The most important conceptual distinction is between:

Looking outward and around

and:

Looking inward or at

These correspond to two fundamentally different spherical relationships.

Sphere of Vision Perspective

Sphere of Vision Perspective is observer-centred.

The observer, eye, camera or viewpoint is imagined at or near the centre of a sphere and looks outward towards the surrounding environment.

It is a:

looking-out / looking-around system

The sphere represents every possible viewing direction around the observer.

It can include:

the forward field;
the rearward field;
left and right directions;
upper and lower directions;
the complete circular horizon;
every intermediate direction.

The Sphere of Vision is therefore a model of a total surrounding visual field rather than one isolated view.

A complete Sphere of Vision may be produced or represented through:

a spherical painting;
a 360-degree photograph;
spherical video;
a virtual-reality environment;
an equirectangular panorama;
a cube map;
a dome display;
a six-point graphical construction.

INSERT IMAGE 21 — SPHERICAL PERSPECTIVE / SPHERE OF VISION

Suggested caption: Sphere of Vision Perspective places the observer within a complete surrounding field and represents looking outward and around.

Sphere of Revolution Perspective

Sphere of Revolution Perspective is object-centred.

An eye, camera or other viewing system moves around a three-dimensional object or restricted spatial region, obtaining views from positions distributed across a surrounding sphere.

It is a:

looking-in / looking-at system

The object remains near the centre while the viewpoint revolves around it.

Sphere of Revolution Perspective may provide:

front, rear and side views;
upper and lower views;
close and distant views;
successive moving views;
a complete set of views from all surrounding directions.

It is relevant to:

object photography;
turntable imaging;
photogrammetry;
three-dimensional scanning;
digital modelling;
product visualisation;
medical and scientific imaging;
archaeological recording;
interactive models.

The resulting views may be kept separate, joined into a moving sequence or combined into a reconstructed three-dimensional model.

INSERT IMAGE 22 — SPHERE OF REVOLUTION

Suggested caption: Sphere of Revolution Perspective explores an object through viewpoints distributed around it, producing a looking-in or looking-at system.

Sphere of Vision and Sphere of Revolution compared

The distinction can be expressed simply:

Sphere of Vision

observer-centred;
viewpoint at or near the centre;
environment surrounds the observer;
viewing proceeds outward;
describes looking around a scene.

Sphere of Revolution

object-centred;
object at or near the centre;
viewpoints surround the object;
viewing proceeds inward;
describes looking at or around an object.

The same physical or digital system may sometimes combine the two.

For example, a person may rotate a glass ball while also moving around it. The ball displays a wide reflected or refracted environment while its orientation and the observer’s viewing direction continually change.

The Dictionary therefore allows mixing between Sphere of Vision and Sphere of Revolution processes rather than forcing every spherical image into one completely isolated class.

Sphere of Rotation Perspective

Sphere of Rotation Perspective concerns a spherical image or display that is itself rotated or navigated so that different parts of its represented space become visible.

Examples include:

an ordinary terrestrial globe;
a celestial globe;
a rotating painted sphere;
an interactive digital globe;
a touch-controlled spherical map;
a globe-shaped display;
a digital model that can be rotated and enlarged.

The viewing position may remain comparatively fixed while the represented sphere rotates.

This differs from Sphere of Revolution Perspective, where the viewpoint moves around an object.

The distinction is:

Sphere of Revolution — the viewpoint revolves around the subject.

Sphere of Rotation — the spherical subject or display rotates before the viewer.

In interactive digital systems, both may appear equivalent because rotating the model can simulate moving the viewpoint around it.

Three principal Spherical View types

Your revised Dictionary also identifies three useful classes of Spherical View.

Type 1 — Unfolding optical sphere

A glass, metal or mirror ball produces a miniature spherical or panoramic view of its surroundings.

As the ball, viewer or viewing direction changes, the image appears to unfold across the surface.

This form may combine:

Sphere of Vision;
Sphere of Revolution;
optical reflection or refraction;
changing viewpoint;
panoramic image formation.

Type 2 — Unfolding globe or digital model

A globe or digital sphere presents a complete object or world whose different regions become visible through rotation, movement, zooming or navigation.

This is principally a Sphere of Revolution or Sphere of Rotation form.

Type 3 — Complete Sphere of Vision

A total surrounding spatial field is organised around a central observer or viewpoint.

This is the fullest looking-out and looking-around form and can represent an actual, virtual or constructed 360-by-180-degree environment.

The three types distinguish optical unfolding, object-centred unfolding and observer-centred all-direction viewing.

Spherical Perspective is multi-directional

Conventional Linear Perspective normally organises an image around one principal frontal viewing direction.

Spherical Perspective organises multiple or unlimited directions around a sphere.

A complete spherical system should therefore not normally be classified as Central or Frontal Perspective.

It may contain many individual local views, each possessing its own centre of vision or optical axis. However, no single frontal axis organises the entire sphere.

The complete spherical field is:

multi-directional;
all-direction;
rotational;
observer-centred or object-centred;
unrestricted to one frontal picture plane.

A spherical panorama may be opened at one selected direction for display, but this seam or chosen front does not turn the complete system into one ordinary frontal perspective.

Spherical geometry

Spherical Perspective uses relationships belonging to spherical rather than purely planar geometry.

Important elements include:

the centre of the sphere;
radius;
diameter;
antipodal points;
great circles;
small circles;
spherical angles;
meridians;
parallels;
hemispheres;
poles;
spherical coordinates.

Great circles

A great circle is formed when a plane passes through the centre of the sphere.

Examples include:

the equator;
meridians on a globe;
principal directional circles within a Sphere of Vision.

Great circles provide the shortest paths between two points on a spherical surface.

Small circles

A small circle is formed when a plane intersects the sphere without passing through its centre.

Parallels of latitude other than the equator are familiar examples.

Antipodal points

Two points at opposite ends of a diameter are antipodal.

In spherical vanishing geometry, one spatial direction and its opposite correspond to antipodal vanishing points.

Vanishing points in all directions

In conventional Linear Perspective, attention is often concentrated upon one, two or three principal vanishing points.

Spherical Perspective reveals that vanishing directions continue around the complete observer.

Every spatial direction can be associated with a point on a directional sphere. Its opposite direction corresponds to the antipodal point on the other side.

Vanishing does not therefore terminate at the left or right edge of a conventional picture.

It continues:

around the horizon;
above and below;
in front and behind;
throughout the complete spherical field.

INSERT IMAGE 18 — VANISHING POINTS IN ALL DIRECTIONS

Suggested caption: Every spatial direction has a corresponding geometrical vanishing point, with the opposite direction represented by its antipodal point.

Six-point Spherical Perspective

A cubical or orthogonal spatial framework contains three principal sets of mutually perpendicular parallel lines:

left–right;
forward–backward;
up–down.

Each set has two opposite vanishing directions.

This produces six principal vanishing points:

left;
right;
front;
back;
above;
below.

These correspond to the six vertices of an octahedral directional structure.

Six-point perspective is especially useful for representing:

a complete cubical environment;
an all-direction room;
a painted spherical world;
a complete Sphere of Vision;
opposite directional pairs.

The six points are not the only vanishing points within the complete spherical system. They are the six principal points associated with one orthogonal cubical framework.

Every other spatial direction also possesses its own corresponding spherical vanishing point and opposite point.

Volume 1 connects six-point Spherical Perspective with the work of Dick Termes and the representation of a complete surrounding room or environment.

Five-point and six-point perspective

Five-point and six-point perspective should be distinguished.

Five-point perspective

Five-point perspective generally represents one hemisphere.

Its principal directions are:

forward or centre;
left;
right;
above;
below.

It can represent a field approaching 180 degrees but does not necessarily include the complete rearward hemisphere.

Six-point perspective

Six-point perspective represents a complete spherical or all-direction field by adding the opposite or rearward direction.

It can therefore describe a full 360-by-180-degree environment.

A five-point image is normally hemispherical and curvilinear.

A six-point image is a fuller spherical structure.

Nevertheless, the terminology varies between authors, and the number of named points should not be mistaken for the complete number of possible vanishing directions.

Spherical and Curvilinear Perspective

Spherical and Curvilinear Perspective overlap, but they are not identical.

Curvilinear Perspective is defined primarily by curved graphical or projected lines.

Spherical Perspective is defined primarily by spherical or all-direction spatial organisation.

A five-point hemispherical drawing is both:

curvilinear, because spatial straight lines may be represented by curves;
spherical, because the field is organised through a hemisphere.

An equirectangular 360-degree image is spherical in its underlying organisation, even though its flattened grid may contain straight vertical and horizontal coordinate lines.

A spherical display can also present an image without requiring the observer to see a conventionally curved flat drawing.

The correct classification therefore depends upon the complete geometry and process, not simply whether the finished image contains curves.

Spherical and Panoramic Perspective

Spherical Perspective and Panoramic Perspective are also related but distinct.

Panoramic Perspective concerns the extension of the represented field beyond one normal frame.

Spherical Perspective concerns the organisation of that field through a sphere or complete all-direction geometry.

A panorama may be:

rectilinear;
cylindrical;
spherical;
multi-image;
stitched;
interactive.

A spherical panorama normally contains:

360 degrees horizontally;
up to 180 degrees vertically;
information above and below;
a complete environmental field.

A cylindrical panorama may extend around the full horizon but omit the upper and lower poles.

INSERT IMAGE 20 — PANORAMIC PERSPECTIVE

Suggested caption: Panoramic Perspective expands the represented field, while Spherical Perspective can extend that field into a complete 360-by-180-degree environment.

Spherical and Linear Perspective

Linear and Spherical Perspective use fundamentally different image structures.

Linear Perspective

usually uses one flat picture plane;
preserves straight spatial lines as straight;
represents a limited frontal field;
uses one principal viewpoint and viewing direction;
expands scale strongly near the margins of very wide views.

Spherical Perspective

organises space through a sphere or spherical mapping;
can represent all directions;
uses curved, transformed or segmented mappings;
may include opposite vanishing points;
can surround the observer or object;
usually requires transformation when flattened.

Linear Perspective is highly suitable for moderate fields, architecture and measured planar images.

Spherical Perspective is better suited to complete environments, panoramas, immersive media and multi-directional viewing.

Neither system preserves every spatial property.

Mapping a sphere onto a flat surface

A spherical surface cannot be flattened onto one plane without transformation.

This is familiar from cartography: every flat world map changes some combination of:

area;
shape;
angle;
direction;
scale;
distance;
continuity.

The same problem applies to spherical photographs and graphical perspectives.

Common mappings include:

equirectangular;
stereographic;
fisheye;
equidistant;
equisolid-angle;
orthographic;
cubical or cube-map;
cylindrical;
azimuthal.

Each selects different properties to preserve or distribute.

A spherical image should therefore not be judged as though it were intended to follow the rules of one flat rectilinear picture plane.

Equirectangular Spherical Perspective

Equirectangular mapping represents spherical longitude and latitude through a rectangular grid.

It is widely used for:

360-degree photography;
spherical video;
virtual-reality environments;
environment maps;
digital panoramas;
texture mapping.

The complete spherical field is stretched into a rectangle.

The equator occupies the central horizontal line, while the upper and lower poles extend across the top and bottom edges.

The format is convenient for storage and computation, but shapes become increasingly stretched towards the poles.

When viewed through an interactive spherical viewer, only a local portion is normally displayed at one time, reducing the conspicuousness of the flattened distortion.

Cube-map Spherical Perspective

A cube map divides a complete surrounding field into six square faces:

front;
back;
left;
right;
top;
bottom.

Each face can be rendered as a local rectilinear view.

Together, the six views form a complete environmental representation.

Cube maps are commonly used in:

computer graphics;
reflections;
virtual reality;
games;
lighting calculations;
environmental imaging.

This structure corresponds closely to the six principal directions of a cubical or six-point spherical system.

The complete cube map is multi-directional even though each individual face uses a local frontal projection.

Glass Sphere Perspective

A transparent glass sphere refracts light from the surrounding scene.

It can produce:

a miniature wide-angle view;
an inverted central image;
strong curvature and compression;
views extending beyond an ordinary frontal field;
changing appearances as the sphere or viewer moves.

The glass ball does not merely display a pre-existing image. Its curved surfaces and refractive material form an optical perspective system.

A complete image may involve:

surrounding Natural Perspective
↓
refraction through the glass sphere
↓
camera or eye imaging
↓
photographic or retinal image
↓
visual interpretation

Glass Sphere Perspective can combine features of Sphere of Vision and Sphere of Revolution because it shows a wide environment while also changing as the ball, camera or viewer moves.

INSERT IMAGE 01 — VISUAL PERSPECTIVE / GLASS BALL

Suggested caption: A glass sphere forms a compressed panoramic view through refraction, while the camera and viewer add further stages to the final perspective image.

Mirror- and Metal-Ball Perspective

A polished metal or mirrored sphere reflects a large part of its surrounding environment.

It may show:

the space in front of the ball;
lateral surroundings;
regions behind the camera or observer;
the camera and observer themselves;
a compressed, curved environmental image.

A mirror ball forms an optical Sphere of Vision around its own position.

Its image changes when:

the sphere moves;
the observer moves;
the surrounding environment changes;
the viewing direction changes.

Artists have used spherical reflections to represent spatial environments within one compact form.

M.C. Escher’s reflective-sphere images are prominent examples of spherical reflection, viewpoint and represented space.

Spherical Perspective and the human eye

Spherical Perspective is often compared with human vision because:

the retina is curved;
viewing directions extend around the observer;
the eyes and head rotate;
perception is assembled through successive fixations;
peripheral vision extends beyond one narrow frontal image.

However, the Sphere of Vision should not be confused with the anatomical retina.

The Sphere of Vision is a directional and spatial model representing looking outward into the surrounding world.

The retina is a curved light-sensitive surface inside the eye onto which optical information is formed.

Human visual experience is not perceived as a small spherical image inside the eye. Retinal information is processed, integrated and interpreted by the visual system.

Spherical Perspective can model the organisation of all-direction viewing without claiming that perceived visual space is literally one visible spherical screen.

Spherical Perspective and Visual Perspective Type 2

Visual Perspective Type 2 concerns retinal and perceptual appearance through the human visual system.

Spherical Perspective may contribute to understanding:

observer-centred direction;
eye and head movement;
panoramic visual exploration;
the relationship between central and peripheral fields;
the complete horizon;
successive local views;
the integration of spatial information over time.

Direct human vision does not normally present the complete surrounding sphere simultaneously at equal clarity.

The observer explores the sphere through:

eye movement;
head movement;
body rotation;
changing attention;
movement through space;
memory and perceptual integration.

A spherical graphical or virtual image can bring these successive directional relationships into one organised system.

Spherical Perspective and movement

Movement is fundamental to many spherical systems.

A spherical view can unfold through:

rotating the observer;
rotating the camera;
rotating the object;
rotating the sphere;
moving around an object;
changing the direction of a virtual view;
navigating through an interactive environment.

Static Spherical Perspective attempts to contain several or all directions within one fixed image or object.

Dynamic Spherical Perspective reveals the same field sequentially through movement.

This distinction connects spherical geometry with:

time;
multiple viewpoints;
moving-image perspective;
interactive media;
spatial navigation.

Spherical displays

A spherical display places images upon:

the interior of a dome or sphere;
the exterior surface of a globe;
a partly spherical screen;
a digital or physical rotating sphere.

Interior displays

Interior spherical or dome displays surround the viewer.

Examples include:

planetariums;
dome cinemas;
immersive theatres;
simulation environments;
hemispherical visualisation rooms.

They create a Sphere of Vision or Hemisphere of Vision by placing the observer inside projected image space.

Exterior displays

Exterior spherical displays place images on the outside of a sphere.

The viewer stands outside and looks at or around the display.

Examples include:

globes;
spherical LED displays;
advertising spheres;
rotating digital models;
artworks such as Termespheres.

The same image content may be perceived very differently according to whether the viewer is inside or outside the spherical image surface.

Inside and outside spherical image space

Spherical Perspective creates an important distinction between two modes of experience.

Viewer inside the image

The image surrounds the observer.

This supports:

immersion;
simulation;
virtual environments;
planetariums;
dome theatres;
Sphere of Vision systems.

Viewer outside the image

The viewer looks at a spherical object or display.

This supports:

globes;
mirror balls;
glass balls;
spherical sculptures;
exterior spherical screens;
Sphere of Rotation and Revolution systems.

An exterior spherical image can be complete in itself, but the observer normally sees only part of its surface at one time.

An interior display may provide a much larger immediate field, but complete 360-by-180-degree inspection still involves movement and changing attention.

Spherical Perspective in art

Artists have used spherical and all-direction systems to:

represent complete environments;
combine opposite directions;
explore multiple viewpoints;
place the viewer within depicted space;
paint globes and spherical surfaces;
construct immersive images;
investigate reflection and self-representation;
question the dominance of the rectangular picture plane.

Important modern developments include:

spherical reflection in the work of M.C. Escher;
systematic curvilinear studies by Albert Flocon and André Barre;
five- and six-point systems;
the spherical paintings and Termespheres of Dick Termes.

Volume 1 places these developments within a longer history extending from ancient optics and Renaissance investigations to contemporary immersive and digital media.

Historical development

Spherical Perspective has roots in several historical traditions.

Antiquity

Ancient optics examined visual rays, angular appearance and the organisation of viewing around the eye.

Euclid discussed aspects of visual geometry and the apparent curvature or variation of visual space, although he did not produce a modern spherical pictorial system.

Renaissance

Leonardo da Vinci and other Renaissance theorists investigated:

wide fields of vision;
lateral distortion;
cylindrical and spherical projection;
changing appearances across the visual field.

Linear Perspective remained dominant, but it did not exhaust interest in curved or all-direction representation.

Panoramic and optical developments

Painted panoramas, cylindrical displays, globes, optical balls and curved screens expanded the field beyond one framed planar image.

Twentieth century

Artists and theorists developed systematic curvilinear and spherical constructions.

Five- and six-point systems, hyperbolic constructions, panoramic cameras and dome theatres increased interest in all-direction representation.

Digital era

Digital photography, CGI, environment mapping, virtual reality, spherical video and interactive displays have made spherical images common.

Spherical Perspective is now central to many systems of navigation, simulation, visualisation and immersion.

Spherical Perspective in photography and cinema

Spherical image capture may use:

several cameras;
multiple lenses;
fisheye lenses;
panoramic mirrors;
rotating cameras;
stitched image sequences;
specialised 360-degree cameras.

The separate views are mapped into one spherical environment.

Spherical video extends the system through time. The viewer may choose where to look while the event continues.

This changes the traditional relationship between filmmaker and audience.

In conventional cinema, the rectangular frame determines the view.

In spherical cinema, the recorded environment surrounds the viewer, and attention can move within it.

Spherical Perspective in virtual reality

Virtual reality is one of the clearest contemporary applications of Sphere of Vision Perspective.

A complete digital environment is organised around the user.

As the user turns:

the displayed view changes;
the local perspective is recalculated;
new parts of the spherical environment become visible;
the visual field remains centred on the changing head direction.

A stored environment may use:

an equirectangular map;
a cube map;
a three-dimensional scene;
a dynamic rendered world.

The user does not normally see the entire sphere flattened at once. Instead, the system presents a moving local window into the complete spherical environment.

Spherical Perspective in computer graphics

Computer graphics uses spherical structures for:

environment maps;
reflection maps;
sky domes;
sky boxes;
global illumination;
panoramic rendering;
virtual cameras;
texture mapping;
immersive simulation.

A spherical environment may first be stored mathematically and later displayed through local rectilinear views.

This process can be expressed as:

Three-dimensional digital environment
↓
Spherical or cube-map representation
↓
Selected viewing direction
↓
Locally projected screen image
↓
Optical display and visual experience

The complete system is spherical even where the final instantaneous image is rectangular.

Spherical Perspective in mapping and astronomy

Cartography maps spherical or approximately spherical worlds onto flat surfaces.

Astronomy represents:

celestial spheres;
star positions;
orbital directions;
all-sky surveys;
hemispherical observations.

Spherical mappings are also used for:

Earth observation;
meteorology;
planetary science;
navigation;
satellite imaging;
geographical information systems.

Different projections are selected according to whether the priority is:

area;
direction;
angle;
distance;
shape;
continuity;
regional or global coverage.

Spherical Perspective in scientific and technical imaging

Spherical Perspective is useful where spatial information surrounds an observer, instrument or object.

Applications include:

robotic navigation;
autonomous vehicles;
computer vision;
archaeological recording;
architectural surveys;
environmental monitoring;
industrial inspection;
medical imaging;
three-dimensional scanning;
photogrammetry;
remote presence.

A robotic system may combine several cameras into one spherical model so that obstacles and landmarks can be detected in every direction.

A scanning system may instead orbit an object and combine Sphere of Revolution views into a three-dimensional reconstruction.

Spherical Perspective as a composite system

A single spherical image may involve several perspective categories.

For example, a 360-degree photograph viewed through a VR headset may contain:

Natural Perspective in the source environment;
Optical Perspective through the camera lenses;
Instrument Perspective through the camera system;
Mathematical Perspective through spherical mapping;
New Media Perspective through stitching and processing;
Graphical or displayed Visual Perspective Type 1;
Optical projection from the headset display;
Visual Perspective Type 2 through the viewer’s eyes and perception.

This is an example of category chaining.

Spherical Perspective describes the all-direction geometry of the system, but it does not replace the need to identify the other processes participating in the final experience.

Strengths of Spherical Perspective

Spherical Perspective can:

represent a complete surrounding environment;
include front, rear, left, right, upper and lower directions;
unite opposite vanishing directions;
support panoramic and immersive experience;
record objects from many viewpoints;
provide environmental maps for computing;
connect viewing, imaging, projection and display;
place the observer inside represented image space;
move beyond the restrictions of one frontal picture plane.

It is particularly valuable for all-direction subjects, environments and interactive systems.

Limitations of Spherical Perspective

Its principal limitations include:

unavoidable transformation when flattened;
curvature or distortion of some spatial lines and forms;
uneven scale under many mappings;
difficulty of representing the complete field legibly at once;
seams, poles or discontinuities in some formats;
dependence upon viewing software or curved displays;
greater complexity than ordinary planar projection;
difficulty of direct measurement;
possible confusion between Sphere of Vision and Sphere of Revolution systems.

A spherical representation is not automatically more accurate than a planar representation.

Its value depends upon:

the purpose;
the chosen mapping;
the field required;
the intended viewing conditions;
the properties that need to be preserved.

Why Spherical Perspective matters

Spherical Perspective demonstrates that perspective is not restricted to a framed view in front of a stationary observer.

Spatial reality surrounds us.

It extends:

above;
below;
behind;
around;
and through movement.

Spherical Perspective provides systems for organising this complete directional structure.

It connects:

natural and visual experience;
graphical and mathematical projection;
optics and instruments;
panoramic photography;
cartography and astronomy;
cinema and immersive display;
computer graphics and artificial intelligence;
virtual and augmented reality.

It also establishes a fundamental distinction between:

looking outward into a surrounding world

and:

looking inward at an object from surrounding viewpoints.

The Sphere of Vision and Sphere of Revolution are therefore not merely two drawing techniques. They describe two basic ways in which viewpoint, object and surrounding space can be organised.

Spherical Perspective in The Art and Science of Perspective

Volume 1, The Past, Present and Future of Visual and Optical Perspective, introduces Spherical Perspective within its historical, graphical, optical and technological contexts.

Volume 2, Dictionary of Perspective, defines Spherical Perspective and distinguishes Sphere of Vision, Sphere of Revolution, Sphere of Rotation, Spherical View, spherical projection, spherical displays and related forms.

Volume 3, Natural and Visual Perspective, will examine the observer-centred Sphere of Vision and its relationship to natural space, human vision, viewpoint, movement, visual fields and all-direction perception.

Volume 4, Graphical and Mathematical Perspective, will provide a deeper account of spherical and curvilinear construction, mapping and projection geometry.