Primary and Secondary Geometry

Primary and Secondary Geometry distinguish two fundamentally different geometrical levels involved in perspective.

Primary Geometry concerns the three-dimensional arrangement of objects, structural lines and projection lines within Object or Target Space.

Secondary Geometry concerns the resulting relationships between points, lines and shapes within the two-dimensional Image or Perspective Space.

The basic relationship can therefore be expressed as:

Primary Geometry in Object Space → Perspective Process or Projection → Secondary Geometry in Image Space

This distinction helps separate the geometry of the original spatial reality from the geometry of its resulting Perspective Image or Representation.

Primary and Secondary Geometry within Perspective

Perspective relates spatial objects and scenes to images, views, measurements and representations.

In doing so, two different geometrical structures may be considered:

Primary Geometry — the geometry of the spatial object, scene and projection relationships in Object Space;
Secondary Geometry — the geometry of the resulting image or representation in Image Space.

These geometries are connected, but they are not identical.

A Perspective Process may systematically transform Primary Geometry into Secondary Geometry while changing how size, shape, direction, position, parallelism, depth and other spatial relationships appear.

What is Primary Geometry?

Primary Geometry is the three-dimensional geometry associated with the original Perspective Object / Scene and the projection relationships used to form an image of it.

It includes the arrangement of:

object points;
structural lines;
object planes;
spatial Forms;
projection lines or rays;
the Viewpoint or Centre of Projection;
the Picture or Projection Plane; and
other geometrical relationships existing in Object or Target Space.

Primary Geometry therefore concerns the geometrical structure from which a perspective image, view, match or representation is derived.

What is Secondary Geometry?

Secondary Geometry is the geometry of the resulting image or representation.

It concerns the relationships between:

image points;
lines;
shapes;
projected planes;
vanishing relationships;
constructed dimensions; and
other geometrical structures on the Picture or Image Surface.

Secondary Geometry therefore belongs principally to Image Space.

It describes how the spatial relationships of Primary Geometry become organised within a two-dimensional Perspective Image or Representation.

Primary Geometry belongs to Object Space

The most important spatial distinction is:

Primary Geometry → Object or Target Space

Object Space contains the original three-dimensional object, scene or spatial information towards which the Perspective Process is directed.

Primary Geometry describes relationships within this spatial domain before or during their projection into an image.

These may include:

actual object dimensions;
object positions;
orientation;
parallel and perpendicular relationships;
spatial distances;
angles;
planes;
structural frameworks; and
the paths of projection lines or rays through three-dimensional space.

Secondary Geometry belongs to Image Space

The corresponding distinction is:

Secondary Geometry → Image or Perspective Space

Secondary Geometry describes the spatial relationships as represented upon or within the Perspective Image.

A three-dimensional arrangement may therefore become a two-dimensional pattern of:

points;
straight or curved lines;
shapes;
projected surfaces;
vanishing points;
vanishing lines;
foreshortened dimensions; and
other image-space relationships.

The resulting geometry can correspond systematically with Primary Geometry without being geometrically identical to it.

Structural Lines

Structural Lines help describe the Forms and spatial organisation of objects within Primary Geometry.

They may correspond to:

object edges;
boundaries;
axes;
intersections of surfaces;
frameworks;
contours; and
other spatial structures.

Such lines provide simplified geometrical descriptions of objects and scenes.

When represented in a Perspective Image, these Object-Space structural relationships become part of the Secondary Geometry of the image.

Projection Lines

Projection Lines or rays establish the geometrical relationship between Object Space and the Image or Projection Plane.

They may be:

parallel;
converging; or
diverging,

depending upon the Perspective or Projection System involved.

In Primary Geometry, these projection lines extend through three-dimensional space between object points and the relevant projection or imaging system.

Where they intersect the Image or Picture Plane, they help determine the resulting Secondary Geometry.

Primary Geometry and Central Projection

In a central or perspectival projection, projection lines associated with points of the spatial object or scene pass through three-dimensional space in relation to a finite Centre of Projection or Viewpoint.

The Picture Plane intersects this geometrical arrangement.

The resulting intersections determine corresponding positions in the Perspective Image.

In simplified form:

Object Point → Projection Line → Picture Plane → Image Point

The complete three-dimensional arrangement belongs to Primary Geometry, whereas the pattern of resulting Image Points belongs to Secondary Geometry.

Primary Geometry and Parallel Projection

Primary Geometry can also employ parallel projection lines.

In a Parallel Perspective or Projection System, the projection lines remain parallel rather than converging towards a finite Centre of Projection.

The distinction between parallel and central projection therefore begins within the Primary Geometry of the projection system.

The different arrangements then produce different forms of Secondary Geometry in the resulting image.

From Primary to Secondary Geometry

A Perspective Process establishes a relationship between Primary and Secondary Geometry.

In simplified form:

3-D Object-Space Geometry → Projection / Perspective Transformation → 2-D Image-Space Geometry

Some relationships may be preserved while others are transformed.

For example:

three-dimensional depth may become represented on a flat surface;
parallel Object-Space lines may become convergent Image-Space lines;
equal spatial lengths may project at different image sizes;
rectangular planes may become foreshortened shapes;
circles may appear elliptical; and
hidden or overlapping Forms may no longer be fully represented.

Secondary Geometry is therefore a geometrical transformation or representation of Primary Geometry rather than simply a duplicate of it.

The Perspective Window

The classical Perspective Window demonstrates the relationship particularly clearly.

An observer views a three-dimensional object or scene from a fixed Viewpoint through an imaginary or actual transparent Picture Plane.

Projection or visual lines extend between the object and the eye and intersect the window.

The complete spatial arrangement of object, rays, Viewpoint and Picture Plane belongs to Primary Geometry.

The pattern traced where those relationships meet the Picture Plane becomes Secondary Geometry.

Linear Perspective as Secondary Geometry

One of the most important developments of Renaissance Linear Perspective was the creation of a practical Secondary Geometry that allowed artists to construct convincing Perspective Images directly on a two-dimensional surface.

Instead of physically tracing every projection ray through three-dimensional space, geometrical rules could be applied to the drawing itself.

These could determine such features as:

vanishing points;
horizon lines;
receding lines;
depth divisions;
measuring relationships; and
projected object positions.

The artist could therefore construct the Secondary Geometry of an image without having to reproduce physically the complete Primary Geometry of the original projection arrangement.

Secondary Geometry can exist without known Primary Geometry

Primary and Secondary Geometry are connected because one may be derived from the other.

However, Volume 1 emphasises an important qualification:

a Perspective Image may be constructed using Secondary Geometry even when the precise original Primary Geometry is unknown.

An artist can construct an imaginary spatial scene through Linear Perspective rules without first possessing a corresponding physical three-dimensional object or scene.

The resulting image may appear to represent a coherent spatial reality even though its Primary Geometry exists only implicitly, approximately or as an imagined spatial construction.

Constructing Imagined Space

Secondary Geometry is therefore particularly important for constructing Perspective Images of things that do not yet exist physically.

These may include:

imaginary architectural spaces;
proposed buildings;
fictional environments;
invented objects;
stage designs;
graphical models; and
other constructed spatial scenes.

The rules of Secondary Geometry allow a coherent Perspective Image to be produced directly within Image Space.

The implied Primary Geometry may then be inferred from, or represented by, the completed image.

Natural and Optical Perspective

Natural and Optical Perspective commonly involve the Primary Geometry of spatial reality.

Examples discussed in Volume 1 include:

natural vision;
the human eye;
photography;
microscopes;
telescopes; and
other optical Perspective Instruments.

These systems form images through optical relationships involving actual or modelled spatial objects, rays, imaging systems and Image Spaces.

The resulting images nevertheless possess Secondary Geometry once those spatial relationships have been mapped into Image Space.

Graphical and Mathematical Perspective

Graphical and Mathematical Perspective can operate strongly through Secondary Geometry.

Linear Perspective provides the clearest example.

Rules applied directly to the Picture Plane can construct a geometrically organised image without reproducing the complete three-dimensional projection apparatus physically.

This makes Secondary Geometry an extremely powerful analytical and constructive tool.

It converts complex spatial projection relationships into practical operations that can be performed within a two-dimensional drawing or image.

Computer-Generated Imagery

Computer-Generated Imagery can combine Primary and Secondary Geometry.

A digital system may first construct an explicit three-dimensional model containing:

points;
edges;
surfaces;
objects;
coordinates;
camera positions; and
projection relationships.

This constitutes modelled Primary Geometry.

The system can then transform that three-dimensional model into a two-dimensional rendered image possessing Secondary Geometry.

Thus:

3-D Model Geometry → Rendering / Projection → 2-D Image Geometry

AI-Generated Perspective Images

Volume 1 notes an interesting contemporary case involving AI-generated images.

An AI system does not necessarily construct an explicit three-dimensional Primary Geometry before producing a perspective image.

Its internal method for generating image geometry may not correspond directly to a conventional geometrical model of the represented scene.

Nevertheless, the resulting two-dimensional image possesses Secondary Geometry because its visible points, lines, shapes and spatial relationships exist within Image Space.

This illustrates especially clearly that Secondary Geometry can exist without an explicitly constructed Primary Geometry being available to the observer.

Deriving Secondary Geometry from Primary Geometry

Where the Primary Geometry is known, Secondary Geometry can often be derived systematically through a defined Perspective or Projection Process.

Relevant information may include:

object dimensions;
object position;
Viewpoint;
Viewing Direction;
Picture Plane position;
projection type;
projection lines; and
scale relationships.

These can be used to calculate where Object-Space points and lines will appear in Image Space.

This is the forward geometrical problem of perspective:

Primary Geometry → Secondary Geometry

Recovering Primary Geometry from Secondary Geometry

The reverse problem is more difficult:

Secondary Geometry → Primary Geometry

A Perspective Image contains only the spatial information preserved by the particular Perspective Process.

Some three-dimensional properties may therefore be:

lost;
concealed;
ambiguous;
foreshortened;
distorted; or
represented only indirectly.

Recovering the original Primary Geometry may therefore require additional assumptions, measurements, known geometrical structures or multiple views.

This connects the distinction directly with the wider problems of spatial correspondence, equivalence and reconstruction.

Primary and Secondary Geometry and Perspective Transformation

The distinction is closely related to Perspective Transformation.

A Perspective Transformation maps relationships from Primary Geometry into Secondary Geometry.

Thus:

Primary Geometry → Perspective Transformation → Secondary Geometry

The transformation determines which geometrical properties are:

preserved;
altered;
reduced;
compressed;
projected;
foreshortened; or
lost.

Different Perspective Methods can therefore produce different Secondary Geometries from the same Primary Geometry.

Parallel and Perspective Projection

The distinction between Primary and Secondary Geometry helps explain why Parallel Perspective and central or convergent Perspective can produce superficially similar representations while operating through different geometrical relationships.

In many Parallel Perspective systems:

parallel directions remain parallel in the image;
distance-dependent diminution is absent; and
relative measurements can often be obtained directly from the representation.

In central Perspective:

projected size varies with distance;
receding parallel directions may converge towards vanishing points; and
distance-dependent perspectival foreshortening alters image measurements.

The Secondary Geometry therefore reveals the characteristic transformation associated with the underlying projection system.

Axonometric and Central Perspective

The distinction is particularly useful when comparing Axonometric Perspective with central or Linear Perspective.

Both may produce convincing representations of three-dimensional objects on two-dimensional surfaces.

However, their Secondary Geometries are formed according to different projection principles.

Axonometric systems preserve parallelism and permit useful proportional measurement along defined axes, whereas central Perspective introduces distance-dependent diminution and convergence.

Understanding the Primary Geometry and its transformation therefore prevents visually similar images from being incorrectly treated as geometrically equivalent.

Primary Geometry is not Primary Projection

Primary Geometry should not be confused with the separate term Primary Projection.

Primary Geometry refers broadly to the three-dimensional geometry of the target and projection relationships in Object Space.

Primary Projection is a particular technical drawing concept associated with principal orthographic views such as:

plans;
front elevations;
side elevations; and
sections.

The similar use of the word primary does not make the two concepts synonymous.

Primary and Secondary Geometry are analytical concepts

The terms Primary and Secondary Geometry identify two analytical levels of the perspective relationship.

They should not be understood simply as indicating that one kind of geometry is more important than the other.

Primary Geometry is essential for understanding the original three-dimensional spatial arrangement.

Secondary Geometry is essential for understanding, constructing and analysing its two-dimensional representation.

Perspective depends upon the relationship between the two.

Why Primary and Secondary Geometry matter

The distinction between Primary and Secondary Geometry helps clarify one of the most fundamental problems in perspective: the relationship between spatial reality and its representation.

It helps distinguish:

Object Space from Image Space;
three-dimensional spatial geometry from two-dimensional image geometry;
structural lines from their projected representations;
projection processes from the resulting image construction;
actual spatial dimensions from apparent image dimensions;
parallel from central projection;
Primary Geometry from Primary Projection;
the forward construction of an image from the reverse reconstruction of spatial reality; and
an explicitly modelled three-dimensional scene from a two-dimensional image that merely possesses coherent Secondary Geometry.

Primary Geometry is therefore the three-dimensional arrangement of structural and projection relationships in Object or Target Space, while Secondary Geometry is the corresponding arrangement of points, lines and shapes in the resulting two-dimensional Image or Perspective Space.