Modelled Form is the idealised or assumed geometrical Form used to analyse, describe, calculate or construct the relationship between Physical Form and Apparent Form.
It is neither the Physical Form itself nor another visual appearance of that Form. Rather, it is the geometrical description through which spatial reality and its perspective appearance are related within a particular perspective system.
Modelled Form can therefore be understood as an assumed geometrical limit or approximation towards which Apparent Form may be interpreted under specified conditions of scale, geometry, viewpoint and projection.
Modelled Form Is a Geometrical Approximation
Physical reality is often much more complex than the geometry used to analyse it.
A natural or manufactured object may contain irregular surfaces, small-scale structure, curvature, texture and material detail. Perspective analysis commonly reduces this complexity to simpler geometrical Forms that can be described and calculated.
A Physical Form may therefore be modelled as a point, line, plane, rectangle, polygon, cube, sphere, surface or another defined geometrical structure.
The resulting geometrical Form is the Modelled Form.
Physical Form, Modelled Form and Apparent Form
The three Forms describe different aspects of the same perspective relationship:
- Physical Form — the actual structure existing in Object Space;
- Modelled Form — the assumed geometrical approximation used to analyse that structure and its perspective relationship;
- Apparent Form — the resulting perspective appearance.
They belong together, but they should not simply be assumed to coincide. Their degree of correspondence varies with scale, geometry, viewpoint, Projection Mode and other perspective conditions.
Modelled Form as an Assumed Geometrical Limit
Modelled Form may be understood as an assumed geometrical limit through which Apparent Form is interpreted.
For example, a physical surface may be sufficiently flat over a particular range of scale to be modelled as an ideal plane. Physical edges may be sufficiently straight to be modelled as mathematical straight lines. A three-dimensional object may be sufficiently regular to be represented by an ideal cube, sphere or other defined solid.
The Modelled Form therefore represents the geometry that the perspective analysis assumes to be sufficiently close to the underlying Physical Form for the purpose and scale concerned.
It is an approximation of physical structure, not a replacement for physical reality.
Modelled Form Is Not an Arbitrary Choice
Modelled Form should not be understood as a freely interchangeable model that is continually changed merely to obtain a desired result.
Once the geometrical assumptions of a particular perspective analysis are established, the Modelled Form provides the framework through which the relationship between Physical Form and Apparent Form is examined.
Its adequacy can be tested, questioned or refined, but it remains the geometrical assumption defining that particular analysis.
This distinction is important because otherwise perspective modelling can be mistaken for choosing whatever geometry happens to fit an observed image after the event.
Production of Apparent Form
For analytical purposes, the relationship between the three Forms can be expressed in the direction of perspective production or appearance:
Physical Form + Modelled Form → Apparent Form
This does not imply a simple temporal sequence. It expresses a relationship of dependence: the physical spatial structure is analysed through a geometrical model under particular perspective conditions, producing or explaining its Apparent Form.
Interpreting Apparent Form
The same relationship can also be considered in reverse.
Starting from an observed or represented Apparent Form, an observer or analyst may attempt to infer the possible Physical Form and the Modelled Form capable of explaining that appearance.
This inverse relationship can be represented as:
Apparent Form → possible Physical Form + appropriate Modelled Form
The word possible is important. A single Apparent Form does not necessarily identify one unique Physical Form.
One Apparent Form Can Fit More Than One Physical Form
Perspective images are frequently geometrically underdetermined. The same or very similar Apparent Form can sometimes be produced by different arrangements in Object Space.
Modelled Form provides the geometrical framework for analysing these possibilities, but it cannot automatically remove the ambiguity contained in a single view.
Additional viewpoints, known dimensions, contextual information, depth cues or other geometrical constraints may be needed to determine which Physical Form provides the most adequate explanation.
Modelled Form and Object Space
Physical Form belongs directly to Object Space. Modelled Form is the geometrical abstraction used to describe or analyse aspects of that Object-Space structure.
Modelled Form should therefore not be confused with the spatial object itself.
The distinction can be expressed as:
Physical Form → actual Object-Space structure
Modelled Form → assumed geometrical description of that structure
Apparent Form → Perspective-Space appearance
Modelled Form and Scale
The adequacy of a Modelled Form depends strongly upon scale.
A physical surface that can reasonably be represented as flat at one scale may reveal significant curvature or irregularity at another. An edge that appears straight at architectural scale may reveal detailed material structure at a much smaller scale.
The same Physical Form can therefore support different levels of geometrical description depending upon the scale and purpose of the analysis.
This does not mean that Modelled Form is arbitrary. It means that the validity of a geometrical approximation has a range of applicability.
The Scale–Shape–Size Problem
The dependence of Modelled Form upon scale is central to the Scale–Shape–Size Problem.
A geometrical description that sufficiently approximates Physical Form at one range of scales may cease to be adequate when the relevant scale changes.
Perspective systems therefore operate under assumptions about which geometrical features of Physical Form are significant and which can reasonably be neglected.
The Shape-Sufficiency Problem
The Shape-Sufficiency Problem asks whether the assumed Modelled Form remains sufficiently close to Physical Form across the range of scale and conditions for which the perspective system is being used.
For example, Linear Perspective remains a useful geometrical system where the relevant physical surfaces, edges and directional relationships can be sufficiently approximated as planar, straight or parallel.
Where those assumptions cease to provide an adequate approximation, the Modelled Form may no longer be sufficient for the intended analysis.
Modelled Form and Projection Mode
The relationship between Modelled Form and Apparent Form also depends upon the Projection Mode used by the perspective system.
The same Modelled Form can produce different Apparent Forms under central, parallel, spherical, cylindrical, anamorphic or other projection arrangements.
The Modelled Form describes the assumed geometry of the spatial structure; the Projection Mode describes how that geometry is related to the resulting perspective image or view.
Modelled Form and Perspective Geometry
Perspective geometry necessarily works through abstractions.
Points possess no physical size, mathematical lines possess no physical thickness, and ideal planes possess no physical irregularity. Nevertheless, such abstractions provide powerful Modelled Forms through which spatial relationships can be analysed.
The important distinction is that mathematical exactness within the model does not imply that the Physical Form itself possesses that same exact geometry.
A geometrical model can be exact as a model while remaining only an approximation of physical reality.
Modelled Form and Perspective Phenomena
Perspective Phenomena act upon or shape the relationship between the spatial structure being represented and its resulting Apparent Form.
Foreshortening, apparent diminution, vanishing, changes of aspect and other perspective effects can therefore be analysed by comparing the assumed Modelled Form with the resulting Apparent Form.
This allows perspective geometry to distinguish a change in appearance from a change in the underlying spatial structure.
Why Modelled Form Matters
Modelled Form provides the essential geometrical bridge between Physical Form and Apparent Form.
Without this distinction, the geometry used to analyse perspective can easily be mistaken either for physical reality itself or for the visible appearance produced from it.
Modelled Form makes explicit that perspective analysis depends upon assumptions concerning geometry, scale and spatial structure. Those assumptions can be highly accurate and useful without being identical to the complete Physical Form.
The three concepts can therefore be summarised as:
Physical Form — what exists
Modelled Form — how its geometry is represented or assumed for analysis
Apparent Form — how it appears through perspective