Perspective is closely tied to the mathematical concept of proportion. Whereby correct scaling, proportions, and arrangement of (possibly) linked images are set up to form a coherent, unified, often interactive, copy of a spatial reality.
In his Elements, Euclid explored the basic properties of ratios and proportions of lines and surfaces, as well as their equivalents and transformations. In the 13th century, interest in these problems was revived by Leonard of Pisa (Fibonacci), who introduced them into the curriculum of the Abaco school.
In 1430, Leon Battista Alberti applied these geometrical principles to perspective in his Elements of Painting. Piero della Francesca (c.1480) developed this approach, devoting the first two books of On the Perspective of Painting to these geometrical demonstrations based on proportional diminution. Ergo, the apparent size of objects diminishes in an exact proportion to their distance from the observer, so that all quantities are measurable (ref. single unified spatial scale).
Let us now briefly explore the concepts of perspective and proportion.
Perspective of Proportion
Linear perspective is the perspective of lines / outlines. Ergo, the graphical form of Linear perspective is both a mathematical and a graphical perspective simultaneously.
In a way, a graphical perspective method such as linear perspective is a quantitative construction of space—because we employ pre-determined knowledge of spatial forms—or the use of known, often designed, and regular scene geometry. That is, we employ a symmetrically proportioned metric grid to enable the construction of a realistic perspective image using the Legitimate Construction (for example).
Image Proportion Types
In terms of perspective, we have several different kinds of proportion as follows:
- Perspective of Proportion (Type 1: Apparent changes-to/loss-of proportions due to contrast blurring): Changes to, or loss of, proportions due to diminution of outline (Type 1: loss of outline due to optical attenuation).
- Perspective of Proportion (Type 2: Changes to, or loss of, proportions due to reduced image magnification/scale and attendant reduced projection scale resolution): Changes to, or loss of, proportions due to diminution of outline (Type 2: loss of outline detail due to reduced image magnification/scale and attendant reduced projection scale resolution).
- Perspective of Proportion (Type 3: Apparent changes-to/loss-of proportions due to perspective of disappearance [Type 1, 2: disappearing objects / forms / geometric-elements [entirely]): Changes to, or loss of, proportions due to perspective of disappearance (Type 1,2: disappearing objects / forms / geometric-elements).
Perspective Phenomena
An individual perspective category/form demonstrates, and is recognised by, certain perspective phenomena. Perspective phenomena refer to the apparent, generalised changes in the visual features of objects and scenes that occur according to a particular type of perspective and its inherent processes.
Hence, certain forms of graphical perspective can facilitate processes such as viewing, matching, and representation of, or illusion/immersion into, 3-D space on a flat or 2-D picture surface and by employing flat images that evoke specific perspective phenomena, or depth/ shape/size/position cues, including
- Aspect of View: projection-angle / viewing-angle image warping.
- Diminution of Size: remote objects are smaller.
- Diminution of Form: remote objects are less distinct.
- Degradation of Form: size/shape changes with aspect/distance.
- Diminution of Colour/Contrast: Aerial Perspective.
- Optical lighting, reflection and shadow effects: gradient of Chiaroscuro, etc.
- ‘Natural looking’ vanishing points, horizon line [See: horizon line (types)].
- ++ other depth / size /shape cues, etc.
It is important to note that not all forms of graphical perspective employ/exhibit every single type of perspective phenomenona. Parallel perspective (for example) uses only aspect of form or aspect foreshortening (type 1) but not diminution of size, perspectival foreshortening (type 2 or size/ distance law foreshortening), horizon line, or vanishing points.
Image Distortion
A perspective method/process can be defined in terms of the degree of apparent shape distortion introduced to a spatial object’s shape. That is, the changes made to the apparent proportions of a spatial object/scene.
Distortion refers to image shape/form structural apparent defects that arise due to viewpoint geometry relative to object structures in the environment or object space (i.e. aberrations not due to imaging system characteristics alone). Distortion or incorrect image shape can (often) be classed as a perspective image distortion, in which the projected image shape is incorrectly structured/proportioned, scaled, and/or incorrectly positioned, etc.
Many times this class of aberration, or distortion, in a perspective image cannot be objectively measured, because:
- Shape/form is subjective and/or dependent on perspective type: it may be unclear what the correct or most natural shape for an image/view is, and which types of perspective it relates to. For example, is a visual, linear, curvilinear, or spherical projection more or less real than other types?
- Shape/form is multifaceted: apparent image shape depends on the perspective process or method, and relates to a number of factors, including viewpoint, viewing angle, projection surface shape/position, projection scale, and projection scale resolution.
- Shape/form is multifaceted and relates to ouctomes or multiple interacting perspective phenomena.
To conclude this short section, we can state that all types of perspective relate to concepts of proportion and factors influencing image proportion(s) in their various kinds.