The term Shape Grammars refers to a general requirement/capability to understand and systematically model, apparent transformations of spatial object/scene apparent shapes with viewing position, aspect, and projection scale, plus true scale, etc.
Due to the shape equivalency problem that is inherent to all perspective systems, natural and artificial methods, etc., humans need to find ways to rapidly, adequately, and accurately interpret the shapes seen in perspective images/views, and thus be able to (for example) map such visual information back to ordinal, determinative, and subsumptive data present in objective reality. Said shape recognition capabilities also extend to artificial systems, including robotic vision systems and AI.
Order and Complexity
Geometry is the overlaying of abstract shapes (Forms) onto nature, and often in an approximate way to represent (and simplify) the overwhelming complexity of reality (ref. physical forms). Geometry corresponds to the measurement/representation of the degree of order, and (by absence/omission) its direct corollary, disorder/complexity, present in any spatial reality.
Order is the degree of regular, repeating, redundant, predictable, symmetrical, periodic, monotonous, crystal-like patterning, plus reducible, compressible data present in any region of space. Perspective is a highly ordered visual method that seeks out order/disorder (regular/irregular spatial Forms).
The term Geometric Object Model, refers to a mathematical, geometry-based model of a perspective target spatial object or scene, which approximates the irregularly shaped objects present in physical reality. All artificial perspective method(s), for example, such as a linear perspective construction, are based on the assumption that a geometric model adequately represents or models reality. Kim Veltman has said that: ‘perspective is a halfway station where geometry and physical reality meet—it is a link between the ideal and the actual.’
Shape and Perspective
Geometry refers to an Object’s 2-D, or 3-D visible or physical/geometrical Spatial Form/Figure and/or 1-D / 2-D Contour/Perimeter (ref. External Surface Form). In Physical Space a measured Object’s Absolute Shape or Form may depend on the resolution of the sampling instrument employed. The visible Shape or outer external Form of an Object (ref. Contour/Surface-Form) can be contrasted with the SOLID Physical or Material Form and/or SOLID or ‘filled in with matter’ Geometrical Form of a Thing.
A penny is not more real when seen from above (circle shape) than when seen from the side (ellipse shape), but the frontal view just happens to be the one that gives us the most information, and this aspect, which we call the ’characteristic shape’ of the object, is the one that exhibits the distinctive features by which we classify and name the things of our world.
Shape / Scale / Size Problem (Extrinsic Simplicity)
Measuring size implies measuring also the dimension of shape—wherein perceived object shape is a variable quantity that becomes fixed (or quantified) only at a specific projection scale or optical magnification. Thus shape and measured size are both function(s) of the projection scale and associated projection scale resolution; which is named the scale/ shape/size problem of optical/technical perspective.
Equivalence Problem
The so-called ‘correspondence’ and/or ‘equivalence’ problem(s) of monocular perspective; refer to the fact that (typically) a 2-D image of a 3-D object does not contain sufficient information to unambiguously identify the geometry of said 3-D object (ref. object shape/size/location/ orientation). Several visual analysis techniques are used to tackle this problem; including modelling of perspective phenomena, use and knowledge of shape grammars, perspective frameworks. etc.
Shape Grammars
Consider a natural or environmental optical view captured with a camera, and then presented on a television monitor for a human to view with visual perspective. In this example, the optical image ‘chain’ includes both natural and artificial processes that may alter the perspective Forms present in the final image. And these changes can be significant, including: spatial projection/construction/distortion (in several object cases); mathematical calculation/modelling (due to the design of the camera/TV setup),etc.
We can conclude that the best way to understand the origins of, and the processes that formed, a perspective image/view is to acknowledge that the vast majority of these are both structurally and procedurally composite and can have myriad origins. The only exceptions are certain natural perspective processes, including environmental optics such as shadow casting and astronomical seasons.
Overall, we can conclude that accurate perception, comprehension, and understanding of a perspective image/view is only possible when we can make a series of pre-established decoding procedures, and well-founded guesses, about the nature of the forms under consideration, including:
- Target Spatial Reality (nature of)
- Original Forms – Lines / Solids (regular polygons, etc)
- Original Framework Structures (regular metric grid, etc)
- Visual Frameworks: Sight & Height lines, Visual Element of the system.
- Imager: Assembly, Projection, and Observation Modes
- Image Chain (Categories + Phenomena)
But if complexity were the dominant factor in human vision, then we would not be able to make much sense of spatial images. Rather, when looking at a perspective image, human vision seeks to identify well-known geometrical/optical transformations of common shapes, such as regular flat shapes, regular polygons, straight and regularly curved lines, converging lines and vanishing points, and a horizon line. Finally, metric grids and other framework structures are ways to rapidly and accurately encode/decode an image of spatial reality.
Leonardo da Vinci
Ludo Geometrico, or Geometric Games, is a reference to the numerous, exhaustive, and diverse geometrical and perspectival investigations made by Leonardo da Vinci, and recorded in 10,000 images on more than 6000 pages of his notebooks and his book on painting.
Leonardo’s manuscripts contain more information on geometrical and perspective problems than any other subject, and his investigations of the transformation of the appearance of volumes as a result of optics and vision, plus the recording of related phenomena, in drawings and paintings, reached such a high point of sophistication as to never again be reproduced or performed by anyone again in all subsequent artistic and scientific history.
It is not that Leonardo understood correctly every optical law, or had discovered, or even recognised very single visual or optical principle or perspective phenomenon, but rather that the breadth and depth of his knowledge, considered as a unified body of knowledge, plus his ability to apply such knowledge in his paintings and sculptures, exceed others in the shear totality of vision/knowledge and practical skill. Indeed, there is much that Leonardo’s writings and drawings can teach even the optical, perspective, and vision experts of today.
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