Perspective drawing, is equally a graphical and a mathematical process. Wherein perspective is complex as it reconciles 3-D physical space with 2-D visual/represented space. Achieving perfect equality is often difficult because there is no 1:1 correspondence among the dimensions involved.
Patently, information may be lost in this process due to the inherent optical limitations of a single point of view, plus aspect of form changes and scale/size relations, resulting in reduced (or concealed/confused) visual information structures/details. Overcoming aspects of this geometric correspondence—or equivalence—problem is a key ‘goal’ of visual/optical/technical perspective.
In terms of a linear perspective drawing, we have basically two methods: either we use a graphical/mathematical construction method (for example, a legitimate construction), or, more generally, we copy a view directly from nature using the artistic method..
Linear Perspective (Graphical Class)
Type of graphical perspective commonly employed by artists, engineers, and designers for the depiction of a spatial scene/object on a 2-D picture plane with sufficient realism (esp. depth); and normally by employing co-planar or sets of parallel orthogonal lines, or else a metric grid, which often exists on the ground plane (and sometimes with related co-planes) and that (when viewed in one-point perspective) typically provides an accurately placed primary central vanishing point (unified vanishing point).
Linear perspective (graphical type) is a familiar type of perspective which is an application of projective geometry in which the drawing is such as would be made upon a transparent vertical plane (or plane of delineation) interposed in the proper position between the eye and the object, and by projecting straight lines from the position of the eye (point of sight) to the several points of the object, and then recording the results (as object/scene outlines). However, linear perspective does not necessarily reflect all aspects of optical reality with perfect accuracy or realism. In particular, the artist using linear perspective (often) only depicts a small number of primary vanishing points existing on certain planes (ground, co-planar, normal planes, etc); whereas in actuality (physical reality) secondary vanishing points potentially exist in all directions within the depicted space on an infinite number of (angular/tilted) object planes.
Linear perspective embodies the perspective of lines/outlines, referring to the projected outlines of forms in spatial reality, and includes the perspectives of points, angles, lines, planes, solids, etc. Whereby the apparent shape/size/position of object/scene element(s) relate to perspective phenomena such as aspect of view (including object foreshortening [1, 2]), degradation of form/proportion, plus diminution of size and vanishing points, etc. In terms of an expansive spatial scene, whether due to accident or by design, many perspective images represent a particular kind of spatial geometry, containing groups of parallel lines forming standard (or primary) vanishing points.
Legitimate Construction
The costruzione legittima (legitimate graphical construction) is a rigorous 15th-century Italian geometric method for creating accurate linear perspective, popularised by Alberti and Brunelleschi. It uses a central vanishing point for orthogonal lines and a separate, precise method to determine the spacing of horizontal “transversal” lines to map 3-D space. The method for determining transversals works as follows: using a side view and looking at the scene from the side, draw rays from the viewer’s eye to the grid lines on the ground to determine where the transversal lines should appear on the canvas.
It was a practical perspective method explained, used, and depicted by all of the most famous Renaissance artists/scholars including Filarete (named as method 2) in his famous book ‘On Architecture’ and also by Piero della Francesca (named as method 2), Francesco Di Giorgio Martini (named as method 2) and Leonardo Da Vinci (named as method 1) plus by Durer in his Instruction. Others discussed the method during this time, such as Serlio (named as method 1A), and Dante and Vignola (named as method 1A).
Artistic Method
Let us consider how an artist produces a perspective drawing/painting of a three-dimensional scene. Patently, the artist must first select a subject to paint (an object/scene) to be viewed from a fixed location, the so-called station or viewpoint. One might assume that the artist can begin by simply sketching what he/she sees whilst looking at the scene in a somewhat ad hoc manner. But normally, it is necessary to fix the viewpoint, viewing angle, and field of view in some way, because there are a large number of possible viewing angles and contrasting scene aspects or points of view that can be observed, each from a slightly different viewpoint and/or viewing angle, etc.
The problem is that the artist’s viewing location, plus angle-of-view, can (possibly) change whilst the drawing is under construction. To solve this problem, a ‘transparent’ frame is overlaid on the spatial scene, called the perspective window or windowpane (which may be purely imaginary and can even be wholly or partially unconscious). This perspective window is normally rectangular, and planar (or flat) in form, and serves the purpose of fixing the artist’s head position (location), plus angle of vision and field of view (monocular eye position), relative to the spatial scene/object in question.
Ergo, the artist begins by looking at a scene through an imaginary perspective window (or even a real glass window or semi-transparent veil), and the drawing/painting is made as if coincident with this flat and translucent windowpane. Whereby, for a representation created in such a manner, each painted object in the scene is thus depicted as a flat, scaled-down image of the object on the other side of the window. Hence, the technique of graphical perspective works by representing the light that passes from a scene through an imaginary rectangle (represented as the plane of the drawing/painting), to the viewer’s eyes (ostensibly a single eye).
What happens when another person looks at a painting created in this way? Well, it’s as if the viewer were (him or herself) looking through the same perspective window and hence viewing a representation identical to the original scene (at least in some senses, and not including binocular optical/perceptive effects). This happens because each portion of the painted object lies on the same straight line from the viewer’s eye to the equivalent portion of the real object it represents. Henceforth, the viewer ostensibly sees no difference between the painted scene on the perspective window and the real-world view of the spatial scene/object.
For a painting so constructed, the picture plane can be thought of as the glass of the notional windowpane through which the viewer looks into the representation of spatial reality that lies beyond. In practice, the picture plane is the same as the physical surface of the painting produced in this manner. When the resulting picture or representation is viewed from the same spot as the windowpane was created (or from an identical station point located the same distance in front of the picture), then the depicted image is geometrically identical to what was seen through the unpainted window (by the artist).
Technical Drawing
Technical drawing, drafting, or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed. Technical drawing is essential for communicating ideas in industry and engineering and is normally associated today with Computer Aided Design (CAD) systems. Within the drawing framework, spatial objects can be projected according to the rules of recession (linear perspective) or parallel perspective, often both are used to explore the model.
We have parallel perspective drawings or orthographic / primary drawings which are not distorted like perspective drawings, as explained below.
Parallel Perspective
A type of graphical perspective in which the viewer’s position is taken to be at infinity, that is, the depicted scale of the object does not depend on viewer’s location). A parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the projection plane or image plane, where the rays, known as lines of sight or projection lines, are parallel relative to one another. It is a basic tool in descriptive geometry.
Technical Note: The projection is named orthographic if the (depicted/imaginary) rays are perpendicular (orthogonal) to the projection plane, and oblique or skew if they are not. For the orthographic class, we have primary/multi-view when the face of an object remains parallel to the projection plane, and auxiliary/axonometric [isometric, dimetric, trimetric] where the entire object is tilted relative to the plane.
Whereas for an oblique projection we have cabinet, cavalier, and military types, where the individual faces of an object are tilted separately one from another, being either parallel or tilted relative to the projection plane (in optically impossible ways). All of these different categories of parallel perspective are types of graphical or mathematical perspective. It is worth noting that none (ordinarily) falls under natural, visual, or instrument perspectives, etc., because such images are in a sense unreal (they do not exhibit spatial recession; but do show perspective aspect effects such as spatial foreshortening).
Ergo, parallel perspective does not occur in the natural world (at least locally or with a close centre of projection). Nevertheless, the indispensability of parallel perspective within technical and engineering drawing, includes applications such as machine engineering, architecture, construction, etc.
Descriptive Geometry
Descriptive geometry is the branch of geometry that allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are employed for engineering, architecture, design, and art. All geometric aspects of the imaginary object are accounted for in true-size/to-scale and shape, and can be imaged as seen from any position in space. All images are represented on a two-dimensional surface.
Descriptive geometry uses the image-creating technique of imaginary, parallel projections emanating from an imaginary object and intersecting an imaginary plane of projection at right angles. The cumulative points of intersections create the desired image.
Of course, today we have computer-based drawing methods, or CAD systems.
Geometry
Geometry is the science of visual forms; and deals with comprehensive organisation structure for geometrical forms, whence we pay attention to order, complexity, hierarchy and relations as exhibited (typically) by physical forms existing in a physical reality. Knowledge of geometrical form(s) enables systematic prescription of real-world physical form(s)/structure(s)—and this is the source of the accurate modelling powers for all of the quantitative science(s).
It is only by functioning at a certain (abstract) level of reality that geometry (line) and arithmetic (number) can become a vehicle for scientific contemplation, understanding, and finally knowledge.
The rise of descriptive geometry (parallel perspective) in the 18th century, which claimed to offer universal principles for representation, led to the first serious claims that representation and (human) vision must be co-incident. However, in the 19th century, physiologists such as Helmholtz became aware of serious problems with earlier analogies between the eye and camera obscura (see: Leonardo, Kepler, Scheiner). Helmholtz also discovered that curved lines, when seen from nearby, appear straight and hence suggested a distinction between physical space, which was Euclidean (or flat), and visual space, which was non-Euclidean (or curved).
Art historian Sir Ernst Gombrich, for example, referred to visual space as the optical world or the mirror and to geometrical space as the physical world, the experienced world, or the map. The psychologist J. J. Gibson called visual space the visual field and linked geometrical space with the visual world.
In the 19th century, it was assumed that visual space was subjective and geometrical space was objective. In Gibson’s 20th-century approach, both are susceptible to measurement and hence, in a sense objective. Kim Veltman said that: “Perspective is a halfway station where geometry and physical reality meet—it is a link between the ideal and the actual.” Hence, our views of physical reality are limited because data is inevitably lost when capturing the visual details of a spatial scene or object, and when analysing and interpreting it.
Geometrical Forms
- Obey Euclid’s Laws (objectively).
- Represent an abstraction of nature’s physical forms.
- Allow reduction of a wealth of physical forms into simple and calculable, abstract geometrical fragments.
- Can exist as 1-D shapes: points, lines.
- Can exist as 2-D shapes: including regular planar shapes: a circle, triangle, square, rectangle, etc.
- Can exist as 3-D shapes: regular, semi-regular, and irregular solids, etc.
- May have symmetry, axial or radial centrality, etc.
- Structural parts may be independent, dependent, and/or co-dependent.
- Enable use of Cartesian Coordinates (frames-of-reference).
Primary and Secondary Geometry
Primary geometry is the arrangement of lines of projection from a 3-D object/scene to a plane of projection. Refers to the geometry of the target space for a perspective image/view, match, or representation. Secondary geometry is the relationships between the points, lines, and shapes of the drawn projection on a 2-D surface. Refers to the geometry of the image space for a perspective image/ view, match, or representation.
CAD Drawing
Computer-Aided-Design (CAD) perspective refers to a wide-ranging and sophisticated set of New Media methods for producing a variety of perspective images of real and imaginary objects in the process of being designed. Typically, the methods of technical drawing or plan/elevation and parallel perspective views are employed within the digital system, and views can be rapidly and automatically produced on a computer monitor. Alternatively, linear perspective images (e.g., multi-view or multi-vanishing point images) can be produced of the represented object/scene, etc.
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