- Drawing as Transformation: From Primary Geometry to Secondary Geometry: Howard Riley
Abstract
A distinction is made between primary geometry, the arrangement in space of lines of projection from a 3-D object to a plane of projection, and secondary
geometry, the relationships between the points, lines and shapes of the drawn projection on a 2-D surface. Drawing projection systems, such as those
classified under British Standard 1192, are illustrated, and are shown to be defined in terms of primary geometry.
It is argued that John Willats' re-classification of projection systems in terms of secondary geometry enables first-year students of drawing to relate
more easily such systems of geometry to their observational experiences. Student drawings illustrate the argument.
Drawing Conventions
Following the criteria of David Marr's [1] definition of a representation as a "formal system for making explicit certain entities or types of
information, together with a specification of how the system does this", it may be argued that projective geometry is such a means of representation,
because it provides a formal systematic procedure for making explicit information about the three-dimensional attributes of objects and spaces upon
a two-dimensional surface. There are other formal geometric systems which have been devised to represent such information. The various sets of
rules which specify how the procedure may operate are termed drawing conventions. British Standard 1192 [2] categorises these conventions:
Figure 1. B.S. 1192 categories of projection types.
In this classification, all orthographic and oblique projections may be specified as parallel projection systems, since their projectors, those
lines of projection that link salient features of the object to points on the plane of projection, are parallel. Perspective projections may be
classified as convergent since their projectors converge on a point in front of the plane of projection, assumed to be a viewer's eye.
Orthographic projection systems
- Multi-plane orthographic projection
This allows several views of an object to be projected upon several planes, assumed to be at right angles to each other: Projectors are
parallel and are perpendicular to the planes of projection. Each object face is parallel with its plane of projection.
- Axonometric, or single-plane orthographic projection
Projectors are parallel and perpendicular to the plane of projection, and all object faces are inclined to the plane of projection.
Isometric Projection is a unique case of axonometric in which foreshortening on all three axes is the same. Dimetric projection is a
special case of axonometric in which scales along two axes are equal, the third axis being different. Trimetric projection is the general
case of axonometric and occurs when all three axes are randomly orientated and are each of different scales.
Oblique projection systems
Oblique projections all have one face of the object parallel to the plane of projection, and the projectors, although parallel to each other,
are inclined to the plane of projection in various ways.
- Cavalier oblique projection
The front face of the object is parallel with the plane of projection, while the projectors from the front face are perpendicular to the
plane of projection. The projectors from the other two visible faces, although parallel, are inclined to the plane of projection so that
the receding edges are represented at the same true scale as the front face.
- Cabinet oblique projection is similar to Cavalier, except receding edges are drawn to half the scale of the true front face projection.
- Planometric oblique projection is a special case of oblique projection, often inaccurately called 'axonometric', where the plan face of
the object is parallel to the plane of projection (and usually rotated through 45º) and projectors are inclined obliquely to the plane of projection.
Two other forms of oblique projection, not identified in the British Standard have been codified by Fred Dubery and John Willats [3]. They are:
- Horizontal oblique projection. One face of the object remains parallel to the plane of projection and projectors are parallel, but are inclined to
the plane of projection in the horizontal direction only.
- Vertical oblique projection. One face of the object is parallel to the plane of projection, the projectors are parallel but inclined to the
plane of projection in the vertical direction only.
Perspective Projection
This family of projection conventions as defined by BS 1192 differs from orthographic and oblique projections because the projected lines
from the object to the plane of projection are not parallel, but converge to a point, generally regarded as the position of an observer's eye.
The picture is formed by the intersection of all these projectors with the plane of projection, usually termed the picture plane in perspective
projections. Parallel edges on the object appear in the projected picture as orthogonals converging to a point, known as a vanishing point.
- Parallel perspective
The object has its face parallel to and at right angles to the picture plane. Projectors converge to a point.
- Angular (2-point) perspective
Vertical faces of the object are inclined to picture-plane, horizontal faces remain normal to the picture-plane:
- Three-point perspective
All the object's faces are inclined to the picture-plane. There are three vanishing points
Primary geometry and secondary geometry
Peter Jeffrey Booker [4] made the distinction between primary geometry, the arrangement in space of lines of projection from the three-dimensional
object to the plane of projection, and secondary geometry, the relationships between the points, lines and shapes of the drawn projection on a
two-dimensional surface.
The projection types of B.S. 1192 discussed above are defined in terms of primary geometry, but perhaps do not relate easily to students' observational
experiences.
Figure 2. John Willats' Re-classification of B.S. 1192 in terms of secondary geometry.
John Willats [5] has usefully re-classified B.S. 1192 in terms of secondary geometry.
For example, in the original B.S. 1192, axonometric drawings showing three faces of an object have to be classified with orthographic projections
which show only one face, because their primary geometries have parallel, perpendicular projectors in common. Willats suggests it would be beneficial
to re-classify the axonometrics under oblique projections, thus recognising their obvious similarities of secondary geometry, which are the number
of faces shown in the drawings, and, the directions of their orthogonals.
This re-classification of drawings in terms of their secondary geometry provides a way of understanding those drawings which do not depend upon
the drawer's position defined by primary geometry but which, in their secondary geometry, explicate features of the object that are known, but not
necessarily visible to the drawer.
Viewer-Centred and Object-Centred Representations
These terms derive from the investigations of Marr and Nishihara [6] into the representation and recognition of the spatial orientation of objects.
The two categories are implicit in the classification of projection types. Therefore it may be useful to review those again, this time relating primary
and secondary geometries to viewer - and object-centred representations.
According to Marr and Nishihara, vision is the processing of information derived from two-dimensional retinal images (viewer-centred) so as to produce
information that allows us to recognise three-dimensional objects (object-centred descriptions).
The organic visual system receives at the retinae constantly changing arrays of light reflected from surfaces and objects in the world from which we derive
representations of those surfaces and objects that are consistent, as well as unchanging across varying viewpoints and lighting conditions.
Such representations may take the visible form of drawings not readily classifiable under the rules of primary geometry which are based upon specific
assumed viewing positions. Willats' work over a period of time has synthesised aspects of Marr's theory into a unique approach to the understanding of
children's drawings and others whose drawings cannot be defined in terms of primary geometry, but may be understood as examples of the following three
categories:
Divergent perspective
This term describes drawings in which the orthogonals diverge. Although strange to
Western eyes, Willats points out that this system, together with horizontal oblique projection, was the most commonly used in Byzantine art and Russian
icon painting during a period of over a thousand years. Figure 3 illustrates a more recent example, Picasso's Woman and Mirror, 1937.
Figure 3. Woman and Mirror, Pablo Picasso, 1937.
Topological geometry
Drawings which map spatial relations such as connections, separation, and enclosure, rather than resemblance and accurate scale, make use of topological
geometry. Such drawings may be more easily understood in terms of an object-centred secondary geometry.
Australian aborigine art is often constructed using topological geometry. Figure 4 illustrates the artist Uta Uta Tjingala's painting Kaakurnatintja
(not dated) which represents the spatial connections between water-holes and other important locations.
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Figure 4.
Kaakurnatintja
Uta Uta Tjingala |
"Fold-out" drawings and multiple-view drawings
These drawings display information about various aspects of objects and spaces simultaneously. This is not possible in drawings dependent on single-plane
projections based on primary geometry. In Figure 5, Bhawani Das' Aurangzeb and Courtiers, c1710, the ground plane has been folded down in orthographic
projection in order to convey information otherwise not available from a viewer's position perpendicular to the picture-plane. In the same drawing, the
canopy has been rendered in axonometric projection, allowing the viewer a top-view which, whilst inconsistent with the obliquely-projected footstool,
affords extra information about the scene.
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Figure 5.
Aurangzeb and Courtiers
Bhawani Das, c1710 |
To continue with the review of projection types in relation to viewer-centred or
object-centred representations:
Multi-plane orthographic projection
These drawings are independent of any single viewing position, and are useful for describing the true proportions and relationships between faces of
a three-dimensional object. This projection has become the standard for engineers and architects.
Oblique projections
These may be constructed to describe properties of either an object or interior spaces which would not be visible from certain viewer-centred positions.
Figure 6, a Punjabi painting The Gale of Love, c1810, shows interiors of rooms left and right, which would not be possible in a viewer-centred description.
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Figure 6.
The Gale of Love
Punjabi painting, c1810 |
Types of oblique projection are evident in drawings from various cultures and periods. In the West, an early description of oblique projection was
given by Cennino Cennini [7] who advised the artist to
...put in the buildings by this uniform system: that the mouldings which you make at the top of the building should slant downward from the edge next
to the roof; the moulding in the middle of the building, halfway up the face, must be quite level and even; the moulding at the base of the building
underneath must slope upward, in the opposite sense to the upper moulding, which slants downward.
That this advice had already been understood by painters is apparent from Figure 7 painted by Giotto in the Capella degli Scrovegni at Padua between 1304 and 1308.
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Figure 7.
Painting by Giotto, 1304-1308 |
One-point, Artificial Perspective
This is a projection system whose primary geometry is based upon what James J. Gibson [8] termed the natural perspective of an array of light reflected
from surfaces and converging on the eye. It assumes the viewing position is singular, and static. In terms of secondary geometry, all orthogonals converge
on a point known as the vanishing point. Its invention was the culmination of a long-standing desire to produce what Martin Kemp [9] described as "the
imitation of measurable space on a flat surface". As such, it may be understood as a more rational codification of the former, loose method practised by
Giotto and described by Cennini.
Most authorities agree that linear, one-point perspective was invented by Filippo Brunelleschi in Florence. Kemp [10] cites a source which suggests the
date of 1413. It is certain that the system was codified and published in Latin by Leon Battista Alberti in 1435. The Italian version of 1436 had a
prologue addressed to Brunelleschi and explained the primary geometry of light rays reflected from surfaces regarded as the base of a pyramid and converging
to an apex at the painter's fixed eye.
Students' Drawings
Each one of the ways of drawing discussed above makes certain information about three-dimensional objects and spaces explicit, but at the expense of
other information which is obscured.
Therefore the choice of a particular way of drawing will depend upon what specific information about the scene, as well as the viewer's position
relative to the scene, is deemed important enough to be represented in the drawing. Moreover, such decisions will vary according to the intended
purpose of the drawing, for whom it is intended, and according to the socially-conditioned ways that people construe the relationship between themselves
and their environment at different ages and in different periods of history.
It is these relationships between drawing and social context that are explored in the drawing studio.
The studio drawing project afforded students the opportunity to relate the concepts of primary geometry and secondary geometry to those of viewer-
and object-centred representations through their drawing practice. It may be pertinent to note here that few first-year undergraduates came to the
programme with a firm grasp of any geometry , so that for many, this project became an opportunity to explore such basics as orthographic, oblique
and perspective projection systems of secondary geometry.
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Figures 8(left), 9 and 10 (below)
illustrate examples of such exploration, undertaken as part of a pilot study. |
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Drawings from the 'Geometries of Vision' project.
Figure 11 illustrates student inquiry into the assumption implicit in perspective projection, that of the fixed, single point of viewing. Here is
an attempt to break out from such ontological constraints, and to invent a way of representing the information in the light received at both eyes.
Focusing upon the wooden framework with each eye in turn, but paying attention to the primary geometry of the scene, the student shares the experience
of both eyes in the one drawing. The primary geometry of the scene is transformed into a secondary geometry rarely explored.
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Figure 11. |
The drawing illustrated in Figure 12 evolved from the student's study of projective geometry systems in common usage. An awareness that all of those
assumed a flat plane of projection stimulated inquiry into the possibility of projecting onto a non-flat plane. Discussion around the notion of a
'cone of vision' developed into the idea of inventing a system for geometrically projecting what was noticed in the cone of vision onto a cone of
projection. A paper cone was duly constructed and arranged at eye level, apex pointing to eye. With one eye closed, so as to flatten the cone
perceptually, the student proceeded to mark the cone at appropriate distances from the eye, the marks representing the salient scene primitives
(corners and edges). When the paper cone (or pyramid, to be precise) was laid out as a surface development, an original projection system was revealed.
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Figure 12. |
Figures 13 and 14 illustrate two students' efforts to explore the inter-relationships between primary geometry, secondary geometry, and viewer-centred
and object-centred representations. Figure 13 attempts to employ a secondary geometry constructed from the combination of a viewer-centred
representation (that of the figure itself) and several views of the wooden frame which made up the subject-matter. Such multiple views of a single
object have the effect of increasing our information of the object as if we were able to move forwards around it. Such object-centred representations
combined with a viewer-centred representation of the figure produces a drawing in which the viewer's position is ambiguous.
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Figures 13(far left)
and 14(left). |
Figure 14 attempts a further complication. Here the figure itself is represented as mirrored, and the wooden frame appears in front of the figure and
behind the figure simultaneously, as well as forming the geometry of the space within which the figure exists. The effect upon the viewer is that of
a shattered image, dynamic and excited. This exercise stimulated the student to further explore the possibilities of combining viewer- and object-centred
representations in a drawing.
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Figures 15(left)
15a-d(below). |
The series of drawings, Figure 15, a, b, c and d, illustrates a systematic approach to the exploration of a possible transition from a viewer-centred
representation to an object-centred one. Figure 15 was drawn from a (relatively) fixed position and indicates the student's grasp of the transformation
process from primary geometry to a viewer-centred secondary geometry. As the series progresses (15 a, b, c & d) lines and contrast-boundaries between
tones representing the salient edges in the scene become interlocked, producing a complex web of compositional axes. This pictorial device enables
the viewer to see relationships between those edges defining the space which are not available from a fixed viewing position. As more information about
spatial relationships is added, less is revealed of the viewer-centred representation of the figure within the space. Finally, in Figure 15d, the
figure is transformed through geometry into pure organic form.
The research is ongoing. Critical comment is welcome.
References
1. Marr, D. 1982 Vision. A Computational Investigation into the Human Representation and Processing of Visual Information. New York: W. H. Freeman
2. Recommendations for Building Drawing Practice. B. S. 1192 1969 London: British Standards Institution pp. 31-34
3. Dubery, F. & Willats. J. 1983 Perspective and other Drawing Systems. London: The Herbert Press
4. Booker, P. J. 1963 A History of Engineering Drawing. London: Chatto Windus
5. Willats, J. 1997 Art and Representation. New Principles in the Analysis of Pictures. New Jersey: Princeton U.P.
6. Marr, D. & Nishihara, H. K. 1978 Representation and recognition of the spatial organisation of three-dimensional shapes. Proceedings of the Royal Society, London Series B200 pp. 269-294
7. Cennini, C. 1390 The Craftsman's Handbook. Transl. by Thompson, D. V. 1933 New York: Yale U.P. Reprinted 1960, Dover Publications
8. Gibson, J. J. 1979 The Ecological Approach to Visual Perception. Boston, Mass: Houghton Mifflin
9. Kemp, M. 1990 The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven, Conn: Yale U.P.
10. Ibid. p.9
