Abstract
A plane projection transforms a given figure in a plane to an image in the same plane. A plane projection has four parameters, and falls into one of 15 classes according to the incidence relations of these parameters. An example of each class is shown. In the Euclidean plane, projection preserves incidences but not lengths: an arithmetic scale projects to a harmonic scale. In the harmonic plane, contained in a square, there are 15 classes of projection corresponding to the Euclidean case; and in the elliptic plane, contained in a circle, also 15. A space projection transforms a given figure in space to an image in a plane. A space projection has four parameters, and falls into one of 15 classes according to the incidence relations of these parameters. A space projection can be constructed geometrically, or computationally by matrix multiplication. Ten classical space projections are defined, with a geometric and computational construction, and example, for each. A program to produce several classes of space projection is briefly described (full listings on Springer website). Some of the incidence classes produce relief projections, transforming a given space figure not to a plane image but to a space image. The lens projection is of special interest. All the Euclidean projections have corresponding projections in harmonic and in elliptic space.
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Notes
- 1.
The study of projective properties was originated by Jean-Victor Poncelet (1788–1867). Served as military engineer under Napoleon, was taken prisoner in Russia 1812–1814 where he thought out the fundamentals of synthetic projective geometry, published after his return to France as Professor at Metz and at Paris. Poncelet’s general term for projection was homologie, which later took on a more specialized meaning in topology.
- 2.
The four distances can be permuted in six different ways to calculate the cross-ratio. We choose this form as standard.
- 3.
Axonometric properly means that all three axes have the same length unit, whereas in general each axis has a different length unit.
- 4.
The fundamental theorem of orthogonal axonometry [5], due to Gauss, considers unit lengths on the three space axes as a tripod, and the image plane as the complex number plane. If the leg ends of the tripod project to complex points a, b, c and the origin projects to the null point, then \(a^2+b^2+c^2=0\).
- 5.
The fundamental theorem of oblique axonometry [5], proposed in 1853 by K.W. Pohlke and proved in 1864 by H.A. Schwartz, says that any three distinct segments meeting at a point in the plane of projection are the oblique image of a space tripod.
- 6.
Systematic use of two orthographic projections to represent solid objects was originated about 1770 by Gaspard Monge (1746–1818), Professor at Mezieres military school 1768–1792, Minister of the Marine 1792–1793, founder and Principal of Ecole Polytechnique 1794–1809, with Bonaparte in Italy, Egypt and Syria 1796–1799, Senator of the Consulate 1800, Comte de Peluse 1808, title annulled at fall of Bonaparte.
- 7.
Invented about 1820 by William Farish (1759–1837), Jacksonian Professor at Cambridge. For his lectures on mechanical engineering, Farish developed a model system of interchangeable parts, which he illustrated in isometric projection.
- 8.
Expounded by Thomas Sopwith (1803–1879), civil engineer and geologist, from 1834 in John Weale’s superb range of technical publications. Sir Thomas Sopwith (1888–1989), aviator and yachtsman, was his grandson.
- 9.
In common use since the seventeenth century for drawing fortifications and street plans. It is easy to construct by first drawing a true plan, then rotating it by a suitable angle, usually \(45^\circ \) or \(30^\circ \), then drawing verticals at the corners of the plan, and adding two elevations to the same scale as the plan. Systematically developed for drawings of historical architecture by Auguste Choisy (1841–1909), Professor of Architecture at l’Ecole Normale de Ponts et Chausees 1877–1901, notably in his Histoire de l’Architecture, 1899, where he writes “Le lecteur a sous ses yeux, a la fois, le plan, l’exterieur de l’edefice, sa coupe et ses dispositions interieures.” Adopted in the 1920s as the projection of choice for architectural presentation by Van Doesberg, Le Corbusier, Gropius, and many others. Often loosely called axonometric projection, but more correctly called oblique parallel.
- 10.
In common use in Chinese and Japanese landscapes and interiors since early times, and in Europe as an easily-constructed alternative to perspective. First draw a true front elevation, then draw obliques at a suitable angle, usually \(45^\circ \) or \(30^\circ \), at the corners of the elevation, and add top and side elevations to the same scale as the front elevation. It is like military projection, but with true front elevation instead of true plan. A cavalier is a platform in a fortification, some ten feet higher than the surrounding works, used as an observation or gunnery post; so a cavalier projection approximates to a view from a cavalier. The supposed connection with the mathematician Bonaventura Cavalieri (1598–1647) of Bologna is nonsense. Sometimes called Chinese perspective.
- 11.
Possibly named from the older sense of cabinet as a room, or from the newer sense as a piece of furniture.
- 12.
Filippo Brunelleschi (1377–1446), architect of the dome of Florence cathedral and other profoundly original buildings, demonstrated (according to Vasari) about 1415 an astoundingly accurate perspective picture of the Baptistery of Florence. The painting had a small hole at the vanishing point; the viewer looked through the hole from the back, either at the actual Baptistery or at a mirror reflecting the painting. However, his precise method is not known and the picture is lost. His friends, such as Donatello, were evidently in possession of the method in the 1420s. A construction for the one-point perspective image of given objects on a squared pavement was first published [1] by Leon Battista Alberti (1404–1472), Florentine architect, painter, poet and scholar: see Sect. 13.5.
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Parkin, A. (2016). Projection Geometry. In: Digital Imaging Primer. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85619-1_8
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