Orthographic projection
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Orthographic projection (also orthogonal projection and analemma)
The term orthographic sometimes means a technique in
A lens that provides an orthographic projection is an object-space telecentric lens.
Geometry
A simple orthographic projection onto the plane z = 0 can be defined by the following matrix:
For each point v = (vx, vy, vz), the transformed point Pv would be
Often, it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as
For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector Pv would be
In
The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).
The orthographic transform can be given by the following matrix:
which can be given as a scaling S followed by a translation T of the form
The inversion of the projection matrix P−1, which can be used as the unprojection matrix is defined:
Types
Three sub-types of orthographic projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[2][4] Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown to be vertical.
In isometric projection, the most commonly used form of axonometric projection in engineering drawing,
In dimetric projection, the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction is determined separately.
In trimetric projection, the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Trimetric perspective is seldom used in technical drawings.[4]
Multiview projection
In multiview projection, up to six pictures of an object are produced, called primary views, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: first-angle or third-angle projection. In each, the appearances of views may be thought of as being projected onto planes that form a six-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as
Cartography
An orthographic projection map is a
The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions.[7]
Vitruvius also seems to have devised the term orthographic – from the Greek orthos ("straight") and graphē ("drawing") – for the projection. However, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon of Antwerp promoted its present name in 1613.[7]
The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian).[7]
Notes
References
- ^ Sawyer, F., Of Analemmas, Mean Time and the Analemmatic Sundial
- ^ ISBN 0-8014-7280-6.
- ^ Thormählen, Thorsten (November 26, 2021). "Graphics Programming – Cameras: Parallel Projection – Part 6, Chapter 2". Mathematik Uni Marburg. pp. 8 ff. Retrieved 2022-04-22.
- ^ ISBN 1-55860-659-9.
- ISBN 81-8431-558-9.
- ^ Snyder, J. P. (1987). Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395). Washington, D.C.: US Government Printing Office. pp. 145–153.
- ^ ISBN 0-226-76746-9.