Parallel projection

In three-dimensional geometry, 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 to each other. It is a basic tool in descriptive geometry. The projection is called orthographic if the rays are perpendicular (orthogonal) to the image plane, and oblique or skew if they are not.

In three-dimensional geometry, 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 to each other. It is a basic tool in descriptive geometry. The projection is called orthographic if the rays are perpendicular (orthogonal) to the image plane, and oblique or skew if they are not.

Overview

A parallel projection is a particular case of projection in mathematics and graphical projection in technical drawing. Parallel projections can be seen as the limit of a central or perspective projection, in which the rays pass through a fixed point called the center or viewpoint, as this point is moved towards infinity. Put differently, a parallel projection corresponds to a perspective projection with an infinite focal length (the distance between the lens and the focal point in photography) or "zoom". Further, in parallel projections, lines that are parallel in three-dimensional space remain parallel in the two-dimensionally projected image. A perspective projection of an object is often considered more realistic than a parallel projection, since it more closely resembles human vision and photography. However, parallel projections are popular in technical applications, since the parallelism of an object's lines and faces is preserved, and direct measurements can be taken from the image. Among parallel projections, orthographic projections are seen as the most realistic, and are commonly used by engineers. On the other hand, certain types of oblique projections (for instance cavalier projection, military projection) are very simple to implement, and are used to create quick and informal pictorials of objects. The term parallel projection is used in the literature to describe both the procedure itself (a mathematical mapping function) as well as the resulting image produced by the procedure.

Types

Oblique projection

. The presence of one or more 90° principal angles is usually a good indication that the perspective is oblique.]] In an oblique projection, the parallel projection rays are not perpendicular to the viewing plane, but strike the projection plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles separating the coordinate axes as well as the foreshortening factors (scaling) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection, creating a truly-formed, full-size image of the chosen plane. Special types of oblique projections include military, cavalier and cabinet projection. If the image plane is given by equation \Pi:~\vec{n}\cdot\vec{x}-d=0 and the direction of projection by \vec v , then the projection line through the point P:~\vec p is parametrized by g:~\vec x = \vec p+t\vec{v} with t\in\mathbb{R} . The image P' of P is the intersection of line g with plane \Pi ; it is given by the equation P':~\vec{p}' = \vec p + \frac{d-\vec p\cdot\vec{n}}{\vec{n}\cdot\vec{v}}\;\vec{v} \ . In several cases, these formulas can be simplified. (S1) If one can choose the vectors \vec n and \vec v such that \vec{n}\cdot\vec{v}=1 , the formula for the image simplifies to \vec p' = \vec p + (d-\vec p\cdot\vec{n})\;\vec{v} \ . (S2) In an orthographic projection, the vectors \vec n and \vec v are parallel. In this case, one can choose \vec v = \vec n ,\; |\vec n|=1 and one gets \vec p' = \vec p + (d-\vec p\cdot\vec{n})\;\vec{n} \ . (S3) If one can choose the vectors \vec n and \vec v such that \vec{n}\cdot\vec{v}=1 , and if the image plane contains the origin, one has d=0 and the parallel projection is a linear mapping: \vec p' = \vec p - (\vec p\cdot\vec{n})\;\vec{v}=\vec p-(\vec v\otimes \vec n)~\vec p = (I_3-\vec v\otimes \vec n)\;\vec p \ . (Here I_3 is the identity matrix and \otimes the outer product.) From this analytic representation of a parallel projection one can deduce most of the properties stated in the previous sections. File:Optical-grinding engine model.jpg|Optical-grinding engine model (1822), drawn in 30° isometric perspective File:Axonometric example.gif|Example of a dimetric perspective drawing from a US Patent (1874) File:Bank_of_China_Tower_massing_model.svg|Example of a trimetric projection showing the shape of the Bank of China Tower in Hong Kong. File:Sanguo2.PNG|Example of dimetric projection in Chinese art in an illustrated edition of the Romance of the Three Kingdoms, China, c. 15th century CE. File:Along the River During the Qingming Festival (detail of original).jpg|Detail of the original version of Along the River During the Qingming Festival attributed to Zhang Zeduan (1085–1145). Note that the picture switches back and forth between axonometric and perspective projection in different parts of the image, and is thus inconsistent.

See also

Projection (linear algebra)

References

Schaum's Outline: Descriptive Geometry, McGraw-Hill, (June 1, 1962),

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