Параллель проекция: Үш өлшемді нысанды жазыққа түсіру

Projection of a 3D object onto a plane via parallel rays

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.

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.

Orthographic projection is derived from the principles of descriptive geometry, and is a type of parallel projection where the projection rays are perpendicular to the projection plane. It is the projection type of choice for working drawings. The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane (or the paper on which the orthographic or parallel projection is drawn). However, the term primary view is also used. In multiview projections, up to six pictures of an object are produced, with each projection plane perpendicular to one of the coordinate axes. However, when the principal planes or axes of an object are not parallel with the projection plane, but are rather tilted to some degree to reveal multiple sides of the object, they are called auxiliary views or pictorials. Sometimes, the term axonometric projection is reserved solely for these views, and is juxtaposed with the term orthographic projection. But axonometric projection might be more accurately described as being synonymous with parallel projection, and orthographic projection a type of axonometric projection. The primary views include plans, elevations and sections; and the isometric, dimetric and trimetric projections could be considered auxiliary views. A typical (but non obligatory) characteristic of multiview orthographic projections is that one axis of space usually is displayed as vertical. When the viewing direction is perpendicular to the surface of the depicted object, regardless of the object's orientation, it is referred to as a normal projection. Thus, in the case of a cube oriented with a space's coordinate system, the primary views of the cube would be considered normal projections.

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.

Objects drawn with parallel projection do not appear larger or smaller as they lie closer to or farther away from the viewer. While advantageous for architectural drawings, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike perspective projection, this is not how human vision or photography normally works. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right. This visual ambiguity has been exploited in op art, as well as "impossible object" drawings. Though not strictly parallel, M. C. Escher's Waterfall (1961) is a well known image, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the law of conservation of energy. Oscar Reutersvard is credited with discovery of the impossible object, an example of the impossible triangle (top) shown in this mural by Paul Kuniholm.