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.
Properties
Every parallel projection has the following properties:
It is uniquely defined by its projection plane Π and the direction v → {\displaystyle {\vec {v}}} of the (parallel) projection lines. The direction must not be parallel to the projection plane. Any point of the space has a unique image in the projection plane Π, and the points of Π are fixed. Any line not parallel to direction v → {\displaystyle {\vec {v}}} is mapped onto a line; any line parallel to v → {\displaystyle {\vec {v}}} is mapped onto a point. Parallel lines are mapped on parallel lines (or on a pair of points if they are parallel to v → {\displaystyle {\vec {v}}} ). The ratio of the lengths of two line segments on a line or on two parallel lines stays unchanged. As a special case, midpoints are mapped on midpoints. The centroid of a set of points in space is mapped to the centroid of the image of those points The length of a line segment parallel to the projection plane remains unchanged. The length of any line segment is not increased if the projection is orthographic. Any circle that lies in a plane parallel to the projection plane is mapped onto a circle with the same radius. Any other circle is mapped onto an ellipse (or a line segment if direction v → {\displaystyle {\vec {v}}} is parallel to the circle's plane). Angles in general are not preserved. But right angles with one line parallel to the projection plane remain unchanged. Any rectangle is mapped onto a parallelogram (or a line segment if v → {\displaystyle {\vec {v}}} is parallel to the rectangle's plane). Any figure in a plane that is parallel to the image plane is congruent to its image.
Types
Orthographic projection
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.
Oblique projection
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