User:Pat Kelso/sandbox

Axonometric projection of a scheme displaying the relevant elements of a vertical picture plane perspective. The standing point (P.S.) is located on the ground plane π, and the point of view (P.V.) is right above it. P.P. is its projection on the picture plane α. L.O. and L.T. are the horizon and the ground lines (linea d'orizzonte and linea di terra). The bold lines s and q lie on π, and intercept α in Ts and Tq respectively. The parallel lines through P.V. (in red) intercept L.O. in the vanishing points Fs and Fq: thus one can draw the projections s' and q', and hence also their intersection R', the projection of R.

Main article: Axonometric projection

Axonometric projection is a type of parallel projection used to create a pictorial drawing of an object, where the object is rotated along one or more of its axes relative to the plane of projection. There are three main types of axonometric projection: isometric, dimetric, and trimetric projection.

The three axonometric views.

Isometric projection

In isometric pictorials (for protocols see isometric projection), the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 60° between them. As the distortion caused by foreshortening is uniform the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing.

Dimetric projection

In dimetric pictorials (for protocols see 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 (vertical) is determined separately. Approximations are common in dimetric drawings.

Trimetric projection

In trimetric pictorials (for protocols see 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. Approximations in Trimetric drawings are common.

Oblique projection

In oblique projections the parallel projection rays are not perpendicular to the viewing plane as with orthographic projection, 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 among the axes as well as the foreshortening factors (scale) 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 thereby creating a true shape, full-size image of the chosen plane. Special types of oblique projections are:

Cavalier projection

In cavalier projection (sometimes cavalier perspective or high view point) a point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x" and y". On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an orthographic projection, as the third axis, here y, is drawn in diagonal, making an arbitrary angle with the x" axis, usually 30 or 45°. The length of the third axis is not scaled.

Cabinet projection

The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry. Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typical 30° or 45° or arctan(2)=63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.

Perspective projection

Perspective of a geometric solid using two vanishing points. This is incorrect. Near face shows as square; should show as a trapezoid per correct example below. Recommend removal

Two point perspective of beveled cube

Computer graphics geometric solid

Main article: Perspective (graphical)

Example of perspective protocol: a projection using three vanishing points

Example #1: undistorted image and object appear identical to a viewer from the Station Point

Example #2: distorted image and object appear identical to a viewer from the Station Point

A perspective projection is a manually drawn, linear projection that graphically produces three dimensional scenes on a Projection Plane. The practice is now largely replaced by computer graphics which may generate scenes from any direction within the virtual space of the computer.

Perspective projection is accomplished by a geometric protocol which exhibits the inevitable distortion of three-dimensional space when “projected," i.e., drawn, on a two-dimensional surface. Projected/projection here refers to manually drawn lines of the protocol which simulate line-of-sight traces from a (simulated) viewer's Station Point (eye location) to the (simulated) viewer's conceptualized scene in (simulated) three dimensional space; accordingly, these line-of-sight traces create images at their (simulated) points-of-intersection with a (simulated) projection plane. It is impossible, however, to perfectly map the eye's imagery of a three dimensional scene on a projection plane because it is impossible to perfectly develop a retina-sphere into a flat surface. In the one special case in which perspective imagery appears the same to the eye as does normal vision, an actual viewer's eye must view the finished, manually drawn perspective imagery from precisely the corresponding, actual spatial location of the station point. See Examples 1 and 2.

The distinctive characteristic of perspective projection is the application of two components: horizon lines and vanishing points which, when taken together, produce the phenomenon of foreshortening. Note that ‘foreshortening' is a result not a cause.

Horizon lines in a perspective projection designate the infinite limit of given planes within a depicted scene. For practical purposes the earth is considered planar (exception: at high altitudes and outer space where the curvature of the earth becomes obvious); taken to absurdum ad nauseam, the earth is considered a fictitious plane tangent to the actual earth at (a ground level view) or above (a bird's eye view)/below (a worm's eye view) the station point. Under routine conditions the earth's horizon line appears at eye level. In perspective projection, as in human vision, there may exist an unlimited number of geometrically defined planes in different positions, with an accompanying unlimited number of horizon lines –- only one of which may actually be horizontal. Note that parallel planes share a common horizon line.

The most significant feature of the defined planes are parallel lines which may be there on, and which, contrary to Euclid, i.e., "parallel lines never intersect," proceed toward intersection at respective points on the respective horizon lines. (The physiological justification for this is beyond the scope of this paper. See:http://www.scribd.com/robert_kelso_2) There may be numerous sets of parallel lines on a given plane, bearing in differing directions, all of which intersect at differing points on the plane's horizon line. Often the plane itself is defined by parallel lines.