3D projection
A 3D projection is a design technique used to display a three-dimensional object on a two-dimensional surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane.
3D projections use the primary qualities of an object's basic shape to create a map of points, that are then connected to one another to create a visual element. The result is a graphic that contains conceptual properties to interpret the figure or image as not actually flat, but rather, as a solid object being viewed on a 2D display.
3D objects are largely displayed on two-dimensional mediums. As such, graphical projections are a commonly used design element; notably, in engineering drawing, drafting, and computer graphics. Projections can be calculated through employment of mathematical analysis and formulae, or by using various geometric and optical techniques.
Overview
In order to display a three-dimensional object on a two-dimensional surface, a projection transformation is applied to the 3D object using a projection matrix. This transformation removes information in the third dimension while preserving it in the first two. See Projective Geometry for more details.If the size and shape of the 3D object should not be distorted by its relative position to the 2D surface, a parallel projection may be used.
Examples of parallel projections:If the 3D perspective of an object should be preserved on a 2D surface, the transformation must include scaling and translation based on the object's relative position to the 2D surface. This process is called perspective projection.
Examples of perspective projections:
Parallel projection
In parallel projection, the lines of sight from the object to the projection plane are parallel to each other. Thus, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a perspective projection with an infinite focal length, or "zoom".Images drawn in parallel projection rely upon the technique of axonometry, as described in Pohlke's theorem. In general, the resulting image is oblique ; but in special cases the result is orthographic. Axonometry should not be confused with axonometric projection, as in English literature the latter usually refers only to a specific class of pictorials.
Orthographic projection
The orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is a parallel projection. It is the projection type of choice for working drawings.If the normal of the viewing plane is parallel to one of the primary axes, the mathematical transformation is as follows;
To project the 3D point,, onto the 2D point, using an orthographic projection parallel to the y axis, the following equations can be used:
where the vector s is an arbitrary scale factor, and c is an arbitrary offset. These constants are optional, and can be used to properly align the viewport. Using matrix multiplication, the equations become:
While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as it would be recorded photographically or perceived by a viewer observing it directly. In particular, parallel lengths at all points in an orthographically projected image are of the same scale regardless of whether they are far away or near to the virtual viewer. As a result, lengths are not foreshortened as they would be in a perspective projection.
Multiview projection
With multiview projections, up to six pictures of an object are produced, 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 6-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a 3D object. These views are known as front view, top view, and end view. The terms elevation, plan and section are also used.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 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 (45°)
In cavalier projection 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 axonometric 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 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. Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.Military projection
A variant of oblique projection is called military projection. In this case, the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the xy-plane and a vertical translation an amount z.Axonometric projection
Axonometric projections show an image of an object as viewed from a skew direction in order to reveal all three directions of space in one picture. Axonometric projections may be either orthographic or oblique. Axonometric instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in instrument drawings of pictorials great liberties may then be taken for economy of effort and best effect.Axonometric projection is further subdivided into three categories: isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. A typical characteristic of orthographic pictorials is that one axis of space is usually displayed as vertical.
Isometric projection
In isometric pictorials, the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. The distortion caused by foreshortening is uniform, therefore 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, 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. Approximations are common in dimetric drawings.Trimetric projection
In trimetric pictorials, 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.Limitations of parallel projection
Objects drawn with parallel projection do not appear larger or smaller as they extend closer to or 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 our eyes or photography normally work. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right.In this isometric drawing, the blue sphere is two units higher than the red one. However, this difference in elevation is not apparent if one covers the right half of the picture, as the boxes are then obscured.
This visual ambiguity has been exploited in op art, as well as "impossible object" drawings. M. C. Escher's Waterfall, while not strictly utilizing parallel projection, is a well-known example, 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. An extreme example is depicted in the film Inception, where by a forced perspective trick an immobile stairway changes its connectivity. The video game Fez uses tricks of perspective to determine where a player can and cannot move in a puzzle-like fashion.