Image of a three-dimensional object inside a box with transparent walls

Orthographic projection is a means of representing a three-dimensional (3D) object in two dimensions (2D). It is a form of parallel projection, where the view direction is orthogonal to the projection plane. It is further divided into multiview orthographic projections and axonometric projections. With multiview orthographic projections, up to six pictures of an object are produced, with the projection plane parallel to one of the coordinate axes of the object.^[1] This is how the term is applied in the rest of this article. Rio de Janeiro birds-eye view. ... A worms-eye view is a view of an object from below, as though the observer were a worm. ... Grand Theft Auto Top-down perspective, also sometimes referred to as birds-eye view or helicopter view, is a view used in computer and video games that shows the player and the area around him or her from above. ... The Mercator projection shows courses of constant bearing as straight lines. ... 2-dimensional renderings (ie. ... Example of a dimetric axonometric drawing from a US Patent (1874). ...

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.

Quadrants in descriptive geometry

Example of orthographic drawing from a US Patent (1913), showing two views of the same object. Third angle projection is used.

Modern orthographic projection is derived from Gaspard Monge's descriptive geometry. Monge defined a reference system of two viewing planes, horizontal H ("ground") and vertical V ("backdrop"). These two planes partition 3D space into 4 quadrants, which he labeled: Image File history File links Download high-resolution version (754x730, 86 KB) Example of orthographic drawing, from US Patent 1,054,412, a “Mechanical Power Apparatus” (February 25, 1913). ... Image File history File links Download high-resolution version (754x730, 86 KB) Example of orthographic drawing, from US Patent 1,054,412, a “Mechanical Power Apparatus” (February 25, 1913). ... Gaspard Monge. ...

• I: above H, in front of V
• II: above H, behind V
• III: below H, behind V
• IV: below H, in front of V

These quadrant labels are the same as used in 2D planar geometry, as seen from infinitely far to the "left", taking H and V to be the X-axis and Y-axis, respectively.

The 3D object of interest is then placed into either quadrant I or III (equivalently, the position of the intersection line between the two planes is shifted), obtaining first- and third-angle projections, respectively. Quadrants II and IV are also mathematically valid, but their use would result in one view "true" and the other view "flipped" by 180° through its vertical centerline, which is too confusing for technical drawings.

Monge's original formulation uses 2 planes only, and obtains the top and front views only. The addition of a 3rd plane to show a side view (either left or right) is a modern extension. The terminology of quadrant is a mild anachronism, as a modern orthographic projection with three views corresponds more precisely to an octant of 3D space. Principal façade of the Panthéon, Paris, by Jacques-Germain Soufflot An elevation is an orthographic projection of a 3-dimensional object from the position of a horizontal plane beside an object. ...

First-angle projection

In first-angle projection, the object is conceptually located in quadrant I, i.e. it floats above and before the viewing planes, the planes are opaque, and each view is pushed through the object onto the plane furthest from it. (Mnemonic: an "actor on a stage".) Extending to the 6-sided box, each view of the object is projected in the direction (sense) of sight of the object, onto the (opaque) interior walls of the box; that is, each view of the object is drawn on the opposite side of the box:

A two-dimensional representation of the object is then created by "unfolding" the box, to view all of the interior walls: Image File history File links Views of object being projected according to first-angle File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

This produces two plan views and four side views. Image File history File links First-angle projection being unfolded File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Views of object shown according to first-angle projection File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Archaeological plan In an archaeological excavation, a plan is a drawn record of features (and artefacts) in the horizontal plane. ... Principal façade of the Panthéon, Paris, by Jacques-Germain Soufflot An elevation is an orthographic projection of a 3-dimensional object from the position of a horizontal plane beside an object. ...

Third-angle projection

In third-angle projection, the object is conceptually located in quadrant III, i.e. it lurks below and behind the viewing planes, the planes are transparent, and each view is pulled onto the plane closest to it. (Mnemonic: a "shark in a tank", esp. that is sunken into the floor.) Using the 6-sided viewing box, each view of the object is projected opposite to the direction (sense) of sight, onto the (transparent) exterior walls of the box; that is, each view of the object is drawn on the same side of the box. The box is then unfolded to view all of its exterior walls.

Additional information

First-angle projection is as if the object were sitting on the paper and, from the "face" (front) view, it is rolled to the right to show the left side or rolled up to show its bottom. It is standard throughout Europe and Asia.

Third-angle is as if the object were a box to be unfolded. If we unfold the box so that the front view is in the center of the two arms, then the top view is above it, the bottom view is below it, the left view is to the left, and the right view is to the right. It is standard in USA and Canada.

Both first-angle and third-angle projections result in the same 6 views; the difference between them is the arrangement of these views around the box.

A great deal of confusion has ensued in drafting rooms and engineering departments when drawings are transferred from one convention to another. On engineering drawings, the projection angle is denoted by an international symbol consisting of a truncated cone, respectively for first-angle (FR) and third-angle (US): It has been suggested that this article or section be merged into Technical drawing. ... This article is about the geometric object, for other uses see Cone. ...

Image File history File links No higher resolution available. ...

The 3D interpretation of the symbol can be deduced by envisioning a solid truncated cone (Mnemonic: a "gift-wrapped megaphone"), standing upright with its large end on the floor and the small end upward. The top view is therefore two concentric circles ("donut"). In particular, the fact that the inner circle is drawn with a solid line instead of dashed disambiguates this view as the top view, not the bottom view.

• In first-angle projection, the "top" view is pushed down to the floor, and the "front" view is pushed back to the rear wall; the intersection line between these two planes is therefore closest to the large end of the cone, hence the first-angle symbol shows the cone with its large end open toward the donut.
• In third-angle projection, the "top" view is pulled up to the ceiling, and the "front" view is pulled forward to the front wall; the intersection line between the two planes is thus closest to the small end of the cone, hence the third-angle symbol shows the cone with its large end away from the donut.

Multiviews without rotation

Orthographic multiview projection is derived from the principles of descriptive geometry and may produce an image of a specified, imaginary object as viewed from any direction of space. Orthographic projection is distinguished by parallel projectors emanating from all points of the imaged object and which intersect a plane of projection at right angles. Above, a technique is described that obtains varying views by projecting images after the object is rotated to a desired position. Descriptive geometry builds on a practice, evolved over centuries, of displaying two images of an object, one as seen in one direction and a second image as seen from a direction 90° rotated (e. ...

Descriptive geometry customarily relies on obtaining various views by imagining an object to be stationary, and changing the direction of projection (viewing) in order to obtain the desired view.

See Figure 1. Using the rotation technique above, note that no orthographic view is available looking perpendicularly at any of the inclined surfaces. Suppose a technician desired such a view to, say, look through a hole to be drilled perpendicularly to the surface. Such a view might be desired for calculating clearances or for dimensioning purposes. To obtain this view without multiple rotations requires the principles of Descriptive Geometry. The steps below describe the use of these principles in third angle projection.

• Fig.1: Pictorial of imaginary object that the technician wishes to image.
• Fig.2: The object is imagined behind a vertical plane of projection. The angled corner of the plane of projection is addressed later.
• Fig.3: Projectors emanate parallel from all points of the object, perpendicular to the plane of projection.
• Fig.4: An image is created thereby.
• Fig.5: A second, horizontal plane of projection is added, perpendicular to the first.
• Fig.6: Projectors emanate parallel from all points of the object perpendicular to the second plane of projection.
• Fig.7: An image is created thereby.
• Fig.8: A third plane of projection is added, perpendicular to the previous two.
• Fig.9: Projectors emanate parallel from all points of the object perpendicular to the third plane of projection.

Image File history File links Ten_through_Seventeen_Step_by_Step_Orthographic_Auxillary_Projection. ...

• Fig.10: An image is created thereby.
• Fig.11: A fourth plane of projection is added parallel to the chosen inclined surface, and per force, perpendicular to the first (Frontal) plane of projection.
• Fig.12: Projectors emanate parallel from all points of the object perpendicularly from the inclined surface, and per force, perpendicular to the fourth (Auxiliary) plane of projection.
• Fig.13: An image is created thereby.
• Fig.14-16: The various planes of projection are unfolded to be planar with the Frontal plane of projection.
• Fig.17: The final appearance of an orthographic multiview projection and which includes an "Auxiliary view" showing the true shape of an inclined surface.

Pictorials

Main article: Axonometric projection

Within orthographic projection there is an ancillary category known as Pictorials. Pictorials show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in one picture. Orthographic pictorial instrument drawings are often used to approximate Graphical Perspective projections, but there is attendant distortion in the approximation. Because Pictorial projections innately have this distortion, in the instrument drawing of Pictorials, great liberties may then be taken for economy of effort and best effect. Orthographic Pictorials rely on the technique of axonometric projection ("to measure along axes"). Example of a dimetric axonometric drawing from a US Patent (1874). ...

See also

• Cross section (geometry)
• Engineering drawing
• First angle projection
• Orthographic projection (cartography)
• Plans (drawings)

A 3-D view of a beverage-can stove with a cross section in yellow. ... It has been suggested that this article or section be merged into Technical drawing. ... Example of orthographic drawing from a US Patent (1913), showing two views of the same object. ... Orthographic projection (equatorial aspect) of the hemisphere 30W–150E Orthographic projection is a map projection of cartography. ... For other uses, see Plan (disambiguation). ...

References

1. ^ Ingrid Carlbom, Joseph Paciorek (Dec. 1978), Planar Geometric Projections and Viewing Transformations, vol. v.10 n.4, ACM Computing Surveys (CSUR), pp. p.465-502, DOI 10.1145/356744.356750

The Association for Computing Machinery, or ACM, was founded in 1947 as the worlds first scientific and educational computing society. ...

External links

• Educational website describing the principles of first and third angle projection — University of Limerick