# Multiview projection

In technical drawing and computer graphics, a multiview projection is a technique of illustration by which a standardized series of orthographic two-dimensional pictures is constructed to represent the form of a three-dimensional object. Up to six pictures of an object are produced (called primary views), 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 six-sided box around the object, although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as front view, top view and end view. Other names for these views include plan, elevation and section.

The terms orthographic projection and orthogonal projection are sometimes reserved specifically for multiviews. However, orthographic and orthogonal more correctly refer to the right angle formed between the projection rays and the projection plane, versus the angle formed between the subject of the drawing and the projection plane. Thus, orthographic projections include axonometric or auxiliary views in addition to multiviews.

## Overview

Comparison of several types of graphical projection, including elevation and plan views

To render each such picture, a ray of sight (also called a projection line, projection ray or line of sight) towards the object is chosen, which determines on the object various points of interest (for instance, the points that are visible when looking at the object along the ray of sight); those points of interest are mapped by an orthographic projection to points on some geometric plane (called a projection plane or image plane) that is perpendicular to the ray of sight, thereby creating a 2D representation of the 3D object.

Customarily, two rays of sight are chosen for each of the three axes of the object's coordinate system; that is, parallel to each axis, the object may be viewed in one of 2 opposite directions, making for a total of 6 orthographic projections (or "views") of the object:[1]

• Along the x-axis: The left and right views, which are known as elevations (because they often show the features along the "vertical" length of an object such as a building).
• Along the y-axis: The top and bottom views, which are known as plans (because they often show the features within a cross section of the object, such as a floor in a building).
• Along the z-axis: The front and back views, which are also known as elevations, following the same reasoning.

These six planes of projection intersect each other, forming a box around the object, the most uniform construction of which is a cube; traditionally, these six views are presented together by first projecting the 3D object onto the 2D faces of a cube, and then "unfolding" the faces of the cube such that all of them are contained within the same plane (namely, the plane of the medium on which all of the images will be presented together, such as a piece of paper, or a computer monitor, etc.). However, even if the faces of the box are unfolded in one standardized way, there is ambiguity as to which projection is being displayed by a particular face; the cube has two faces that are perpendicular to a ray of sight, and the points of interest may be projected onto either one of them, a choice which has resulted in two predominant standards of projection:

1. First-angle projection: The points of interest are projected in the same direction as the ray of sight; the points are projected onto the plane that lies behind the object, as determined by the ray of sight; the plane of projection acts like the top of a table over which the points of interest are dropped into place.
2. Third-angle projection: The points of interest are projected in the opposite direction of the ray of sight; the points are projected onto the plane that lies in front of the object, as determined by the ray of sight; the plane of projection acts like a glass table from under which the observer sees the points of interest after they have been dropped into place on the top of the table.

## Primary views

Multiview projections show the primary views of an object, each viewed in a direction parallel to one of the main coordinate axes, these primary views are called plans and elevations. Sometimes they are shown as if the object has been cut across or sectioned to expose the interior: these views are called sections.

### Plan

A plan view of Millbank Prison, 1828

A plan is a view of a 3-dimensional object seen from vertically above (or sometimes below), it may be drawn in the position of a horizontal plane passing through, above, or below the object. The outline of a shape in this view is sometimes called its planform, for example with aircraft wings.

The plan view from above a building is called its roof plan. A section seen in a horizontal plane through the walls and showing the floor beneath is called a floor plan.

### Elevation

Principal façade of the Panthéon, Paris, by Jacques-Germain Soufflot.

An elevation is a view of a 3-dimensional object from the position of a vertical plane beside an object; in other words, an elevation is a side-view as viewed from the front, back, left or right (and referred to as a front elevation, [left/ right] side elevation, and a rear elevation).

An elevation is a common method of depicting the external configuration and detailing of a 3-dimensional object in two dimensions. Building façades are shown as elevations in architectural drawings and technical drawings.

Elevations are the most common orthographic projection for conveying the appearance of a building from the exterior. Perspectives are also commonly used for this purpose. A building elevation is typically labeled in relation to the compass direction it faces; the direction from which a person views it. E.g. the North Elevation of a building is the side that most closely faces true north on the compass.[2]

Interior elevations are used to show detailing such as millwork and trim configurations.

In the building industry elevations are a non-perspective view of the structure, these are drawn to scale so that measurements can be taken for any aspect necessary. Drawing sets include front, rear and both side elevations, the elevations specify the composition of the different facades of the building, including ridge heights, the positioning of the final fall of the land, exterior finishes, roof pitches and other architectural details.

#### Developed elevation

A developed elevation is a variant of a regular elevation view in which several adjacent non-parallel sides may be shown together, as if they have been unfolded. For example, the north and west views may be shown side-by-side, sharing an edge, even though this does not represent a proper orthographic projection.

### Section

A section, or cross-section, is a view of a 3-dimensional object from the position of a plane through the object.

A section is a common method of depicting the internal arrangement of a 3-dimensional object in two dimensions. It is often used in technical drawing and is traditionally crosshatched, the style of crosshatching often indicates the type of material the section passes through.

With computed axial tomography, computers construct cross-sections from x-ray data.

## Auxiliary views

An auxiliary view or pictorial, is an orthographic view that is projected into any plane other than one of the six primary views.[3] These views are typically used when an object has a surface in an oblique plane. By projecting into a plane parallel with the oblique surface, the true size and shape of the surface is shown. Auxiliary views tend to make use of axonometric projection.

## Multiviews

Gaspard Monge's four quadrants and two planes.

Modern orthographic projection is derived from Gaspard Monge's descriptive geometry.[citation needed] Monge defined a reference system of two viewing planes, horizontal H ("ground") and vertical V ("backdrop"). These two planes intersect to partition 3D space into 4 quadrants, which he labeled:

• 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. (In cases where such a view is useful, e.g. a ceiling viewed from above, a reflected view is used, which is a mirror image of the true orthographic view.)

Monge's original formulation uses two planes only, and obtains the top and front views only. The addition of a third 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.

### 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, this produces two plans and four elevations. A simpler way to visualize this is to place the object on top of an upside-down bowl. Sliding the object down the right edge of the bowl reveals the right side view.

### Third-angle projection

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

In third-angle projection, the object is conceptually located in quadrant III, i.e. it is positioned 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. A simpler way to visualize this is to place the object in the bottom of a bowl. Sliding the object up the right edge of the bowl reveals the right side view.

Here is the construction of third angle projections of the same object as above. Note that the individual views are the same, just arranged differently.

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 (excluding Japan). First-angle projection was widely used in the UK, but during World War II, British drawings sent to be manufactured in the USA, such as of the Rolls-Royce Merlin, had to be drawn in third-angle projection before they could be produced, e.g., as the Packard V-1650 Merlin. This historical position of the UK means that some British companies completely adopted third angle projection. BS 308 (Part 1) Engineering Drawing Practice, gave the option of using both projections, but generally every illustration (other than the ones explaining the difference between first and third-angle) was done in first-angle, after the withdrawal of BS 308 in 1999, BS 8888 offered the same choice since it referred directly to ISO 5456-2, Technical drawings – Projection methods – Part 2: Orthographic representations.

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 the USA (ASME Y14.3-2003 specifies it as the default projection system), Japan (JIS B 0001:2010 specifies it as the default projection system), Canada, and Australia.

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 and third-angle:

Symbols used to define whether a projection is either First Angle (left) or Third Angle (right)

The 3D interpretation of the symbol can be deduced by envisioning a solid truncated cone, 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 identifies 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 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 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.

Figures one through nine.
• 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.
Figures ten through seventeen.
• 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.

## Territorial use

First-angle is used in most of the world.[4]

Third-angle projection is most commonly used in America,[5] Japan (in JIS B 0001：2010);[6] and is preferred in Australia, as laid down in AS 1100.101—1992 6.3.3.[7]

In the UK, BS8888 9.7.2.1 allows for three different conventions for arranging views: Labelled Views, Third Angle Projection, and First Angle Projection.