Axonometric projection

Introduction
The term axonometric projection includes four types of pictorial drawing.

These are as follows:-
(a) Isometric Projection *
(b) Dimetric Projection **
(c) Trimetric Projection ***
(d)Oblique Axonometric Projection

The first three of the above, namely Isometric, Dimetric and Trimetric Projections are categorised as Orthogonal Axonometric Projections because the projectors from the object are perpendicular to the plane of projection.
Orthogonal Axonometric Projection is in fact classified as Orthographic Projection. The following definition establishes this relationship:-

Definition: "Orthographic projection is a system of views of an object formed by projectors from the object perpendicular to the desired planes of projection".

Elevations and Plan views are the "regular" orthographic projections, whereas Isometric, Dimetric and Trimetric projections are pictorial projections.
Pictorial projections show several faces of an object at once, approximately as they appear to the observer.
Figure 2 below shows the following:-

• An elevation projected on the plane AOC
• An end-elevation projected on the plane BOC
• An axonometric view projected on the plane ABC
Note:- The plan view which would be on the plane AOB is obscured by the object. Distinction between Isometric Projection and Isometric Drawing
It is quite common to find the terms isometric projection and isometric drawing used incorrectly or treated as if they have the same meaning. It is therefore important to clearly establish the differences.
Isometric Projection, as a true pictorial projection results in lines being foreshortened in a mathematical proportion. Distances in the direction of the axes are Ö2/3 or 0.816 times true size.
Isometric Drawing is not a true pictorial projection. If an isometric drawing is constructed from a three-view drawing at the same scale as the three-view drawing, it will be 1.224 times as large as an isometric projection made from the same three views.
Stages involved in constructing an isometric projection* (axonometric method)

 1. Draw projections of co-ordinate axes as shown in Figure 3. (120° between each)                       2. Draw trace triangle ABC (equilateral) as shown in Figure 4.        BC at right angles to X-axis       AC at right angles to Y-axis       AB at right angles to Z-axis                 3. Draw development of planes of reference. (Note application of the angle in a semi-circle).                           4. Draw any two of the orthographic views.   In the case of the elevation, the view should be drawn with the edges respectively parallel to the lines o¹a¹ and o¹c¹. In the case of the end elevation, the view should be drawn with the edges respectively parallel to the lines o¹b¹ and o¹c¹. In the case of the plan, the view should be drawn with the edges respectively parallel to the lines o¹a¹ and o¹b¹.                 5. Having located the orthographic views, draw projecting lines from each view perpendicular to the corresponding edge of the axonometric plane.                         6. The intersections of these projectors will yield points on the axonometric (isometric) projection.    Stages involved in constructing a dimetric projection**
1. Draw projections of co-ordinate axes. (two angles between axes equal and over 90°)
2. Draw trace triangle ABC. (isosceles)
3-6. As before.

Stages involved in constructing a trimetric projection***
1. Draw projections of co-ordinate axes. (all three angles unequal and not less than 90°)
2. Draw trace triangle ABC. (scalene)
3-6. As before.

Definitions:
* "An isometric projection is an axonometric projection of an object so placed that all three of its axes make equal angles with the plane of projection. Hence, the three axes are foreshortened equally". (isometric means "of equal measure")

** "A dimetric projection is an axonometric projection of an object so placed that two of its axes make equal angles with the plane of projection and the third axis makes either a smaller or a greater angle. Hence, the two axes making equal angles with the plane of projection are foreshortened equally, while the third axis is foreshortened in a different ratio".

*** "A trimetric projection is an axonometric projection of an object so placed that no two axes make equal angles with the plane of projection. In other words, each of the three axes and the lines parallel to them, respectively, have different ratios of foreshortening when projected to the plane of projection".

References:-

Engineering Graphics         Graphics for Engineers
Giesecke & Others            O'Bryant & Dobrovolny
Macmillan Publishers         Stipes Publishing Co.
ISBN 0-12-946326-2            ISBN 0-87563-361-7