Axonometric Projection


Axonometric Drawing

The distinguishing feature between projections and drawings is the unit of measurement used. In projections a scale is constructed and this scale is used to step off measurements. These measurements are not standard units of measurement, e.g. mm, cm, m, etc. In drawings however, standard units of measurement are always used. The scales constructed for isometric, dimetric and trimetric are always smaller than standard units of measurements from which they are derived. This means that axonometric projections are always smaller than axonometric drawings. However, using standard units of measurement rather than scales, a satisfactory axonometric representation of an object can be obtained. An axonometric drawing of an object is slightly distorted but is visually as satisfactory as an axonometric projection of it. No time is wasted constructing scales and then using them in generating an axonometric drawing. Thus, axonometric drawings are generally preferred to axonometric projections. Below is shown an isometric projection and an isometric drawing of a cube. As you can see the drawing is bigger but the same information is available in both.


Axonometric Projection Methods

Axonometric projections can be obtained by a number of methods,

  1. By auxiliary views.
  2. By auxiliary scales.
  3. By two methods of rotation and tilt.
  4. Method of Intersections.

In general the method most widely used is the Method of Intersections. This is a very simple method of producing axonometric projections and was introduced by Rudolf Schuessler in 1905. It was published by Theodor Schmid in 1922 and L. Eckhart in 1937. This method is applicable to isometric, dimetric and trimetric projections. It is particularly useful when orthographic views of an object are available. The question of scale on the various axes is automatically determined. So, if two or more orthographic views of an object are available then this method can be used to obtain an axonometric projection of it.

The simplest way of explaining this method is to work backwards i.e. from the axonometric projection back to the orthographic views. Position the object as shown. Now position the axonometric plane to intersect the 'X','Y' and 'Z' axis at 'A','B' and 'C' respectively. If a view is taken along 'OY' the true shape of the triangle 'AOC' is obtained. This view and the plan and end-view are shown animated in the introduction. It is on this triangle, 'AOC', that the true elevation of the object is projected. Actually it is onto the plane on which the triangle 'AOC' lies that the true shape of the elevation is projected.
The elevation is obtained by projecting lines from the edges of the object onto the triangle 'AOC' parallel to 'OY'. So, if the true shape of 'AOC' can be obtained then the true elevation can be drawn by projecting lines parallel to 'OY'. The true shape of 'AOC' is found using the method for obtaining axonometric scales discussed in "Alternate Isometric Scale" in isometric projection. The triangle 'AOC' is hinged or rotated about line segment 'AC' until it is perpendicular to the line of vision i.e. when the angle 'AOC' equals 90 degrees. If 'AOC' is rotated rebatted about 'AC' in its present position then its true shape will overlap onto other lines in the drawing. To prevent this and maintain a neat legible drawing the line 'AC' is moved a satisfactory distance away from the rest of the drawing so that the true shape of 'AOC' will not interfere with the rest of the drawing. Using the "Angle in a semi-circle theorem", a semi-circle is drawn and the point of intersection of the 'Z' and 'Y' axes is projection onto the semicircle parallel to the line segment 'OY'. The true shape of the triangle 'AOC' is then drawn i.e.'A1O1C1'.

The elevation of the object can now be drawn in this triangle, aligning its edges with the axes 'O1A1' and 'O1C1'. Edges from the axonometric view are projected onto 'A1O1C1' perpendicular to line 'AC'.
If the triangles 'AOB' and 'BOC' were hinged in their present positions to obtain their true shapes 'A3O3B3' and 'B2O2C2' respectively, then they would overlap with other lines on the drawing. For this reason the hinge lines, 'AB' and 'BC', are positioned, 'A3B3' and 'B2C2' so that when the true shape of all three triangles are obtained no over lapping takes place. The true shape of the end-view and plan view are obtained following the same procedure outlined above for the elevation. The illustration on the left shows the axonometric projection and its three orthographic views. The same procedure follows when given the orthographic views of an object to find its axonometric projection except that it is reversed. Lines are projected from the orthographic views perpendicular to the axonometric traces e.g. 'AB', 'AC' and 'BC'. Where corresponding projection lines intersect gives points on the axonometric projection of the object. Indexing corresponding points in the orthographic views greatly simplifies this operation. Notice that only two orthographic views are required to obtain an axonometric projection of an object.
Here is the above example animated, see if you can follow it.

Axes Projections


Axonometric Traces

Given the angles between the projection of the axes, determine the axonometric traces?

This is often the first step in obtaining an axonometric representation of an object. The projection of the axes are shown here. The 'ZX' plane and the 'XY' plane intersect to give the trace known as the 'X' axis. The 'ZX' plane and the 'ZY' plane intersect to give the 'Z' axis. Finally, the 'XY' and 'ZY' planes intersect to give the 'Y' axis. The planes are perpendicular to each other i.e. 90 degrees though they do not appear so in an axonometric projection. The angle between the projection of the axes in an isometric projection is 120 degrees. However, if a view, showing any one of the axes as a true length, is taken, the other axes will always appear perpendicular to it. To appreciate this more fully observe closely the animated plan, elevation and end-view of an object shown in the introduction. If a view is taken looking along the 'X' axis, the 'Z' and 'Y' axes will appear as true lengths. In this view it is important to appreciate that the 'X' axis is perpendicular to the 'Z' and 'Y' axis and perpendicular to the plane on which they lie i.e. the 'ZY' plane.

In orthographic projection the end-view of an object is projected onto the 'ZY' plane parallel to the 'X' axis. The 'X' axis is perpendicular to the plane on which the elevation lies but not perpendicular to the lines in the elevation.
The axonometric trace 'CB' lies on the 'ZY' plane. This line is used to rebate the triangle 'BOC' to get its true shape. Once the true shape of this triangle is established the true end-view is drawn. Lines are drawn from the axonometric view perpendicular to line 'CB' onto this triangle. These lines are drawn perpendicular to 'CB' because it is also on the plane 'ZY' onto which the outline of the end-view is projected.

How does this help position the traces of the axonometric plane?

Well, the axonometric plane intersects the 'ZX', 'ZY' and 'XY' planes. When two planes intersect a line is generated known as a "trace". In an axonometric projection the axonometric plane intersects the 'ZX' plane leaving a trace which runs from a point on the 'Z' axis to a point on the 'X' axis e.g. 'AC'. The 'Y' axis will appear perpendicular to this line. The axonometric plane can be positioned anywhere as long as it is perpendicular to the viewers line of vision. So a line drawn from any point on the 'Z' axis to some point on the 'X' axis which is perpendicular to the 'Y' axis is an axonometric trace. Axonometric trace lines intersect so drawing a line from the point of intersection on the 'X' axis to some point on the 'Y' axis and perpendicular to the 'Z' axis gives the second trace of the axonometric plane in question. The final trace line can now be obtained following the same procedure as before or by simply joining the points where the other trace lines intersect the 'Z' and 'Y' axes. In isometric projection the traces of the axonometric projection form an equilateral triangle. In dimetric they form an isosceles triangle and in trimetric they form a scalene triangle. This information is very useful when determining what type of axonometric projection has been employed to generate a drawing.