Isometric projection is one of the three forms of axonometric projection. In isometric projection the angles between the projection of the axes are equal i.e. 120º. It is important to appreciate that it is the angles between the projection of the axes that are being discussed and not the true angles between the axes themselves which is always 90º.

To explain the
"Projection of the axes" lets take a view of a cube so
that its three principal faces are visible. Lets place a
transparent sheet of perspex in front of the cube and draw lines
where the front edges of the cube meet at a point. The angle
between adjacent edges of a cube is always 90º. After drawing
the outline of the converging edges on the perspex we can measure
the angles between them. We can see that the angle between
adjacent edges is greater than 90º in all three cases i.e
µ>90º,ß>90º and Ø>90º. These are the angles
between the projection of the axes. Theses axes are known as the
axonometric axes. If the angle between all three axes are the
same then an isometric view results ( µ=ß=Ø); if two of the
angles are the same then a dimetric view results
(e.g.µ<>ß, ß=Ø); finally if all three angles are
different a trimetric view results (i.e. µ<>ß<>Ø).
In third angle the planes of projection are in front of the
object so the projection of the axonometric axes will be along
the front corner of the object. The cube in the animation is in
third angle as the axonometric axes intersect at its front
corner. These axes would be used to solve questions in third
angle.

However, in first angle
projection the planes of projection are behind the object and so
the axonometric axes will intersect at the furthest back corner.
First angle projection is generally preferred to third angle
projection in second level schools. Where the three edges of the
cube meet at the furthest corner from the observer are the
axonometric axes used in first angle. The axonometric axes
(isometric axes in this case) for first angle projection are
shown here using a hollow cube. Compare the axonometric axes of
this cube with those of the cube above.

In fact the isometric axes can be placed in any desired position
so that the object will be in the position that best describes
it. However, the angle between the projection of axes must always
be 120º. If the object is considerably long then it is customary
to place the long axis horizontally for best effect. Here are
some typical positions of the isometric axes.

The concept of
foreshortening is a very important one in axonometric projection.
Let us take two orthographic views (first angle i.e elevation
above, plan below) of a pencil which is parallel to the ground,
and take a view perpendicular to its length. In its starting
position the pencil is a true lenght in both plane and elevation.
Holding the pointed end of the pencil steady move the opposite
end away from you, ensuring the pencil remains parallel to the
ground. Keep moving the pencil away from you until you are
looking along the point of the pencil. A view along the point of
the pencil (a point view) is a view parallel to the direction in
which the pencil is pointing. The length of the pencil changed as
you moved it from full length to a point view. Every time you
moved the pencil away from you its length appeared to get shorter
or it **foreshortened**. As the pencil rotates
parallel to the Horizontal plane its elevation becomes
foreshortened. The further you move the end of the pencil away
from you the greater the foreshortening. This is known as the **degree
of foreshortening**. For example, if you moved the pencil
from its starting position (full length) 5º away from you (the
5º is relative to the vertical plane) then the degree of
foreshortening is very small as the pencil appears only a little
smaller than full size. However, if you moved it 85º from its
initial position then it would appear very small (very near a
point view). In this case there is a large degree of
foreshortening.

**Note: **The pencil is initially
parallel to both the Vertival and Horizontal planes. It is then
rotated 5º . The pencil is now no longer parallel to the
Vertical plane. The full 5º rotation can be seen in the plan
view. When fully rotated the pencil is perpendicular to the
Vertical plane and still parallel to the Horizontal plane.

Let us take the same cube as above to
further illustrate this point. Starting with an elevation view of
the cube we see a square. The LOS, in this view, is perpendicular
to the front edges e.g. 'AB' and 'BC' and also to the back edges
e.g. 'EF' and 'HG' ('G' is the lower corner right behind 'C' in
the elevation). Because the LOS is perpendicular to these edges,
they are true lengths. It is also important to note that in the
elevation edges 'AE', 'BF', 'CG' and 'DH' all appear as point
views. This is because we are looking parallel to these lines.

Let us now rotate the cube about 'AD' so that the elevation
and end view are visible together. We are no longer looking
perpendicular to the front face of the cube. What has happened to
the edges 'AB' and 'AE'? The line 'AB' has been rotated just like
the pencil above so it appears shorter in this view, i.e. 'AB'
has foreshortened. However, the line 'AE' has gone from being a
point view to a line view. This line still does not appear as a
true length so it is still foreshortened. How about line 'AD'? We
are still looking perpendicular to this line so it appears as a
true length.

Finally let us rotate the cube vertically up about the line
'DG' while holding point 'D' on the ground. Now, what has
happened to line 'AD'? Well, we are no longer looking
perpendicular to it so it now also has foreshortened. If you
watch the animation you will see the vertical height of line 'AD'
decrease. In fact, all the edges of the cube now appear
foreshortened. If the angles between the projection of the axes
are equal, i.e. 120º, then all the edges foreshorten equally.
This view is known as an isometric view. Also, when the angles
between the projection of axes are equal the axonometric axes are
known as isometric axes. When the cube has been fully rotated a
point view of the body or long diagonal 'DG' results which only
occurs in an isometric view of a cube.

**Isometric
Scale**

We have seen how edges appear shorten (foreshorten) when a view is taken which is not perpendicular to them. So how do we obtain the length of a foreshortened edge in order to draw it on paper? A foreshortened line is a smaller or scaled down version of its true length. Hence, we need to generate a scale to establish the length of the foreshortened edges of an object so that it can then be drawn on paper. In isometric the three angles between the projection of the axes are equal, so the degree of foreshortening along each of the axes is the same. Isometric means "equal measure". This means that only one set of scales is needed to draw an isometric projection of an object. These scales can then be used to draw the edges of a object which are parallel to the axes. How are these scales constructed ?

Let us take an isometric view of a cube. In order
to see the true lengths of the edges that make up the top of the
cube we need to rotate it until we are looking perpendicular to
it. Lets rotate it about the line 'AC'. In its starting position
line 'AB' made an angle of 30º with line 'AC'. However, in its
final position, which shows the true lengths of the lines 'AB'
and 'BC', line 'AB_{1}' makes an angle of 45º with line
'AC'.

A scale is now constructed by stepping
off true measurements along line 'AB_{1}' which is a true
length line. The measurements are then transfered back to line
'AB' to get a smaller scale, in this case an isometric scale
(which is the same procedure used in the division of lines).
Lines drawn using the isometric scale are approximately 80% of
true size. This scale is usually marked off on a piece of paper
and used to step off the foreshortened measurements along the
projection of axes lines and lines parallel to them. Lines
parallel to the projection of axes are known as isometric lines.
Lines which are not parallel to theses axes are known as
non-isometric lines. It is important to note that you can only
use the scales on isometric lines.

**Alternative
Isometric Scale**

**(a)**

The parts of the isometric axes of interest are from where they intersect each other ('O') to where each axis line intersects the axonometric plane ('OX', 'OY' and 'OZ').

**(b)**

The true length of only one of these sections is required as the same scale can be used for all three axes. Let us take the section on the 'Y' axis ('OY'). The true angle between the 'Z' axis and the 'Y' axis is 90 degrees. So, if we hinge the plane that the 'Z' and 'Y' axes lie on, about the line of intersection between this plane and the axonometric plane, the true length of the section of the 'Y' axis is obtained (It is a true length as this section of the 'Y' axis is perpendicular to the line of vision). Rotating (rebatting) part of the side plane 'ZOY' about the line of intersection between the axonometric plane and the 'ZY' plane untilit is perpendicular to the LOS then the true shape of 'ZOY' is obtained.

**(c)**

Mark standard units of
measurement along the line '**O****Y**'
(e.g. 1cm, 2cm, 3cm etc.).

**(d)**

The standard units of measurement are now used to divide the section of the 'Y' axis, i.e.'OY', before it was rotated.

**(e)**

The section of the 'Y' axis line, i.e . 'OY', with the measurements now on it, is the required scale. This scale can now be used to step off measurements parallel to any of the three axes to generate an axonometric view of an object.