Isometric Projection and Isometric View

By the end you should be able to: Distinguish isometric projection from isometric view, apply the isometric scale, and draw isometric views of prisms, pyramids, cylinders, cones, spheres and their combinations.

Why a pictorial view

Orthographic views are exact but require training to read. An isometric drawing shows three faces in a single picture, so shape is grasped immediately — at the cost of not showing any face in true shape.

The isometric axes

The object is tilted so that all three mutually perpendicular edges make equal angles with the plane of projection. Hence isometric, "equal measure".

The three axes are drawn 120° apart: one vertical, and two at 30° to the horizontal.

Because all three make the same angle with the plane, all three foreshorten by the same factor — which is the whole convenience of the method.

The isometric scale — and the distinction the exam tests

The true foreshortening factor works out to

\frac{\text{isometric length}}{\text{true length}} = \sqrt{\frac{2}{3}} = 0.8165 \approx \frac{9}{11}

That gives two different drawings, and the difference between them is the most frequently examined point in this module:

Isometric PROJECTION — lengths reduced by 0.8165. This is the geometrically correct projection: it is what the object would actually look like. Requires an isometric scale.

Isometric VIEW (or isometric drawing) — true lengths used directly, no reduction. Not geometrically exact, but proportionally identical and far quicker to draw.

\text{Isometric view is } \frac{1}{0.8165} = 1.22 \text{ times larger than the isometric projection}

Both are correct answers to different questions. Read the wording: "draw the isometric view" means use true lengths; "draw the isometric projection" means apply the 0.8165 scale. Unless stated, isometric view is intended, since it is standard practice.

Constructing the isometric scale. Draw a horizontal line, then lines at 30° and 45° from one end. Mark true lengths along the 45° line, drop perpendiculars to the 30° line, and read isometric lengths there. The ratio produced is $\cos 45°/\cos 30°$ ... more simply, the 45° line carries true lengths and the 30° line carries isometric ones.

What is preserved and what is not

Preserved:

Lines parallel in the object stay parallel in the drawing.
Lengths along the three isometric axes are measured directly.

Not preserved:

Angles. A 90° corner appears as 60° or 120°.
Lengths not parallel to an isometric axis — these are non-isometric lines and may never be measured directly.

The rule that follows, and it governs the whole topic: measure only along isometric axes. A non-isometric line — a slanted edge, a diagonal — is drawn by locating its two endpoints using isometric measurements and then joining them.

Circles: the four-centre method

A circle on any isometric face becomes an ellipse. Drawing it by plotting points is slow; the four-centre method approximates it with four arcs.

For a circle of diameter $d$ on an isometric face:

Draw the isometric square (a rhombus) of side $d$ enclosing it.
Mark the midpoint of each side.
From the two obtuse-angle corners, draw lines to the midpoints of the two non-adjacent sides. These four lines intersect at two points inside the rhombus.
The two obtuse corners and those two interior points are the four centres.
From each obtuse corner, strike the larger arc between the two nearer midpoints. From each interior point, strike the smaller arc between the remaining midpoints.

The four arcs meet at the midpoints, giving a smooth closed curve.

Check: the ellipse must be tangent to the rhombus at the four midpoints, and its major axis lies along the long diagonal of the rhombus. If your ellipse's long axis lies along the short diagonal, the centres have been swapped.

Spheres — the exception worth knowing

A sphere is the one solid whose isometric appearance is not distorted: it projects as a circle from every direction.

But the radius depends on which drawing you are making. In an isometric projection the circle has the true radius of the sphere; in an isometric view, where all other lengths have been enlarged by 1.22, the sphere is drawn with radius $1.22r$ to stay in proportion.

This is a favourite examination question precisely because it inverts the usual rule.

Procedure for a solid

Draw the three isometric axes from a chosen origin.
Construct the isometric box — a rectangular block enclosing the whole solid, with edges along the axes. This is the single most useful habit in the topic.
Locate each vertex of the solid inside the box using measurements along the axes only.
Join the vertices; erase the construction box.
Apply visibility — hidden edges are usually omitted in pictorial views rather than dashed.

For a cylinder: draw the isometric box, construct the ellipse on the base face by the four-centre method, repeat on the top face, and join with two tangent lines. For a cone: base ellipse, apex located on the vertical axis, then two tangent lines from apex to ellipse.