Development of Surfaces

Syllabus limit: problems with through holes are excluded.

What development means

The development of a surface is the flat figure obtained by unrolling it onto a plane, without stretching or tearing.

This is the topic with the most direct industrial use in the course. Every duct, hopper, chute, funnel, elbow and sheet-metal transition is manufactured by cutting the development from flat sheet and then rolling or folding it. Get the development wrong and the metal is scrap.

The one rule that governs everything

Every line in the development is a TRUE LENGTH.

Unrolling neither stretches nor compresses, so any distance measured on the developed figure equals the distance measured on the actual surface.

The consequence, and the source of nearly every error in this topic: you may never take a length from a foreshortened view. If the length you need does not appear true in either view, you must find its true length first — by the rotation method of Module 1.

Which surfaces can be developed

Only single-curved and plane surfaces: prisms, pyramids, cylinders, cones. These are developable.

A sphere cannot be developed. It is double-curved, and no flat sheet can be wrapped onto it without stretching. This is exactly why every flat map of the Earth distorts something — area, angle or distance — and why sheet-metal spheres are made in approximate segments (gores) rather than developed exactly.

The four methods

1. Parallel line method — for prisms and cylinders, where the lateral edges or generators are parallel.

The development is a rectangle: width equal to the perimeter of the base, height equal to the lateral edge.

• Prism: width $= n \times$ base edge.
• Cylinder: width $= \pi d$.

Divide the width into the same number of parts as there are edges or generators, erect perpendiculars, and mark the true height of each.

2. Radial line method — for pyramids and cones, where the edges or generators converge on an apex.

The development is a sector of a circle of radius equal to the slant height (pyramid) or slant length (cone).

For a cone of base radius $r$ and slant length $l$, the sector angle is

Derived by equating arc length to base circumference: $\dfrac{\theta}{360°}\times 2\pi l = 2\pi r$.

For a pyramid, strike an arc of radius equal to the lateral edge and step off the base edges around it as chords.

Note carefully: for a cone the sector radius is the slant length, measured apex to base circle. For a pyramid, the arc radius is the lateral edge (apex to base corner), not the slant height (apex to edge midpoint). Using the slant height for a pyramid is a standard and costly error.

3. Triangulation — for transition pieces, by dividing the surface into triangles.

4. Approximate — for spheres and other undevelopable surfaces.

Developing a cut solid

The procedure that most exam questions actually ask for.

1. Draw the two views of the solid with the cutting plane, as in the previous topic.
2. Draw the development of the whole, uncut solid first.
3. On the development, mark the position of every edge or generator — the divisions along the base perimeter.
4. For each one, transfer the true length from the apex or base up to the cutting plane.
5. Join the marked points with a smooth curve or straight lines as appropriate.

Step 4 is where the marks are won and lost. For a prism or cylinder the lateral edges are vertical and parallel to the VP, so their cut heights appear true in the front view and can be transferred directly.

For a cone or pyramid, the generators are inclined, so most of them appear foreshortened in both views. Their true lengths must be found first by rotating each generator until it is parallel to the VP — swing the cut point round to the extreme generator, then read the height there.

A quick way to remember which generators are safe: the two extreme generators of a cone, the ones forming the outline of the front view, are parallel to the VP and therefore already true length. All the others are not.