Transition Pieces

Why this matters in the real world

Transition pieces are everywhere in industry: feed hoppers on grain silos and aggregate plants, sand and cement chutes, and the duct fittings that reduce a large HVAC main down to a smaller branch. Every one of them starts life as a flat sheet-metal development, cut and folded or rolled into its final funnel shape.

Two families of transition

A square-to-square (or rectangle-to-rectangle) hopper, aligned and concentric, is really just a truncated pyramid — its development uses the same "virtual apex" and similar-triangles technique as any pyramid, just cut off before the compass reaches the true apex.

A round-to-square transition is more demanding: its surface is neither a true cone nor a true prism, so it can't be developed directly. Instead it is broken into a mesh of small triangles — alternating "corner" triangles (apex at a square corner) and "side" triangles (apex at a circle point) — a technique called triangulation.

The triangulation workflow

• Draw the front and top views, and divide the round end into 12 equal points.
• Join every square corner to its two nearest circle points, and every circle point to its nearest square edge — this is the triangulation mesh.
• Find the TRUE LENGTH of every distinct sloping edge in the mesh (each is, in general, an oblique line).
• Unfold each triangle edge-to-edge in the development, using its three true lengths and the "swing two arcs" method.
• Add a seam allowance along one edge so the finished piece can actually be folded and joined.