Hopper & Transition Builder

The transition is modelled in three dimensions first. The bottom opening sits in the plane z = 0 centred on the origin; the top sits in z = H centred on the offset. Every panel, every view and every DXF comes from those eight corner points, so nothing on this page can disagree with anything else on it.

Why the panels are exactly flat

This is the property the whole tool rests on. Take the front panel: its bottom edge runs along X at the front of the bottom opening, and its top edge runs along X at the front of the top opening. Both are parallel to the X axis, so they are parallel to each other — and two distinct parallel lines always lie in a single plane. The same holds for all four panels, and nothing in that argument mentions the offset or the relative sizes.

So each panel is a planar trapezoid, and developing it is an isometry: the flat pattern has exactly the same edge lengths, diagonals and area as the real panel. There is no flattening error to quote because there is no flattening error.

Where that stops being true

Rotate the top opening and the two edges of a panel stop being parallel. The four corners then do not lie in one plane, and the piece can only be made by triangulating it into facets with genuine break lines, or by accepting a deviation tolerance — which is exactly the choice CAD packages present for a non-developable loft. The tool refuses rotation rather than approximating it. The same applies to round and square-to-round transitions, which are a different family solved by triangulation.

How each panel is developed

By distance geometry, from the six 3D distances between the panel's four corners alone. The first corner goes to the origin, the second onto the X axis at the true bottom-edge length, and the remaining two are placed by intersecting circles of the correct radii. Building it that way makes the result an isometry by construction and means a vertex cannot end up matched to the wrong corner — a failure mode that leaves the area and the overall shape looking perfectly correct while individual edges are wildly wrong.

Offsets, and why they are not restricted

Slice the transition at any height and the cross-section is a rectangle whose width and depth are straight-line interpolations between the two openings, and whose centre moves linearly. Both stay positive, so every slice is a proper rectangle and slices at different heights never meet. The surface cannot pass through itself, however far the top is offset — so large offsets are accepted, including one that puts the top opening entirely outside the bottom footprint.

Dimensions and thickness

Opening dimensions are developed exactly as entered. Thickness and corner-joint compensation are not applied automatically. A correction is derivable — it works out as half the thickness divided by the cosine of each panel's lean — but it differs per panel, stops being symmetric as soon as there is an offset, feeds back into the very leans it depends on, and above all depends on whether the corners are butt-welded, lapped or flanged. Applying one on your behalf would be guessing at your corner detail, so the tool develops what you enter and reports each panel's lean so you can apply your own.

Corner lines and the angle between panels

Each corner line joins a bottom corner to the matching top corner. Its slope from horizontal follows from the two panels that meet on it: each panel keeps one run along its whole length, because its top and bottom edges are parallel, and the two runs are at right angles in plan. So cot²(corner) = cot²(a) + cot²(b) for panel slopes a and b, with or without an offset. The inside angle between the panels is taken square to the corner line, from the two directions that run into each panel; the tests check it a second way, as 180° minus the angle between the panels’ outward normals.

The same relation is published as cot²C = cot²A + cot²B in IS 9178 (Part 1):1979, Fig. 5, where walls at 46° and 67° give a valley of 43.4° read off the nomograph (43.47° by the formula); that example is one of the tests. The standard defines the valley angle (cl. 2.15) and, for square and rectangular hopper bottoms, applies its slope rule for gravity flow to it (cl. 6.5). Sources: IS 9178 (Part 1):1979, Criteria for Design of Steel Bins for Storage of Bulk Materials; E. Maynard (Jenike & Johanson), “Ten Steps to an Effective Bin Design”, Chemical Engineering Progress, November 2013, p. 30.

Panels and files

Panels are named against the plan view: +X is Right, +Y is Back, Front is the −Y face. Each exports its own DXF containing one closed CUT outline and nothing else — no dimensions, no text, no construction lines. The combined file places all four in a fixed arrangement with a constant 10 mm gap — Front and Right on the bottom row, Back and Left above them, as seen in CAD — translated only, with no labels. It is a convenience layout and makes no claim to be a nest.