Eccentric (Offset) Reducer Flat Pattern
Develop an eccentric (offset) reducer — or any oblique cone between two parallel round ends — into one exact flat pattern. Both diameters, the height and the offset in: the flat pattern, a true-length table for laying it out by hand, optional bend lines and a DXF out.
The method behind it: How to lay out an eccentric reducer by hand — step by step, with sources.
Dimensions
Results
One closed outline on CUT; bend lines, if any, on BEND.
Preview
Flat pattern
- Outline — cut
- Bend lines
- Layout lines from the table — not exported
Shown from the outside, opening upward like the cone tool: the large end is the outer edge, the small end the inner one, and the line opposite the seam stands upright in the middle.
The reducer
Lengths for laying it out by hand
Both ends divided into equal parts from the seam. The element joins a point of the large end to the matching point of the small end, and the apex column runs from the apex to the large end: straight lines on the part, so their true lengths are what you lay down. The steps and the diagonal are measured on the flat pattern itself — slightly longer than the straight chords across the round part, which is why stepping the plan chord leaves a pattern about 1 % short. Step these, and every point lands on the exact pattern. How to use the table, step by step.
How this tool is verified
- Geometry checked against hand-derived golden cases and geometric invariants — developed arc equals true circumference, metric and imperial inputs give identical millimetre geometry.
- Every exported DXF is read back and audited by ezdxf, an independent DXF library that had no part in writing it (ezdxf 1.4.4).
- Output is deterministic: the same inputs produce the same file, byte for byte, with a version tag in the header.
The method, the reports and the responsibility boundary are on the Verification page.
Built by Ruthen Systems
Free, with no account, and the dimensions you type never leave your browser. If a result looks wrong, or the option your shop needs is missing, say so — reports of a wrong result are dealt with first.
Report a problem or request a toolAssumptions and limits
Read these before you cut anything. They describe exactly what this tool does and does not account for.
- Both ends are round and lie in parallel planes; the offset is measured in plan, centre to centre.
- Every point of the large end is joined to the point of the small end at the same plan angle — the part you get by dividing both ends into equal parts and triangulating. That surface is an oblique cone.
- Dimensions are developed exactly as entered: enter the surface you want the pattern of (usually the mid-thickness for formed plate). No thickness compensation is applied.
- The surface is developed exactly by numerical integration around the apex, not by the shop approximation of a few flat triangles; the curved edges are sampled polylines held within the maximum deviation you set.
- One seam, along the short side, the long side or halfway round. No seam, lap or flange allowance is added.
- Bend lines, if asked for, are straight generators at equal divisions of the ends: the lines a brake operator bumps along. The facets between them are flat.
- Geometry is nominal. Check the formed part against the real openings before cutting a batch.
Output is a planning and fabrication aid. You are responsible for verifying dimensions, tolerances and material behaviour before production — full disclaimer.
Worked example
Flush reducer, 300 to 150, 200 high: offset (300 − 150) / 2 = 75.
- Short side200.00 mm
- Long side250.00 mm
- Apex above the large end400.00 mm
Hand-checkable. The short side is vertical, so it is the height, 200. The long side runs from (−150, 0, 0) to (0, 0, 200): √(150² + 200²) = 250. The small end is half the large one, so the apex is twice the height up, 400 — and the apex-to-large-end lengths on the short and long sides are 400 and √(300² + 400²) = 500.
Method
An eccentric reducer is a textbook pattern: the classic pattern books treat round, parallel ends placed off-centre as the frustum of a scalene (oblique) cone, and develop it by triangulation from the apex — true-length lines radiating from the apex, laid out triangle by triangle with the division chord between them (Kittredge, The New Metal Worker Pattern Book, 1901, Problem 167, pp. 311–312; the oblique cone itself is Problem 163, pp. 306–308; both public domain).
Without the apex, the same part is triangulated directly between the two ends: both divided into equal parts, joined by elements and diagonals whose true lengths come from right-triangle diagrams, and laid out quadrilateral by quadrilateral (Kidder, Triangulation Applied to Sheet Metal Pattern Cutting, 1917, Ch. VI). That is the practical route when the ends are close in size, because the apex radius gets inconveniently long (Atkins, Practical Sheet and Plate Metal Work, 2nd ed., p. 117). The table on this page gives the lengths for both methods.
Why it is a cone
Divide both ends into the same number of equal parts and join matching points. Because the ends are parallel circles, every one of those lines passes through the same point: the small end is the large end shrunk by D2 / D1 towards it. That point is the apex of an oblique cone, and the reducer is the slice of it between the two end planes. A cone rolls flat without stretching, so there is one exact flat pattern.
Developing it exactly
Around the apex every line to the large end keeps its true length, and the angle between neighbouring lines in the flat is the angle between them in space — added up all the way round by numerical integration to machine precision. The small end lies on the same lines at D2 / D1 of the distance. The hand methods approach this as their divisions get finer.
How this tool is checked
Hand-worked true lengths; the concentric case held to the cone tool's closed-form sector (same sweep, radii and area); invariants of an exact development — both edges unroll to π × D, the pattern's area equals the surface area computed separately in 3D, the sweep equals the apex angle summed from the 3D lines, and moving the seam changes none of them. A second implementation, written separately in Python, never uses the apex: it triangulates between the two ends with 8,192 divisions, unrolls the strip triangle by triangle from true lengths, and must land on the exported DXF, read back with ezdxf, within 0.002 mm at every vertex — after a rotation and a move only, so a pattern drawn from the wrong face would fail. It runs on every release.
Exact versus approximate
The surface is developed exactly by numerical integration around the apex, not by the shop approximation of a few flat triangles; the curved edges are sampled polylines held within the maximum deviation you set.
What is not covered
Ends that are not parallel or not round, thickness and seam allowances, and forming springback. For a square or rectangular end, use the square-to-round tool; for a concentric cone with plate-thickness compensation, the cone tool.
Questions
What is a flush (flat-on-one-side) reducer?
One whose small end is offset by exactly (D1 − D2) / 2, so one side of the part is a straight vertical line — the usual eccentric reducer on a pump suction or a duct run that must keep a flat bottom or top. Tick "Flat on one side" and the offset follows the diameters.
Can the offset be anything else?
Yes. Untick "Flat on one side" and enter any centre-to-centre offset. Zero gives an ordinary concentric frustum — the same pattern as the cone tool. An offset larger than (D1 − D2) / 2 makes the small end overhang the large end in plan; that is still a valid oblique cone, and the tool says so in case it is not what you meant.
Why does it say the part is an oblique cone?
Because it is. Join every point of the large end to the point of the small end at the same angle — what dividing both ends into equal parts and triangulating does — and, with the ends parallel, all those lines meet at one point. The part is the slice of an oblique cone between two parallel planes, so it rolls flat without stretching and can be developed exactly around that apex.
How do I lay it out by hand from the table?
Two ways. From the apex: swing each apex-to-large-end length from one point, stepping the large-edge step between neighbouring lines, and mark the small end on each line one element length in. Or, when the apex is far away, by triangulation between the ends: build the strip quadrilateral by quadrilateral from the elements, the diagonals on the pattern and the two steps. The table gives every length both methods need, as laid out on the sheet; the guide walks through both.
Which dimensions should I enter for plate?
Dimensions are developed exactly as entered: enter the surface you want the pattern of (usually the mid-thickness for formed plate). No thickness compensation is applied. For thin sheet the difference between inside, outside and mid-thickness is small; for heavy plate, decide which surface has to fit and enter that.
Where does the seam go?
Along the short side (the flat side of a flush reducer), the long side, or halfway round between them. Moving the seam changes where the flat pattern is split, so the seam length, the blank size and the order of the table change; the area, the sweep and the lengths of the two round edges stay the same.
What is in the DXF?
One closed POLYLINE on the CUT layer — the whole developed outline — and the bend lines, if you asked for any, as LINEs on the BEND layer. Nothing else: no text, no construction lines.
Missing an option?
A different datum, an offset, a shop allowance, another shape altogether — say what this tool does not cover yet. Requests decide what gets built next, and a report of a wrong result is dealt with before anything new.
Rather have the whole part drawn — a shop drawing or a cut file from your sketch? That is paid drafting, quoted per job: drafting for small shops.
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