Square / Rectangle to Round Transition

A square-to-round is the classic exercise in triangulation. Kittredge sorts pattern problems into parallel forms, regular tapering forms and irregular forms, and develops the irregular ones by triangulation (G. W. Kittredge, The New Metal Worker Pattern Book, 1901, Ch. V, pp. 71–72; public domain).

Four flat triangles and four corners

Each side of the rectangle is joined to the one point of the round whose tangent runs parallel to that side — the front-most point for the front side, and so on. That gives four flat triangles. Between them, each rectangle corner is joined to a quarter of the round: a piece of an oblique cone with its tip at the corner. Because each triangle contains the round's tangent at its apex, it meets the neighbouring cones without a crease, so the whole part rolls flat.

Developing the corners exactly

The shop method cuts each corner into a few flat triangles and lays them out one by one from their true lengths. Here each corner is unrolled exactly instead: around its tip, every line to the round keeps its true length, and the angle between neighbouring lines is the angle between them in space, added up by numerical integration to machine precision. The result is what hand triangulation approaches as the triangles get thinner.

In two or four pieces

How many pieces an article is made in depends, Atkins says, on cutting the sheet economically and easing the bending, weighed against the extra joints (E. A. Atkins, Practical Sheet and Plate Metal Work, 2nd ed., p. 109). Most books we checked draw a square-to-round’s half pattern with its seams at the middles of two opposite sides, each half carrying one whole side, two corners and two half sides (Kittredge (1901), pp. 327–328; E. M. Longfield, Sheet Metal Drafting (1921), pp. 188–190), and two say the part is made in two pieces: J. S. Daugherty, Sheet-Metal Pattern Drafting and Shop Problems (1922), pp. 154–156, a chimney base; Laying Out for Boiler Makers, 3rd ed. (1922), p. 373. Neubecker draws a half pattern whose free edges are the corner lines at two diagonally opposite corners (W. Neubecker, Sheet-Metal Work (1920), pp. 20–22). No book we found shows a square-to-round made in four pieces: the quarter between two mid-side lines is the unit they draft, and the ICS textbook says developing only a half or a quarter is customary because these patterns are usually too large for one piece of metal (ICS, A Textbook on Sheet-Metal Pattern Drafting (1901), §18, pp. 53–55). The tool offers four pieces split at those mid-side lines.

With the round offset the corners all differ; Kittredge develops each corner and side separately (Kittredge, pp. 331–333), and Kidder shows that a round offset along the split line gives two halves that are one blank formed opposite-hand (F. S. Kidder, Triangulation Applied to Sheet Metal Pattern Cutting (1917), pp. 26–32). Here each piece is unrolled from its own seam and turned so its rectangle edge lies level, and whether it is a copy or a turned-over copy of another is decided by comparing the developed outlines. The books add laps or locks on the seam edges (Daugherty, p. 154; Longfield, p. 190); the pieces here have none, so add them for your seam.

How this tool is checked

Hand-worked true lengths; invariants of an exact development — the rectangle edge unrolls to the true perimeter, the round edge to π × D, the pattern's area equals the surface area computed separately in 3D, and moving the seam changes none of them. A second implementation, written separately in Python, does it the shop way with over 4,000 flat triangles per corner, unrolling the surface as one strip with every triangle placed from its true distances; it must agree with the exported DXF, read back with ezdxf, to a few thousandths of a millimetre. It runs on every release, on the one-piece pattern. For the pieces, the tests check that their edges and areas add up to the one-piece pattern, that each seam, measured on the developed outlines, has the same length on both pieces it joins and equals its true length in 3D, and that the seam lengths match those we worked from the dimensions of the ICS and Neubecker examples (13.0000 and 12.0104 in; 3.5089 in).

Exact versus approximate

The corners are developed exactly by numerical integration, not by the shop approximation of a few flat triangles; the round edge is a sampled polyline held within the maximum deviation you set.

What is not covered

Rotated or tilted openings, a round that overhangs only some sides, four pieces split on the corner lines, thickness and seam allowances, and forming springback.