Round to Rectangle on a Bend: flat pattern

A square-to-round whose two ends are at an angle: it changes shape and direction in one piece. A transition between a rectangle and a circle whose ends are at an angle to each other, so that the piece changes shape and direction together.

Configurations

• Rectangle at End A. End A is the rectangle; End B, at the end of the bend, is the circle.
• Round at End A. End A is the circle; End B, at the end of the bend, is the rectangle.

A worked example

The rectangle at end a configuration with the sizes SheetMorph opens with, in mm. One plate: four flat triangles on the sides of the rectangle and four corners joined to the quarters of the circle, between ends at an angle, seam on the middle of the left side, outside the bend: 1 part, 1 seam.

How it is developed

Triangulation (true-length unfolding). This is a triangulated approximation: the curved surface is divided into small flat triangles, each laid out at its true size. The finer the division, the closer the pattern is to the surface; SheetMorph states the deviation it found.

• The piece is a square-to-round transition whose round and rectangular ends are not parallel: it changes shape and turns in one piece.
• Each side of the rectangle is the base of a flat triangle whose point is on the circle, and each corner of the rectangle is joined to a quarter of the circle. The triangles on the inside of the bend are short and those on the outside long.
• Curved parts are divided into narrow triangles between the points of the two ends and unfolded from their true lengths; more divisions bring the plate closer to the true surface.

Checked before it can be exported

Every pattern SheetMorph draws is checked against the 3D part it comes from. For this example:

• Each end lies in its plane (passed): 40 points of the two ends compared with their planes (max deviation 2.84e-14).
• Surface connectivity (passed): 40 facets form one closed band: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 120 facet edges compared between the 3D surface and the flat pattern (max deviation 3.41e-13).
• Shared edges and seam closure (passed): 39 internal edges are shared vertex-for-vertex between adjacent facets; 1 seam(s) have mating edges of equal true length (max deviation 5.68e-14).
• Developed facets consistently oriented (passed): All developed facets are counter-clockwise (outside face up); none is mirrored or degenerate.
• Developed area equals surface area (passed): Facet area in 3D and in the flat pattern agree (difference 1.75e-10).
• Cut outlines closed and non-overlapping (passed): 1 closed outline(s); no outline crosses itself or another outline.

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside sizes are converted to it by one thickness across each opening.
• Curves are chains of chords between division points.
• No seam, lap or weld allowance is added.

Limits of this module

• The piece is a single transition between two ends at an angle, not a segmented bend; for a tight or long bend, make a transition and a bend separately.
• With the ends at an angle the flat triangles stay flat, but the corners are only approximately parts of cones.
• Curves are chains of straight chords, so a curved edge of the plate is slightly short of its true length unless the matched-circumference option is used.
• The wall is cut only on the lines offered under Pattern; other seams, or more pieces, are not offered.
• No seam, lap, flange or weld allowance, and no bend allowance, is added.

Conventions

Coordinate system. End A is at Z = 0, centred on the Z axis, with its width along X and its depth along Y. End B is above it; its centre is moved by the offsets, positive toward +X and +Y.

Points and seam. A<k> and B<k> are the points of End A and End B, counted counter-clockwise seen from +Z.

How the wall is built. Each straight side is joined to the one point of the other end that faces the same way, as a flat triangle; each corner is joined to a whole arc, as a fan of narrow triangles that is part of a cone. Lines where a flat meets a fan are bend lines; the lines inside a fan are forming lines.

Curves and thickness. A circle is divided into 4N equal chords and an oval end into 2N for each half circle. Sizes are inside, mid-thickness or outside sizes; the pattern is developed on the mid-thickness surface and the views show the sizes as entered.

Bend. The centre of End B is carried round an arc of the radius entered, in the XZ plane, from the Z axis toward +X, through the bend angle; End B stays square to the arc. The rectangle has its width in the plane of the bend and its depth along Y: its left side is on the outside of the bend and its right side on the inside.

Pattern. The wall is one plate, cut from the middle of the side of the rectangle chosen (bottom, right, top or left) to the point of the other end that faces the same way; or two halves, cut there and on the opposite side; or four quarters, cut on the middle of every side, for work too big for one sheet.

Change the sizes to your own in SheetMorph, check the drawing, add seam and flange allowances, and export DXF, PDF, SVG, CSV or G-code. Check every pattern against your own standards before cutting.

Other duct bends

• Elbow: Elbows of rectangular duct, rolled to a radius or in mitered segments: two cheeks, a throat and a heel.
• Reducing Elbow: A rectangular bend whose width, or width and depth, change steadily from one end to the other.
• Square-Throat Elbow: A rectangular elbow with a square inside corner, joined on one mitre, with or without turning vanes.
• Offset: A rectangular duct moved sideways between two parallel ends: an ogee of two curves, or a slanted piece on two mitres.