Y-Piece, Round to Rectangle: flat pattern

A round trunk dividing into two branches that change from round at the junction to rectangular at their ends. A round trunk dividing into two branches that change from round at the junction to rectangular at their ends: a Y-piece junction with a square-to-round on each branch.

Configurations

• Equal branches. Both branches have the same end rectangle, angle and length.
• Unequal branches. Each branch has its own end rectangle, angle and length. A branch at 0 degrees runs straight on from the trunk.

A worked example

The equal branches configuration with the sizes SheetMorph opens with, in mm. 3 plates: the trunk, unrolled, and two branches from round to rectangular, each four flat triangles and four corners triangulated over 8 divisions: 3 parts, 6 seams.

How it is developed

Parallel-line development (trunk) and triangulation (branches). 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 trunk and the two branches meet at the junction on three flat cuts, as in the Y-piece. The trunk is a cylinder and its plate is that of a Y-piece of the same angles.
• Each branch starts on its two cuts at the junction, at the trunk's diameter, and ends on a rectangle square to its axis. As on a square-to-round, each side of the rectangle is the base of a flat triangle whose point is the division of the junction that faces it, and each corner is joined to the quarter of the junction between, as a fan of narrow triangles. Lines where a flat meets a fan are bend lines.
• Each branch plate is cut from the middle of the side of its rectangle on the outside of the Y; the trunk on its +X side.
• The branches are alike, so their plates are mirror images.

Checked before it can be exported

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

• End A is a square-cut circle (passed): The 32 points of End A lie on the circle of radius 200.5 at Z = 0 (max deviation 2.84e-14).
• The branch ends are square-cut rectangles (passed): Every corner and side middle of End B and End C lies in its end plane, square to its axis, at the width and depth of its end (max deviation 2.84e-14).
• Junction joints planar, meeting at two points (passed): The three joints of the junction each lie in the plane that bisects their two axes, and all run between the same two points (max deviation 2.84e-14).
• Every side of a rectangle stands on a flat triangle (passed): The two halves of the triangle on each side of each rectangle are in one plane (max deviation 0).
• Surface connectivity (passed): 144 facets form one trunk dividing into two branches, round to rectangular: every edge inside a part joins exactly two facets, every open edge one (V - E + F = -1).
• Developed edges equal true 3D lengths (passed): 432 facet edges compared between the 3D surface and the flat pattern (max deviation 2.10e-12).
• Shared edges and seam closure (passed): 141 internal edges are shared vertex-for-vertex between adjacent facets; 3 seam(s) have mating edges of equal true length (max deviation 1.14e-13).
• 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 2.79e-9).
• Cut outlines closed and non-overlapping (passed): 3 closed outline(s); no outline crosses itself or another outline.
• Symmetric about the XZ plane (passed): Every point has its mirror image across the XZ plane (max deviation 3.42e-13).
• Branches are mirror images (passed): Branch C is the mirror image of branch B (max deviation 0).

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside sizes are converted to it by one thickness across each opening.
• The three axes meet at one point and lie in one plane. Each rectangle has its width in that plane and its depth square to it, and is centred on its branch axis.
• No seam, lap, flange or weld allowance is added.

Limits of this module

• The branches start at the trunk's diameter at the junction; a branch that is smaller right from the trunk wall is a tee or lateral.
• The rectangles are square-cut and centred on their branch axes; a rectangle set off its axis or turned about it is not covered.
• The corners are triangulated, so their part of the plate is a close approximation; the flat triangles are exact.
• No seam, lap, flange or weld allowances.

Conventions

Coordinate system. End A, the open end of the trunk, is the circle at Z = 0. The trunk runs up the Z axis to the junction point. Branch B (End B) leaves toward +X and branch C (End C) toward -X; all three axes lie in the XZ plane, so the Front view shows the Y.

Branches. A branch angle is measured at the junction point between the trunk axis continued and the branch axis, up to 90 degrees; the two together must be at least 20 degrees. A branch length is measured along its axis from the junction point to the middle of its open end, which is square to the axis.

Rectangular ends. Each branch ends on a rectangle centred on its axis: its width W in the plane of the Y, its depth D along Y, so the width shows in the Front view and the depth in the side view.

Junction. The junction is that of a Y-piece of the trunk's diameter: three flat cuts between the trunk and each branch and between the two branches, all running between the same two points beside the junction. Trunk and branches must be longer than those cuts reach.

Development. The trunk is unrolled. Each branch is built as a square-to-round between its junction cuts and its rectangle: a flat triangle on each side and a fan of narrow triangles at each corner, unfolded from true lengths. The listed true lengths are those of the lines from the middles of the sides and from the corners down to the junction, on the +Y half; the -Y half is its mirror image. The lines from a corner are numbered round from the outside of the Y toward the crotch; the first and last are bend lines. Plate S1 is the trunk, S2 branch B and S3 branch C.

Seams. The trunk seam is on its +X side. Each branch seam runs from the middle of the side of its rectangle on the outside of the Y down to the junction.

Material thickness. Say whether the sizes are inside, mid-thickness or outside sizes; the pattern uses the mid-thickness surface, and the views show the sizes as entered.

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 branches

• V-Piece: Breeches pieces: one round inlet dividing into two round branches, equal or unequal, level or angled.
• Y-Piece: A trunk dividing into two branches on flat miter cuts, symmetric or as a lateral, the branches of the trunk diameter or tapering to smaller ends.
• Twin Bend: A trunk dividing into two segmented bends that turn opposite ways: a bullhead at 90°, or any angle you give.
• Twin Bend to Rectangle: A round trunk dividing into two segmented bends that turn opposite ways and change to rectangular duct as they turn: a bullhead at 90°, or any angle you give.
• Twin Bend from Rectangle: A rectangular trunk that changes to round and divides into two round segmented bends that turn opposite ways: a bullhead at 90°, or any angle you give.
• Three- or Four-Way: A round trunk that divides into three or four equal branches spaced evenly round it: a trifurcation or a four-way piece.
• Rectangular Y: A rectangular trunk dividing into two rectangular legs side by side: a trouser piece, with equal or unequal legs.
• Rectangle to Two Rounds: A rectangular inlet that divides into two round outlets side by side, as two square-to-rounds on one dividing edge.