Three-Way Branch: flat pattern

A round trunk that divides into three or four equal branches spaced evenly round it: a trifurcation or a four-way piece. A round trunk that divides into three or four equal branches spaced evenly round it, all of one diameter: a trifurcation or a four-way piece.

Configurations

• Three branches. Three branches 120 degrees apart round the trunk.
• Four branches. Four branches 90 degrees apart round the trunk.

A worked example

The three branches configuration with the sizes SheetMorph opens with, in mm. 4 plates: the trunk and 3 branches that are all alike: 4 parts, 10 seams.

How it is developed

Parallel-line development (unrolled cylinders). 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.

• Every pipe is unrolled to a strip as long as its girth. Each line across it runs from the square end to the joint, where it is cut by the plane it reaches last: under a branch, by the trunk; above, by a neighbouring branch.
• The trunk is cut lowest under each branch and rises to 3 peaks between them. The branches are all alike: cut one and repeat it.
• The joints meet in a point on the axis, the crotch, where all the branches come together.

Checked before it can be exported

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

• Surface connectivity (passed): 256 facets form 4 lengths of pipe set against one another: every edge inside a part joins exactly two facets, every open edge one (V - E + F = 0).
• Developed edges equal true 3D lengths (passed): 768 facet edges compared between the 3D surface and the flat pattern (max deviation 1.54e-12).
• Shared edges and seam closure (passed): 252 internal edges are shared vertex-for-vertex between adjacent facets; 4 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 1.16e-9).
• Cut outlines closed and non-overlapping (passed): 4 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 the thickness.
• Curves are chains of chords between division points.
• No seam, lap, weld-preparation or pressing allowance is added.

Limits of this module

• The trunk and all the branches are of one diameter, and the branches are equal in angle and length; a reducing or unequal piece is not modelled.
• The joints are taken on the mid-thickness surface as plain butt joints. No crotch plate, gusset or reinforcing ring is drawn.
• Each pipe is one plate; the joints between them are not given a common seam allowance.
• Curves are chains of straight chords between division points.
• No seam, lap, weld-preparation or pressing allowance is added, and plates are not nested onto stock sheets.

Conventions

Coordinate system. The part stands on the Z axis with End A at Z = 0. Angles round the axis are measured from +X toward +Y, counter-clockwise seen from +Z.

Divisions and thickness. A circle is divided into 4N equal chords. 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.

Trunk and branches. The trunk stands on the Z axis from End A at Z = 0 up to the point where all the axes meet. The branches leave that point at the angle entered from the axis of the trunk, spaced evenly round it: branch 1 toward +X, the others counter-clockwise seen from above. Their ends are End B, End C and so on.

Joints. All the pipes are of one diameter, so each pair meets in a flat plane, the one that halves the angle between their axes. The trunk is cut lowest under each branch, at its throat, and rises to a peak between each pair; each branch is cut by the trunk below it and by its two neighbours above, which meet in the crotch on the axis.

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.
• 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.
• Flat-Oval Tee: A round or flat-oval branch on the flat side of a flat-oval main: the main with its hole, and the branch cut to the main.