V-Piece: flat pattern

Breeches pieces: one round inlet dividing into two round branches, equal or unequal, level or angled. Breeches piece: one round inlet dividing into two round branches, equal or unequal, each leg developed by triangulation as its own plate.

Configurations

• Equal branches. Two equal branches with level ends, placed symmetrically about the inlet axis.
• Equal branches, angled. Two equal branches, each leaning outward by the branch angle, so their ends are inclined.
• Unequal branches. Each branch has its own diameter, offset, height and angle.

A worked example

The equal branches configuration with the sizes SheetMorph opens with, in mm. 2 plates, one per leg, each triangulated between 32 divisions: 2 parts, 3 seams.

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 V-piece is made of two legs. Each leg joins its half of the inlet, and the crotch line between the legs, to its own branch end.
• The crotch line is a half-ellipse in the plane between the legs, from one side of the inlet over the crotch to the other. Both legs are developed to the same points on it, so they meet exactly.
• A leg is not exactly developable, so it is split into narrow triangles and unfolded from true lengths; the more divisions, the smoother the cut lines.
• The branches are equal, so the two plates are mirror images of each other.
• Division points lie on the true circles, so each developed round edge (a sum of chords) is slightly shorter than pi x D. Increase the divisions, or use the matched-circumference model, if a rolled end must close exactly on its true diameter.
• Each leg has one seam, on the outside of each leg; the crotch line is the third joint.

Checked before it can be exported

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

• Inlet and branch ends are planar circles (passed): All 96 end points lie in their end plane, at radius 200.5, 125.5, 125.5 for End A, B and C (max deviation 2.84e-14).
• Crotch line planar and shared by both legs (passed): All 17 points of the crotch line lie in the plane X = 0 on its half-ellipse, the crotch at height 200 (max deviation 4.44e-14); each of its 16 edges belongs to both legs.
• Each leg on its own side of the centre plane (passed): No point of leg B has X < 0 and no point of leg C has X > 0 (max overshoot 0).
• Surface connectivity (passed): 128 facets form one inlet dividing into two legs: every edge inside a part joins exactly two facets, every open edge one (V - E + F = -1).
• Developed edges equal true 3D lengths (passed): 384 facet edges compared between the 3D surface and the flat pattern (max deviation 1.06e-12).
• Shared edges and seam closure (passed): 126 internal edges are shared vertex-for-vertex between adjacent facets; 2 seam(s) have mating edges of equal true length (max deviation 1.71e-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 1.16e-10).
• Cut outlines closed and non-overlapping (passed): 2 closed outline(s); no outline crosses itself or another outline.
• Symmetric about the plane of the centres (passed): Every point has its mirror image across the XZ plane (max deviation 1.31e-13).
• Legs are mirror images (passed): Branch C is the mirror image of branch B across the centre plane (max deviation 1.08e-13), and the two plates have equal area (difference 4.66e-10).

Assumptions

• The pattern is developed on the mid-thickness surface. Inside or outside diameters are converted to it by one thickness; offsets, heights, angles and the crotch height are not changed.
• The two legs meet on the plane through the inlet centre, square to the line of the branches. The crotch line is a half-ellipse in that plane.
• Each leg is the surface joining matching divisions of its foot and its branch end, approximated by flat triangles.
• Every circle is divided into equal parts; plate edges are straight between divisions.
• No rolling allowance, and no seam, lap, flange, weld or trimming allowance, is added to any outline.
• Plates are drawn outside face up.

Limits of this module

• Round inlet and round branches only, with all three centres in one plane.
• The legs meet on the plane through the inlet centre; a branch that crosses that plane, such as a straight-through run with a side branch, is not covered here; use the Y-piece for that.
• The crotch line is a half-ellipse of the height you enter; it is not calculated from an intersection of two cones.
• Plate edges are straight between divisions, not smooth curves.
• No rolling allowance, and no seam, lap, flange, weld or trim allowances; no straight collars on any end.
• The fit of the crotch joint through the sheet thickness (bevels) is not modelled.
• Sheet nesting, grain direction and stock-size checks are not performed.

Conventions

Coordinate system. Right-handed, Z vertical in the views. End A, the inlet, is the circle at Z = 0, centred on the origin. Branch B (End B) leaves toward +X and branch C (End C) toward -X; all three centres lie in the XZ plane, so the Front view shows the V of the piece.

Branch position. A branch end is located by the offset of its centre from the inlet axis (along X) and the height of its centre above the inlet plane. With equal branches you give the distance between the two centres instead; each is offset by half of it. The whole of a branch end must be on its own side of the plane between the legs.

Branch angle. The branch angle is the outward lean of a branch: the angle between its end plane and the inlet plane, from 0 (level end) to 60 deg. The end turns about the Y axis through its own centre, the outer edge going down and the inner edge up. Two equal branches include twice the branch angle.

Crotch line. The two legs are joined on the plane X = 0. The joint runs from one side of the inlet, over the crotch, to the other side: a half-ellipse as wide as the inlet and as high as the crotch height you enter. The crotch must be lower than the inner edge of both branch ends.

Numbering. Every circle is divided into 4N equal parts, N per quarter, numbered from +X toward +Y: division 0 toward +X, N toward +Y, 2N toward -X, 3N toward -Y. A is the inlet, B and C the branch ends, and J1 to J(2N-1) the points of the crotch line from the +Y side to the -Y side; JN is the crotch. Plate S1 is leg B and plate S2 is leg C.

Development method. Triangulation. Each leg is divided into strips between neighbouring divisions; a strip is split by its shorter diagonal into two triangles, and the triangles are laid out one after another from their true lengths. The listed true lengths are those of the lines joining the foot and the top of each division.

Seams. Each leg is cut along one line to open it flat: on the outside of the leg, or on the crotch side, where the line is shortest. The crotch line itself is the joint between the two plates.

Circle model. Inscribed: division points are on the true circles, and each developed round edge is slightly shorter than pi x D. Matched circumference: points are on marginally larger circles so each developed round edge equals pi x D exactly.

Material thickness. A plate has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your diameters are inside, mid-thickness or outside sizes: inside diameters are increased by one thickness and outside diameters reduced by one thickness to reach the mid-thickness surface. Offsets, heights, angles and the crotch height are the same for every surface. The 3D view and the views always show the diameters 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

• 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.
• 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.
• 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.