Rectangle-to-Rectangle Transition: flat pattern

Hoppers and duct transitions between two rectangular ends, centred, offset or inclined. Transition piece between two rectangular ends, developed by triangulation.

Configurations

• Concentric. Both ends share one centreline. End planes are parallel.
• Centre offset. End B centre is displaced from End A centre by dx, dy. End planes are parallel.
• Eccentric. Selected faces of both ends stay flush; the centre offset is calculated from the end sizes.
• Angled faces. One or both end planes are inclined. End centres stay on the Z axis.
• Combined offset + angle. Inclined end planes together with centre offsets or face alignment.

A worked example

The concentric configuration with the sizes SheetMorph opens with, in mm. Exact development of a planar-panel (polyhedral) surface: 4 parts, 1 seam.

How it is developed

Triangulation (true-length unfolding). This is an exact development: the surface unfolds without stretching, so the pattern closes exactly.

• All four panels are planar. The reference surface is a polyhedral surface that unfolds about its straight bend lines without stretching.
• The panel diagonals used to develop them are not bends, so they are not drawn.
• One-piece wrap-around pattern with 3 corner bend lines and 1 seam.

Checked before it can be exported

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

• End contours are true rectangles (passed): Side lengths and diagonals of both ends match the specified widths and heights (max deviation 0).
• Surface connectivity (passed): 8 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): 24 facet edges compared between the 3D surface and the flat pattern (max deviation 1.71e-13).
• Shared edges and seam closure (passed): 7 internal edges are shared vertex-for-vertex between adjacent facets; 1 seam(s) have mating edges of equal true length (max deviation 0).
• 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): 1 closed outline(s); no outline crosses itself or another outline.
• Expected symmetry (passed): left/right and top/bottom mirror symmetry holds: mirrored corners coincide (max deviation 0) and mirrored panels have equal developed area.
• Panel planarity classified (passed): All four panels are planar within tolerance: the development is exact for the reference surface.

Assumptions

• The pattern is developed on the mid-thickness surface. Inside or outside end dimensions are converted to it by one thickness; mid-thickness dimensions are used unchanged.
• Corners are sharp folds of the mid-thickness surface (equivalent to a neutral axis at mid-thickness); no bend radius or separate bend allowance is applied.
• No seam, lap, flange, weld or trimming allowance is added to any outline.
• Each end is a planar rectangle; end-plane twist about the Z axis is not supported.
• Side panels are either planar or split into two planar triangles along the shorter diagonal.
• The pattern is drawn outside face up.

Limits of this module

• No bend radius or separate k-factor: corners and creases are sharp folds of the mid-thickness surface.
• Inside/outside conversion is one thickness at each end; the extra difference on steeply inclined walls is not modelled.
• No seam, lap, flange, weld or trim allowances.
• No end-plane twist about the Z axis; ends are always planar rectangles.
• Warped panels are faceted with one diagonal crease, not subdivided into finer strips.
• Sheet nesting, grain direction and stock-size checks are not performed.

Conventions

Coordinate system. Right-handed. X is horizontal (width, +X right), Y is vertical (height, +Y top), Z is the reference axis. End A is centred on the origin; End B is centred on (dx, dy, L). Left, right, top and bottom are as seen looking from End B back toward End A.

Length and offsets. L is the distance between the two end centres measured along Z, not the slant distance. dx and dy are End B centre minus End A centre: positive dx moves End B to the right, positive dy moves it up.

End-plane inclination. Each end can be inclined about two axes through its own centre, so the centre does not move. "About X" turns the end about the horizontal axis; positive moves its top edge toward +Z. "About Y" turns it about the vertical axis; positive moves its right edge toward -Z. The rotation about X is applied first. Limit: +/-75 deg. An inclination changes the end plane; it is not an offset.

Face alignment. Aligning a face (left, right, top or bottom) places that face of both ends in one plane of constant X or Y. The centre offset on that axis is then calculated, not entered. Left/right alignment is rejected for an end inclined about both axes, because its side edge is no longer parallel to such a plane.

Corner and panel numbering. Corners on each end: 0 bottom-left, 1 bottom-right, 2 top-right, 3 top-left (A0-A3, B0-B3). Panels: Bottom (0-1), Right (1-2), Top (2-3), Left (3-0). Corner edge Ci joins Ai to Bi.

Development method. Each panel is split along its shorter diagonal into two triangles. Triangles are unfolded in sequence from their true 3D edge lengths, each sharing an edge with the previous one. A panel whose four corners are coplanar is exact and its diagonal is only a construction line, which is not drawn. A panel that is not coplanar is warped: its diagonal becomes a real crease and the pattern is a faceted approximation.

Material thickness. A flat pattern has the length of the sheet's neutral surface, taken as mid-thickness. Say whether your end dimensions are inside, mid-thickness or outside sizes: inside sizes are increased by one thickness and outside sizes reduced by one thickness to reach the mid-thickness surface, on which the pattern is developed. Length, offsets and inclinations are not affected. Sharp folds of the mid-thickness surface correspond to a neutral axis at mid-thickness; bend radii are not modelled.

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 transitions

• Rectangle to Round: Square-to-round transitions with flat sides and conical corners, offset or inclined.
• Round to Oval: A round end to a flat-oval end: flat triangles under the straight sides, curved parts round the ends.
• Rectangle to Oval: A rectangular end to a flat-oval end: flat panels on the long sides and parts of cones at the corners.
• Rectangle to Ellipse: A rectangular end to an elliptical end, or an ellipse to a rectangle: concentric, offset, eccentric or with angled faces.
• Polygon: A regular polygon, such as a hexagon or an octagon, to a circle or to another polygon.
• Twisted Rectangle: A transition between two rectangles, one of them turned about the axis; its twisted sides are split into flat triangles.
• Pyramid Hopper: A four-sided pyramid between two rectangles, such as a hopper or a plenum, with or without flanges turned outward.