Polygon Transition: flat pattern

A regular polygon, such as a hexagon or an octagon, to a circle or to another polygon. A transition from a regular polygon, such as a hexagon or an octagon, to a circle or to another polygon.

Configurations

• Polygon to round. End A is a regular polygon and End B a circle: a flat triangle on every side and part of a cone at every corner.
• Polygon to polygon. Both ends are regular polygons, which may differ in the number of sides and be turned relative to each other. Every panel is flat.

A worked example

The polygon to round configuration with the sizes SheetMorph opens with, in mm. One plate: 6 flat triangles and 6 corners that are parts of cones, seam on the left quadrant line: 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.

• Each side of the polygon is the base of a flat triangle whose point is on the circle. Each corner of the polygon is joined to the arc of the circle between two such points, as a fan of narrow triangles.
• 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): 38 points of the two ends compared with their planes (max deviation 0).
• Surface connectivity (passed): 38 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): 114 facet edges compared between the 3D surface and the flat pattern (max deviation 1.14e-13).
• Shared edges and seam closure (passed): 37 internal edges are shared vertex-for-vertex between adjacent facets; 1 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 1.16e-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

• Polygons are regular; a polygon with unequal sides is not modelled.
• Between two polygons the panels are the flat triangles and four-sided panels a sheet folded round both would form; other ways of dividing the wall are not offered.
• 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.

Polygons. A polygon is regular, with 3 to 12 sides, and is given across its flats (for a triangle, twice the distance from the centre to a side). One flat of End A faces +X. A second polygon may be turned about the axis, counter-clockwise seen from +Z. The centre of End B may be moved by the offsets, which are zero unless entered.

Pattern. The wall is one plate, cut along the line between the two ends on the quadrant line chosen (left, bottom, right or top: where both outlines face -X, -Y, +X or +Y); or two halves, cut there and on the opposite line. A piece is cut only where the two ends have points facing the same way.

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 Rectangle: Hoppers and duct transitions between two rectangular ends, centred, offset or inclined.
• 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.
• 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.