Rectangle to Ellipse: flat pattern

A rectangular end to an elliptical end, or an ellipse to a rectangle: concentric, offset, eccentric or with angled faces. A transition between a rectangular end and an elliptical end, either way up: concentric, offset, eccentric, with inclined ends, or offset and inclined together.

Configurations

• Concentric. Both ends share one centreline and the end planes are parallel.
• Centre offset. The centre of End B is moved from that of End A by dx, dy. The end planes are parallel.
• Eccentric. The rectangle and the ellipse touch one plane on the side or sides chosen, so that side is straight; the centre offset is calculated.
• Angled faces. One or both end planes are inclined. The end centres stay on the Z axis.
• Combined offset + angle. Inclined end planes together with centre offsets or flush sides.

A worked example

The concentric configuration with the sizes SheetMorph opens with, in mm. One plate: four flat triangles on the sides of the rectangle and four corners that are parts of cones, seam on the middle of the bottom side: 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 rectangle is the base of a flat triangle whose point is the point of the ellipse that faces the same way. Each corner of the rectangle is joined to the quarter of the ellipse between two such points, as a fan of narrow triangles.
• The lines where a flat triangle meets a corner are bend lines; the lines inside a corner are forming lines.
• 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): 72 points of the two ends compared with their planes (max deviation 0).
• Surface connectivity (passed): 72 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): 216 facet edges compared between the 3D surface and the flat pattern (max deviation 1.78e-13).
• Shared edges and seam closure (passed): 71 internal edges are shared vertex-for-vertex between adjacent facets; 1 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 4.66e-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

• The ellipse lies with its axes along its end's X and Y, square to the sides of the rectangle; an ellipse turned about the axis is not offered.
• The girth of the ellipse is from Ramanujan's approximation, which is within a few millionths of the true girth for any ellipse a plate is rolled to.
• With inclined ends the flat triangles stay flat, but the corners are only approximately parts of cones.
• 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.

Rectangle and ellipse. The rectangle is End A or End B, as chosen, and the ellipse the other end, a true ellipse; each has its width along its end's X and its height along its Y. Each side of the rectangle is the base of a flat triangle whose point is the point of the ellipse that faces the same way, and each corner of the rectangle is joined to the quarter of the ellipse between two such points, as a fan of narrow triangles that is part of a cone.

Pattern. The wall is one plate, cut from the middle of the side chosen (bottom, right, top or left) to the point of the ellipse that faces the same way; two halves, cut there and on the opposite side; or four quarters, cut on the middle of every side, for work too big for one sheet.

Position and inclination. End B is at Z = L with its centre moved by dx, dy, positive toward +X and +Y. A flush side makes both ends touch one plane of constant X (left, right) or Y (bottom, top), allowing for their inclination. An end is inclined first about its own X axis (+ moves its top toward +Z), then about Y (+ moves its right side toward -Z), about its centre.

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.
• 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.