Round to Ellipse: flat pattern

A cone from a round end to an elliptical end, or from an ellipse to a round: concentric, offset, eccentric or with angled faces. A cone between a round 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 circle 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, curved all round, between a circle and an ellipse, 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 point of the circle is joined to the point of the ellipse that faces the same way, so the wall is the one a sheet takes when it is wrapped round the two ends. Where the ellipse is flatter than the circle, a point of the circle is joined to several points of the ellipse, as a narrow fan.
• 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): 96 points of the two ends compared with their planes (max deviation 0).
• Surface connectivity (passed): 96 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): 288 facet edges compared between the 3D surface and the flat pattern (max deviation 3.46e-12).
• Shared edges and seam closure (passed): 95 internal edges are shared vertex-for-vertex between adjacent facets; 1 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 4.07e-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; 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 wall is joined in the same way, each end's directions taken in its own plane; the parts between are close to, not exactly, 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.

Ellipse. The circle is End A or End B, as chosen, and the ellipse the other end. The ellipse is a true ellipse with its width along its end's X and its depth along its Y; either may be the larger. Its points are where it faces the same way as the points of the circle, and where it is flatter than the circle each step between them is divided further.

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.

Position and inclination. End B is at Z = H 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 cones

• Round to Round: Reducers and cones between two round ends: concentric, eccentric, offset or inclined.
• Cut by a Plane: A right cone with its top, or both ends, cut by an inclined plane.
• Multi-Section: A large cone made of several plates: equal segments round it and courses up it.
• Rain Cap: A conical cap that runs to a point, as over a flue or a vent, with or without a skirt round its rim: the sector it is rolled from.