Multi-Section Cone: flat pattern

A large cone made of several plates: equal segments round it and courses up it. A right cone made of several plates: equal segments round it and, for a tall cone, equal courses up it.

Configurations

• Segments. The cone is divided round its circumference into equal plates.
• Segments and courses. The cone is divided round its circumference and also into courses up its height.

A worked example

The segments configuration with the sizes SheetMorph opens with, in mm. 4 plates: 4 round the cone: 4 parts, 4 seams.

How it is developed

Radial-line development by triangulation. 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 cone is divided into 4 equal segments round its circumference, so that each plate fits the sheet and the rolls.
• Each plate is a sector of an annulus on the developed cone. The plates of one course are identical: cut the number shown from one pattern.
• Lengthwise joints between segments are drawn as seams; girth joints between courses are the edges where two courses meet.

Checked before it can be exported

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

• Every ring is a circle of the cone (passed): All 64 points lie on the cone, on 2 level rings (max deviation 1.14e-13).
• Surface connectivity (passed): 64 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): 192 facet edges compared between the 3D surface and the flat pattern (max deviation 1.33e-12).
• Shared edges and seam closure (passed): 60 internal edges are shared vertex-for-vertex between adjacent facets; 4 seam(s) have mating edges of equal true length (max deviation 2.27e-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 5.12e-9).
• Cut outlines closed and non-overlapping (passed): 4 closed outline(s); no outline crosses itself or another outline.
• Plates of a course are identical (passed): Each of the 1 course(s) is developed as 4 plate(s) of equal area (max deviation 9.31e-10).
• Symmetric about the XZ plane (passed): Every point has its mirror image across the XZ plane (max deviation 3.87e-13).

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside diameters are converted to it by one thickness.
• Segments are equal round the cone and courses equal up its height; lengthwise joints line up from course to course.
• No seam, lap, flange or weld allowance is added.

Limits of this module

• Right cones only, with equal segments and equal courses; joints are not staggered from course to course.
• Plate edges are straight between divisions.
• No seam, lap, flange or weld allowances, and no nesting onto stock sheets.

Conventions

Coordinate system. The cone stands on the Z axis, with its base diameter at End A (Z = 0) and its top diameter at End B (Z = height).

Segments and courses. Segments (1 to 16) are equal parts round the cone; their number must divide the divisions of the circle, 4N. Courses (1 to 8) are equal parts of the height. Ring j is the circle between courses; Rj_k is its division k.

Plates. Each plate is one segment of one course, a sector of an annulus. All the plates of a course are identical and are drawn side by side; plate labels give the course. The blank sizes listed are the bounding rectangle of one plate.

Material thickness. Say whether the diameters are inside, mid-thickness or outside sizes; the pattern uses the mid-thickness surface, and the views show the sizes 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 cones

• Round to Round: Reducers and cones between two round ends: concentric, eccentric, offset or inclined.
• Round to Ellipse: A cone from a round end to an elliptical end, or from an ellipse to a round: concentric, offset, eccentric or with angled faces.
• Cut by a Plane: A right cone with its top, or both ends, cut by an inclined plane.
• 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.