Cone Cut by a Plane: flat pattern

A right cone with its top, or both ends, cut by an inclined plane. A right cone with its top, or both ends, cut by an inclined plane.

Configurations

• Top cut. The base is square; the top is cut at an angle.
• Both ends cut. Both the base and the top are cut, each at its own angle.

A worked example

The top cut configuration with the sizes SheetMorph opens with, in mm. 1 plate: a cone sector with its cut edges, triangulated between 32 divisions: 1 part, 1 seam.

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.

• A plane that cuts a cone at an angle cuts it in an ellipse. Each line of the cone is developed at its true length from the base cut to the top cut, so the cut edges of the plate are smooth curves, drawn as straight lines between divisions.
• The wall is a cone, which is developable; it is unfolded triangle by triangle from true lengths.

Checked before it can be exported

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

• Both ends lie in their cut planes (passed): All 64 end points lie in planes cut at 0 deg (base) and 30 deg (top) from square (max deviation 6.39e-14).
• Every point is on the cone (passed): Every end point is at the radius the cone has at its height (max deviation 2.84e-14).
• 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 4.12e-13).
• Shared edges and seam closure (passed): 63 internal edges are shared vertex-for-vertex between adjacent facets; 1 seam(s) have mating edges of equal true length (max deviation 8.53e-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 1.75e-10).
• Cut outlines closed and non-overlapping (passed): 1 closed outline(s); no outline crosses itself or another outline.
• Symmetric about the XZ plane (passed): Every point has its mirror image across the XZ plane (max deviation 1.31e-13).

Assumptions

• The pattern is developed on the mid-thickness surface; inside or outside diameters are converted to it by one thickness.
• The cone is a right cone on the Z axis; both cuts are tilted about the Y axis.
• No seam, lap, flange or weld allowance is added.

Limits of this module

• Right cones only; the cut planes are tilted about one axis.
• A cut end is drawn as straight lines between divisions.
• No seam, lap, flange or weld allowances.

Conventions

Coordinate system. The cone stands on the Z axis. The base and top diameters are those of the uncut cone at Z = 0 and at Z = height. Both cuts are tilted about the Y axis, so the Front view shows them as straight lines.

Cut angle. A cut passes through the axis, at Z = 0 for the base and at Z = height for the top. Its angle is measured from square, from -75 to 75 degrees; a positive angle raises the +X side. A cut must be less steep than the side of the cone and must not reach the other cut.

Development. Each line of the cone is developed at its true length between the two cuts, by triangulation. The listed true lengths are those of the wall at each division, from +X half-way round.

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.
• 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.