Sphere: flat pattern
A sphere made of plates: gores from pole to pole, or zones round it, with a round cap at each pole. A sphere made of plates: gores from pole to pole, or zones (bands) round it, with a round cap at each pole.
Open the sphere in SheetMorphConfigurations
- Gores. Equal gores between meridians, running from one polar cap to the other. All gores are alike.
- Zones. Bands between circles of latitude, each developed as part of a cone.
A worked example
The gores configuration with the sizes SheetMorph opens with, in mm. 12 gores, all alike, and two round caps: 12 parts, 0 seams.
| Blank of G1 | 2712.6 mm × 523.9 mm |
| Blank of G2 | 2712.6 mm × 523.9 mm |
| Blank of G3 | 2712.6 mm × 523.9 mm |
| Blank of G4 | 2712.6 mm × 523.9 mm |
| Blank of G5 | 2712.6 mm × 523.9 mm |
| Blank of G6 | 2712.6 mm × 523.9 mm |
| Blank of G7 | 2712.6 mm × 523.9 mm |
| Blank of G8 | 2712.6 mm × 523.9 mm |
| Blank of G9 | 2712.6 mm × 523.9 mm |
| Blank of G10 | 2712.6 mm × 523.9 mm |
| Blank of G11 | 2712.6 mm × 523.9 mm |
| Blank of G12 | 2712.6 mm × 523.9 mm |
| Length of cut, every outline and hole | 68585.3 mm |
| Diameter of the sphere | 2001.00 mm |
| Area of the whole sphere | 12,578,940 mm² |
| Latitude of the edge of the caps | 78.46 ° |
| Diameter of the caps, across | 400.20 mm |
| Flat blank of a cap, diameter | 402.92 mm |
| Number of gores | 12 |
| Width of a gore at the equator | 523.86 mm |
| Length of a gore, along its middle | 2740.25 mm |
| Width of a gore at its ends | 104.77 mm |
| G1 Gore, long | 2712.58 mm |
| G1 Gore, wide | 523.86 mm |
| Plates | 12 |
| Triangular facets | 768 |
| Pattern extent X | 2712.58 mm |
| Pattern extent Y | 7726.94 mm |
| Developed area | 12,223,470 mm² |
| Total cut length | 68585.29 mm |
| Longitudinal seam length | 0.00 mm |
How it is developed
Gore (polycylindric) development. 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 gore is laid out flat across and curved along: at every height its width is the girth of the sphere there divided by the number of gores. All gores are alike, so one pattern is cut the number of times shown.
- A gore is pressed or rolled across its width to the curve of the sphere after cutting; the more gores, the less it has to be stretched.
- A round cap closes each pole. It is not drawn: its flat blank is listed.
Checked before it can be exported
Every pattern SheetMorph draws is checked against the 3D part it comes from. For this example:
- Surface connectivity (passed): 768 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): 2304 facet edges compared between the 3D surface and the flat pattern (max deviation 9.81e-13).
- Shared edges and seam closure (passed): 756 internal edges are shared vertex-for-vertex between adjacent facets; 0 seam(s) have mating edges of equal true length (max deviation 0).
- 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 1.68e-8).
- Cut outlines closed and non-overlapping (passed): 12 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 the thickness.
- Curves are chains of chords between division points.
- No seam, lap, weld-preparation or pressing allowance is added.
Limits of this module
- A sphere cannot be laid flat: gores and zones are plates cut flat and then pressed or rolled to the sphere, and the pattern gives their size before pressing.
- The caps are not drawn; the diameter of the flat blank of a cap is listed.
- Gores are equal and run the whole way between the caps; a sphere of zones each divided into petals is not modelled.
- Curves are chains of straight chords between division points.
- No seam, lap, weld-preparation or pressing allowance is added, and plates are not nested onto stock sheets.
Conventions
Coordinate system. The part stands on the Z axis with End A at Z = 0. Angles round the axis are measured from +X toward +Y, counter-clockwise seen from +Z.
Divisions and thickness. A circle is divided into 4N equal chords. 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.
Sphere. The sphere stands on the Z axis. A round cap plate closes each pole; End A and End B are the edges of the two caps, and the gores or zones lie between them. Meridian j is at j x 360 / gores degrees from +X.
Gores and zones. A gore is flat across and curved along: its width at any height is the girth of the sphere there divided by the number of gores, and its length is the arc between the caps. A zone is the band of a cone through two circles of latitude, which are equally spaced in latitude between the caps.
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.
Open the sphereOther vessels
- Shell with Nozzles: A cylindrical shell unrolled to a rectangle, with the holes for up to four radial nozzles marked in their developed shape.
- Tank Shell: The shell of a storage tank or silo in courses and plates from stock, the vertical seams staggered, with a nozzle or manway marked on its plate.
- Silo: A silo, bin or hopper as one job: the body, the cone bottom from its wall angle, the spout, a support skirt and a cone roof, with what it holds.
- Reinforcing Pad: The pad welded round a nozzle on a shell or a pipe: rolled to the shell, with the hole for the nozzle and a tell-tale hole, whole or in two halves.
- Dished Head: A vessel head in petals round a crown plate: torispherical, hemispherical, semi-elliptical (2:1), or a cone with a knuckle.