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Sphere zone band pattern calculator

A band of a sphere between two parallel planes, made from a stack of shallow cones. Enter the sphere diameter, both plane heights and the number of cones; you get one ring pattern per band.

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1:1 PDF and DXF files of this flat pattern are made in Trueflat, ready for print, plasma, laser or waterjet. It works offline on site.

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7,631.9 mm1,248.7 mm
Flat pattern: cut on the bold outline, bend on the dashed blue lines, the thin lines are marking lines

Seam

309.9mm

Approximate, within ±0.9 mm

Cut length
In the app
Sheet size
In the app

Every result is in Trueflat

The free web version shows the main result and the drawing. Trueflat gives you every mark and measurement, and exports a cut sheet. It is free to download.

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How to lay it out

  1. Divide the arc

    In the elevation, divide the sphere's outline between the two planes into equal angular steps, one for each cone.

  2. Draw the chords

    Join neighbouring division points with straight lines. Each line is the side of one cone, and all have the same length.

  3. Extend each chord to the axis

    Extend each side line to the axis to find that cone's apex. The distances from the apex to the two ends of the side line are its pattern radii.

  4. Swing each ring

    For each band, swing the two arcs and step off the circumference of its larger circle on the outer arc. A band that sits exactly on the equator is a straight strip.

  5. Number the pieces

    Mark each piece with its band number and the edge that joins the next band, then add allowances.

Good to know

  • This pattern is approximate; the result card shows how close it is.
  • Source: Kittredge, The New Metal Worker Pattern Book (1917), Problem 124 "To Construct a Ball in any Number of Zones", Figs. 452–458, p. 252. The book does not say whether the cones lie inside or on the sphere; Trueflat puts their ends on the sphere

Method and formula

A sphere cannot be unrolled, but a band of it can be approximated by truncated cones whose ends lie on the sphere. Each cone unrolls exactly; the approximation is only in replacing the curved profile with straight lines.

  • Cone side c = 2R sin(Δφ ÷ 2), the same for every band, where Δφ is the angular step.
  • Pattern radii c × ρ ÷ Δρ for each end, where ρ is the ring radius and Δρ the difference between a band's two ring radii.
  • Distance from the sphere at most R(1 − cos(Δφ ÷ 2)), at the middle of each band.
  • Area short of the true zone by the factor cos(Δφ ÷ 2).

Plane heights are measured from the sphere's centre, negative below it. Each ring must keep a radius of at least 5 mm, so a zone cannot reach a pole.

Worked example

D = 200 mm, planes at −30 and 50 mm, 4 cones: angular step 11.86°, cone side 20.67 mm. The bottom band has radii 477.44 and 498.11 mm and a sector angle of 71.93°; the top band 233.03 and 212.36 mm, 146.81°. The bands stay within 0.54 mm of the sphere, and their area is 0.9946 of the true zone.

Questions

How accurate is a zone pattern?

Each cone is exact, but a stack of cones only approximates a sphere. The bands stay within R(1 − cos(Δφ ÷ 2)) of the sphere, 0.54 mm in the 200 mm example with 4 cones. Doubling the number of cones cuts that to about a quarter.

Zones or gores: which should I use?

Zones roll like plain cones and need no pressing, but show small ridges at the joints. Gores need forming but give a smoother surface and can close at the poles. Use zones for a band, a tank section or a dome you can stack in rings.

How do I enter the plane heights?

As distances from the sphere's centre: positive above, negative below. The lower plane must be below the upper one and both must be inside the sphere.

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