How to lay out a sphere or dome in zones by hand

1. Divide the quarter circle

Draw the half-section of the dome: a quarter circle of the sphere’s radius, 500 here, with the vertical axis through its centre. Neubecker’s rule is to divide the quarter circle into as many parts as there are pieces (p. 23). Four pieces give parts of 90° ÷ 4 = 22.5°; the top part becomes the crown disc and the three below it the zones. Join the division points with straight chords, each 2 × 500 × sin 11.25° = 195.09 long.

2. Produce each chord to the axis

Extend every chord until it meets the axis. That point is the centre of that zone’s pattern, and its distances to the chord’s two ends are the pattern’s outer and inner radii — the same construction in Neubecker (p. 24), Daugherty (p. 152), Teschmacher (pp. 21–22) and Kittredge (p. 251). Worked out from the plan radii r and r′ at the two ends: outer radius r × chord ÷ (r − r′), inner radius r′ × chord ÷ (r − r′).

3. Lay out each zone

Swing the two radii from one centre and mark off the outer arc equal to the girth of the zone’s lower joint — the books step it off the plan circle; computing it is more accurate, or use the sector angle 360° × (r − r′) ÷ chord:

Zone 1, next to the base, is nearly a cylinder, so its centre is far away: 2562.92 mm. Measure that long radius with a trammel, or fix the ends of the arcs by their straight chords — 2948.57 across the outer arc, 2724.12 across the inner — and draw the arcs through them with a thin batten.

4. Cut the crown disc

Neubecker, Daugherty and Teschmacher cut the crown as a flat disc whose radius is the top chord, 195.09 here. By our own arithmetic that disc has exactly the area of the cap it covers, so it can be raised to the curve. Kittredge uses the plan radius of the top joint, 191.34, which fits the joint flat but has less area than the cap.

5. Raise and join

The zones run on the chords, inside the sphere: at mid-zone each is 9.61 mm inside the curve and is raised by that much; the crown rises 38.06 mm. The books add laps for soldering or brazing (Neubecker, Daugherty, Teschmacher); add them to the patterns above, which have none.

Common mistakes

• For a shallow dome, using the span across its base as the sphere’s diameter. Work from the sphere the dome is part of.
• Stepping the girth off the plan circle with too few divider spaces: each step is a chord, shorter than its arc, so the zone comes out short round the joint.
• Raising the crown as little as the zones: it rises about four times as far.
• Forgetting the laps.

How this matches the tool

The sphere and dome in zones calculator does the same for any sphere, rise and number of pieces, and exports every zone and the crown disc as DXF. The worked numbers on this page were produced by its code when the page was built.

Sources

1. W. Neubecker, Sheet-Metal Work (1920), pp. 23–24: divide the quarter circle into as many parts as sections; chords produced to the axis; the top blank’s radius is the chord
2. Neubecker (1920), pp. 105–107: the same rule worked again; laps for brazing or soldering (p. 107)
3. J. S. Daugherty, Sheet-Metal Pattern Drafting and Shop Problems (1922), pp. 150–153: zones preferred to gores; chords extended to the axis; crown disc of the chord radius; soldered, then raised
4. E. Teschmacher, Practical Sheet Metal Work, Vol. VI (1911), pp. 21–22: “It is best to make them from zones”; crown of the chord radius
5. G. W. Kittredge, The New Metal Worker Pattern Book (1901), pp. 251–252: the chord rule, divisions shifted half a step, crown of the plan radius
6. E. A. Atkins, Practical Sheet and Plate Metal Work, 2nd ed. (Pitman, undated), pp. 393–394: a flat dome developed on the chord produced to the axis