Stepped tiers of seats wrap round a circular orchestra, and a band shell of telescoping arches stands behind the stage. Each tier is a strip between two concentric arcs, extruded up to its own height; the shell is a loft through arches that grow wider and taller towards the audience.

A semicircular stepped stone amphitheatre round a round stage

Reference: choose Examples  Stepped amphitheatre on the canvas toolbar to load the finished definition.

Build it

The tiers

  1. Place nine sliders: Tiers: 12 (4 – 24) with Decimals 0, Inner radius: 14 (6 – 30), Tread: 2.2 (1 – 4) and Riser: 0.9 (0.3 – 2) with Decimals 2, Sweep: 210 (90 – 270) and Shell bands: 8 (2 – 16) with Decimals 0, Shell span: 24 (10 – 40), Shell height: 12 (5 – 25) and Shell depth: 10 (4 – 20).

  2. Place two Expression (Maths) components with Sweep → x, sin(radians(x/2)) and -cos(radians(x/2)), and a Multiplication (Maths) of the first by -1.

  3. Place three Construct Point (Vector) components: (−sin, −cos), (0, -1) and (sin, −cos), and an Arc 3Pt (Curve), Unit arc, through them: an arc of radius 1 centred on the stage.

  4. Place a Series (Sets), Tier radii, with Inner radius → S, Tread → N and Tiers → C, and an Addition (Maths) of the radii and Tread.

  5. Place two Scale (Transform) components on the unit arc, one by the radii and one by the radii plus a tread: the inner and outer edge of every tier.

  6. Place a Ruled Surface (Surface), Treads, between them: one flat strip per tier.

  7. Place a Series, Tier heights, with Riser as both S and N and Tiers → C, a Vector XYZ (Vector) with it as Z, and an Extrude (Surface), Tiers: each strip raised into a block one riser higher than the one inside it.

  8. Place a Multiplication, Inner radius × 0.8, and a Cylinder (Surface), Orchestra, of that radius, one Riser high.

The band shell

  1. Place a Range (Maths) from 0 to 1 with Shell bands → N.

  2. Place three Expression components on it: Arch half-span y*(1 + x)/4 with Shell span, Arch height y*(0.45 + 0.55*x) with Shell height, and Arch position y*(1 - x) with Shell depth; and a Multiplication of the half-span by -1.

  3. Place three Construct Point components, (−half-span, position, 0), (0, position, height) and (half-span, position, 0), and an Arc 3Pt, Arches, through them.

  4. Place a Loft (Surface), Band shell, and a Pipe (Surface), Ribs, with R 0.3, both on the arches.

  5. Turn Preview off on the points, arcs, Scales and Treads.

The amphitheatre definition on the canvas
Figure 1. The finished definition
The red preview of the amphitheatre
Figure 2. The preview in FreeCAD’s 3D view

What is happening

Scaling one unit arc about its centre gives concentric arcs that all span the same angle, so each pair bounds a strip of an annulus. Ruled Surface joins matching points of two curves with straight lines, which for two concentric arcs in the same plane is exactly that flat strip. The geometry tests check every tier’s volume against the area of its strip, (Sweep/360)·π·(R2 − r2), times its height, and that the shell’s arches stand on the ground inside the orchestra.

The shell’s arches run from behind the stage (narrow and low) to the middle of the stage (wide and high). Lofting through them gives a flaring hood that opens towards the seats. The whole theatre updates in about a quarter of a second.

Things to try

  • Set Sweep to 270 for a theatre that almost surrounds its stage, or to 120 for a shallow fan.

  • Set Riser to 1.5 and Tiers to 24 for a steep, high bowl.

  • Set Shell bands to 16 for a finely ribbed shell.