Width
Claim
The shape is 2r wide — which is also d, the diameter of the founding circle.
Construction
The widest span across the shape is the head circle's own diameter. Nothing else reaches further.
The diamond's corners lie at distance r from its centre, so corner to corner it is 2 × r/√2 = r√2 ≈ 1.414r across — well inside the head's 2r.
Proof
Every point of the head is within r of its centre, so the head spans exactly 2r horizontally, achieved at the two ends of the horizontal diameter. Every point of the diamond is within r√2/2 ≈ 0.707r of the vertical axis. Since 0.707r < r, the head governs, and the width is 2r = d.
On naming
It is worth fixing d once. Throughout this library d means the diameter of the circle the shape is built from, and equally the width of the shape itself. That coincidence is what lets the area claim be stated as A = d2 rather than A = 4r2, and the second form is the one worth remembering.
Note
Aaron's slot.
Open questions
- Whether r or d should be the library's working unit. The geometry is stated in r and the headline claim in d, and no page has chosen.
Checks
| What | How | Source |
|---|---|---|
| Width of the exact path | Bounding box of the sampled boundary: exactly 200.000 at r = 100 | _src/core.py |
| The diamond is narrower | r√2 ≈ 141.42 against the head's 200 | reference/geometry-constants.md |