# Unrolling cylinders and cones

```{admonition} Status: outline
:class: warning

This chapter is scaffolded, not written. Section 1 and the failure modes below are real;
sections 2 to 5 are placeholders. PRD 15.3 fixes the seven-part shape used here and states
the standard: *a dedicated chapter that takes a reader from novice to able to implement* —
not a literature dump and not a proof, but a constructive path. Do not delete a section
heading to avoid filling it in.
```

## 1. The problem in one paragraph

These are the two cases with exact closed forms, and between them they cover most
real work. A cylinder's parameterization is conformal, so the unroll is literally
`(x, y) = (r * v, r * u)` and is isometric to machine precision. A cone develops into a
circular sector, which makes the template a curved band rather than a rectangle — correct,
not a bug. The trap is entirely in the setup: the cone's apex is not where the CAD kernel's
`origin` is, and using `origin` as the apex produces a template with a plausible, wrong
developed sweep angle (PRD 6.2, 6.3).

## 2. The minimum background

*To be written.* Develop the notation from scratch, defining every symbol at first use.

## 3. The derivation

*To be written.* In steps small enough that a reader can check each one individually.

## 4. The algorithm

*To be written.* Pseudocode using the real function names in this codebase, so a reader can
move from the page to the source without a translation step.

## 5. A worked numerical example

*To be written.* With actual numbers a reader can reproduce, and the expected output printed
in full. This section is what separates a chapter that can be implemented from one that can
only be admired.

## 6. Failure modes

- **Treating `Cone.origin` as the apex.** It is a point on the axis, and
  `Cone.radius` is the radius *at that origin*. Compute
  `apex = origin - (radius / tan(half_angle)) * axis`.
- Not unwrapping the circumferential angle continuously along the ordered curve, so the
  template jumps by a full circumference at the seam (PRD 7.3).
- **Unwrapping, and then trusting the window it leaves** (PRD 7.15). Unwrapping fixes a
  *branch* of the angle, not an *origin*: it anchors the whole run on the curve's first
  sample, which is wherever the CAD kernel's edge happens to start. The reference head-tube
  cope came out spanning `-450°` to `-90°` — every point correct relative to every other,
  and the datum line drawn one full circumference off the sheet. Since adding a whole
  period moves no point on the surface, the window is free to choose and must therefore be
  chosen on purpose: `mighty_miter.core.unroll.cylinder.canonical_wrap` puts `x = 0` on the
  datum, re-indexing a closed curve so a full wrap runs `[0, 2πr]`.
- Failing to guard the near-cylindrical cone, where `sin(half_angle) -> 0` puts the apex at
  infinity. The fallback to the cylinder case must be triggered by a tolerance derived from
  the accuracy budget, not by a hard-coded angle.

## 7. References

- **do Carmo, M. P. (1976)**, chapter 4-2 (isometries and developments). The
  formal statement that these developments are isometries.