# Channel surfaces and the r/R bound

```{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

A bent tube of constant section is a channel surface: the envelope of a sphere of
fixed radius moving along a spine. It is not developable, so a template for it necessarily
carries strain — but the strain has a closed form. Writing the surface in a
rotation-minimizing frame gives a metric whose circumferential term is
`h = 1 - r * kappa * cos(theta - theta_0)`, from which the circumferential wrap is exactly
`2 * pi * r` and the maximum strain is exactly `r / R`, attained on the back of the bend. That
is a bound a fabricator can act on, and it is why Tier D is worth having as a separate tier
(PRD 6.5).

## 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

- **Quoting the `r / R` bound for a face that does not qualify.** The derivation
  assumes a constant section radius. A face whose section morphs from circular to elliptical
  has no single `r`, and a confident wrong number is worse than an honest measured one. The
  constant-radius gate is mandatory; `AnalyticBoundUnavailableError` is what to raise.
- Recovering the spine badly. It is the locus of fitted section centers, and a fit that
  includes points beyond the trim boundary pulls the center off.
- Forgetting that the back edge of the bend has a genuine misfit, not a numerical error.

## 7. References

- **Pottmann, H., Wallner, J. (2001)**, *Computational Line Geometry*,
  Springer. Chapter on channel and canal surfaces. The reference for the geometry; read the
  definition and the envelope construction.