Channel surfaces and the r/R bound

Status: outline

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.