Strain: definition, measurement, meaning

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 template for a non-developable surface cannot be exact, and the user is entitled to know by how much and where. Strain here has one definition across every tier: the relative discrepancy between a distance measured on the surface and the same distance measured on the template, along the same parametric direction. The number that matters to someone holding a tube is not an average — it is the worst local value, where it occurs, and which way it errs.

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

  • Reporting a whole-face average. A real face in the reference model runs from 0.03 % to 60.8 % ovality along its length; an average of that describes nothing (PRD 7.12).

  • Labelling a measured strain as a bound, or a bound as a measurement. FR-5.1a requires the provenance to travel with the number.

  • Producing a template with no strain figures attached at all (FR-10.2).

7. References

  • do Carmo, M. P. (1976), chapter 4-2. Isometry and the precise sense in which a non-developable surface cannot have one.