# Strain: definition, measurement, meaning

```{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 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.