# Developable surfaces and Gaussian curvature

```{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 surface can be flattened onto paper without stretching exactly when its
Gaussian curvature is zero everywhere. That is Gauss's *Theorema Egregium*: curvature is
intrinsic, computable from the metric alone, and therefore preserved by any length-preserving
map. Cylinders, cones and generalized cylinders qualify; a bent tube does not, and no
algorithm can make it. The practical consequence is that classification is a statement about
the measured metric, never about the type name the CAD kernel reports (PRD 6.1, ADR-005), and
that a non-developable face must come with an honest strain figure attached (FR-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

- Classifying by type name. A face Fusion calls a NURBS surface may be exactly
  cylindrical, and a face it calls a cylinder may have been trimmed to something that is not.
- Choosing the developability threshold by feel. The tolerance `epsilon` must be derived
  from the FR-4.1 accuracy budget, not picked because it looked reasonable.
- Averaging curvature over a whole face. A real tube face can run from near-circular to
  strongly oval along its own length; a single number for the face is a fiction (PRD 7.12).

## 7. References

- **do Carmo, M. P. (1976)**, *Differential Geometry of Curves and Surfaces*,
  chapter 4-1. The Theorema Egregium with the proof. Read it once to believe the result, then
  never again.
- **Tang, C., Bo, P., Wallner, J., Pottmann, H. (2016)**, "Interactive design of developable
  surfaces", *ACM TOG* 35(2). <https://doi.org/10.1145/2832906>. Treats developability as a
  numerical condition rather than a classification, which is exactly what the classifier of
  PRD 6.1 needs.