# The first fundamental form

```{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

Unrolling a tube means finding a map from its surface to flat paper that keeps
lengths right. Every length measured on a parametric surface `P(u, v)` is determined by three
numbers at each point — `E = P_u . P_u`, `F = P_u . P_v`, `G = P_v . P_v` — and by nothing
else. That is the whole reason this codebase has one abstraction rather than five unrelated
algorithms: each unrolling tier is a different answer to *integrate this metric*, they share
one input and one output, and so they can be tested against each other (PRD 6.0, ADR-004).

## 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

- Assuming `F = 0`. It is zero for the three analytic surface types measured
  inside Fusion, and that is what keeps the closed forms clean, but it is not a general
  truth and must not be assumed for a NURBS face.
- Confusing the metric's units. `E` is millimeters squared *per unit of `u` squared*, and
  `u` is not always a length: for a cylinder it is `z / r`, dimensionless.
- Treating a degenerate metric as a small one. `EG - F^2 = 0` means the parameterization has
  collapsed — at a cone apex, or a pole — and no amount of tolerance makes the local frame
  meaningful there. `Metric.is_degenerate` exists for this.

## 7. References

- **do Carmo, M. P. (1976)**, *Differential Geometry of Curves and Surfaces*,
  chapter 2-5. The standard treatment. Cited by chapter so any edition, or an equivalent open
  text, can be used. Read it for the definition and for why the form is intrinsic.