# Theory

Complex subjects get a dedicated chapter that takes a reader **from novice to able to
implement** — not a literature dump, and not a proof: a constructive path. Every chapter
follows the same seven-part shape (PRD 15.3):

1. the problem in one paragraph, in physical terms;
2. the minimum background, developed from scratch with notation defined at first use;
3. the derivation, in steps small enough to check individually;
4. the algorithm as pseudocode matching the real function names in this codebase;
5. a worked numerical example with actual numbers a reader can reproduce;
6. failure modes, and how the code guards them;
7. references, open-access preferred, each annotated with *why you would read it*.

Most chapters below are outlines: section 1 and the failure modes are written, the derivation
and the worked example are not. Each chapter is filled in by the milestone that implements it,
because a derivation written before the code is a guess and a worked example written before
the code has no numbers to quote.

```{toctree}
:maxdepth: 1

first-fundamental-form
developable-surfaces
unrolling-cylinders-and-cones
rotation-minimizing-frames
channel-surfaces
general-unfolding
strain
writing-a-pdf-by-hand
fusion-api-field-notes
```