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.

• The first fundamental form
• Developable surfaces and Gaussian curvature
• Unrolling cylinders and cones
• Rotation-minimizing frames
• Channel surfaces and the r/R bound
• General unfolding and least-squares flattening
• Strain: definition, measurement, meaning
• Writing a PDF by hand
• Fusion API field notes