Developable surfaces and Gaussian curvature

Status: outline

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.