# Glossary

Every domain term is defined once, here, and linked to rather than redefined inline
(PRD 15.5). The definitions match PRD section 4.

```{glossary}
Host face
  The face being unrolled — the outer surface of the tube being cut. Resolving it is
  ambiguous when a severing cut produces two faces of identical radius and near-identical
  area, which is why the add-in asks rather than picking the larger one.

Miter curve
  The ordered 3D polyline of the cut edge or edge chain lying on the {term}`host face`.
  Closed for a full cope, open for a partial one; both are in scope.

Datum
  The reference defining circumferential zero, and optionally axial zero. Everything about
  a template's rotational position is measured from it.

Unrolling map
  A map from points on the {term}`host face` to 2D {term}`template coordinates`. Isometric
  — length-preserving — where the surface allows it.

Template coordinates
  The 2D frame a template is drawn in. `x` is arc length around the tube from the
  {term}`datum`, increasing counter-clockwise viewed along the tube axis in the direction of
  the outward-pointing end. `y` is axial distance, increasing away from the cut end. The
  origin is the datum position at the axial reference. This convention is fixed: getting the
  handedness wrong produces a template that looks correct and fits nothing.

Strain
  Relative length distortion introduced by the {term}`unrolling map`. Zero for a developable
  surface. Reported as a measured field, never as a whole-face average, because a real tube
  face can vary from near-circular to strongly oval along its own length.

Developable surface
  A surface with zero Gaussian curvature everywhere, which can therefore be flattened into
  the plane without stretching. Cylinders, cones and generalized cylinders are developable;
  a bent tube is not.

First fundamental form
  The three quantities `E`, `F`, `G` that govern every length measured on a parametric
  surface. Every unrolling {term}`tier` is an answer to "integrate this metric", which is
  what makes the tiers mutually testable.

Template
  A named 2D artifact: the {term}`miter curve`, datum marks, annotations and calibration
  geometry, drawn at 1:1.

Template set
  A named, ordered collection of {term}`template`s exported together, sharing print settings.

Tier
  One of five unrolling algorithms, A through E, chosen by numerical classification of the
  measured metric — never by the type name the CAD kernel reports.

Channel surface
  The surface swept by a sphere of varying radius moving along a spine curve. A bent tube of
  constant section is the case that matters here, and it is what Tier D unrolls.

Rotation-minimizing frame
  A frame carried along a curve that does not twist about the tangent. Also called a Bishop
  frame. Used instead of the Frenet frame, which flips at an inflection point and would put
  a visible discontinuity in a template of an S-bend.

Calibration geometry
  A rectangle of known size printed on every {term}`template`, so a user can measure it and
  confirm the print came out at 1:1 before cutting anything.
```