The frame is parallel-transported, not rebuilt per point from a fixed
up-vector: an initial normal perpendicular to the first tangent is carried
along the path by projecting it onto each new tangent's plane and
renormalizing. A Frenet frame would flip the tube inside out wherever the
curvature vanishes (a straight run) or reverses (an inflection), which is
exactly what a cable, a pipe run, or a §119 winding path is full of.
The tube is open at both ends: capping a swept path means answering what a
cap looks like at an arbitrary orientation, and a caller who needs closed
ends adds a sphere. Normals are the analytic radial direction in the local
frame. Uv is u around the tube and v = i / (path.length − 1) along it.
A circular cross-section swept along a 3D polyline (§53 "tube").
The frame is parallel-transported, not rebuilt per point from a fixed up-vector: an initial normal perpendicular to the first tangent is carried along the path by projecting it onto each new tangent's plane and renormalizing. A Frenet frame would flip the tube inside out wherever the curvature vanishes (a straight run) or reverses (an inflection), which is exactly what a cable, a pipe run, or a §119 winding path is full of.
The tube is open at both ends: capping a swept path means answering what a cap looks like at an arbitrary orientation, and a caller who needs closed ends adds a sphere. Normals are the analytic radial direction in the local frame. Uv is
uaround the tube andv = i / (path.length − 1)along it.