Module sheet 

Sheets: ruled surfaces, lofts and sweeps that bound no volume.

Sections, profiles and rails are edges or wires, open or closed, planar or not. Each section edge is restated as its exact B-spline (a conic as its rational form), the edges the sections pair up are made compatible (one degree, one knot vector), and the skin interpolates their control points in homogeneous coordinates. Every section therefore lies on the skin exactly, not to a fit’s tolerance. A ruled skin is degree one across, so every ruling is a straight line. A loft with guide curves is a Gordon surface over the sections and the guides (see guided).

A sweep places copies of its profile along the path (rigid copies under a frame law, similar copies between two rails) and skins them the same way. The copies lie on the skin exactly; between them the skin departs from the swept profile, and that departure is measured against copies placed halfway, by projection, and held to ten confusions, the copies doubled until it is.

A sheet has one face per section edge (per span between neighbouring sections for a ruled loft, per spine edge for a sweep), the faces meeting on shared edges. Sections with different edge counts are matched by arc length: each section’s edges are split where the others’ breaks fall, as fractions of its length, the pieces keeping their exact form. A section edge that bounds the sheet, belongs to the model and was not split is the caller’s own edge, so the sheet sews to what it was built from. Each face’s normal is its chart’s: u runs along the sections, v across them.

Functions

make_loft_surface

A sheet lofted through sections, in order.

make_ruled

The ruled surface between two curves: each point of a joined by a straight line to the point of b at the same fraction of its parameter.

make_sweep_surface

Sweep a profile along a spine into a sheet.

make_sweep_two_rails

Sweep a profile between two rails into a sheet.