Module chain 

Arc length along any evaluable curve and its inverse, and arc-length chains (#239), re-exported on the same footing as curve and surface. Evaluation of arc-length chains (Chain2).

A chain’s parameter is cumulative arc length. Each piece is evaluated in its own local frame and placed rigidly at the end of the one before it, so reading a point resolves every earlier piece’s end first:

• an ChainPiece2::Intrinsic piece through intrinsic_point and intrinsic_tangent: heading exact, position Gauss-Legendre quadrature, as for Curve2::Intrinsic;
• a ChainPiece2::Parametric piece through the arc-length inverse of crate::arc_parameter: the curve parameter whose arc length from start is the distance asked for, to ARC_LENGTH_TOLERANCE relative.

Accuracy contract

A point at distance s is the exact chain’s point at a distance within ARC_LENGTH_TOLERANCE * max(1, piece length) of s on each parametric piece it crosses, and within the intrinsic quadrature’s accuracy on each intrinsic piece (machine precision on the closed forms it is pinned against, ADR 0060). The tangent is the exact unit tangent at that point: a parametric curve’s derivative or an intrinsic piece’s closed-form heading. Positions after a parametric piece carry its end error, rotated by the remaining placement; nothing is re-fitted.

Refused by name

A malformed chain (no pieces, a non-finite or degenerate start frame, a non-finite or non-positive length, a malformed curvature law), a parametric piece that does not start at its local origin with tangent +x within PIECE_FRAME_TOLERANCE, a piece longer than its curve (named with the arc length available), and a distance outside [0, length].

Constants

PIECE_FRAME_TOLERANCE

How far a parametric piece’s curve may sit from its local frame at its start parameter: PIECE_FRAME_TOLERANCE * max(1, length) from the origin, and PIECE_FRAME_TOLERANCE radians from +x.

Functions

chain_certifies

Whether crate::curve::flatten2 certifies a chain’s chords: every intrinsic piece’s curvature is bounded and every parametric piece’s curve is a certified family with no corner anywhere in its domain (#232).

chain_chord_bound

Certified chord bound of a chain over arc lengths [a, b] inside one piece (#232); None across a join, for a span outside the chain, a piece whose curve has no certified bound, or a chain that does not evaluate.

chain_point

Point on a chain at arc length s from its start. See the module documentation for the accuracy contract and refusals.

chain_tangent

Unit tangent of a chain at arc length s (unit when the start frame is orthonormal; a skewed frame maps it as it maps an intrinsic curve’s).