Module pipe 

Swept disks along chains of straight segments and circular arcs (#232).

A pipe run, a rebar or a cable tray edge is a disk swept along legs joined by bends: authoring formats write it as a disk swept along a composite of lines and arcs, or along a polyline whose corners are rounded to a fillet radius. swept_disk_along_pieces meshes both so that every point of the exact tube lies within the chord budget of the triangles.

The bound

Each piece is bounded on its own, with the split crate::sweep uses for one arc (#231): half the budget c chords the disk, so a ring of n points inscribed in the circle of radius r lies within r (1 - cos(pi/n)) <= c/2 of it, and every point of the exact tube is within c/2 of the tube the ring polygon traces when it is carried rigidly along the exact centreline by the piece’s own frames (each exact section circle and its polygon lie in the same plane). The other half bounds that polygon tube against the triangles:

• Along a segment the frames do not turn, so the polygon tube is a prism whose walls are the mesh’s own planar rectangles: zero.
• Along an arc of radius R the frames are rotation-minimising, which along a circle means they turn with it: consecutive stations are rotations of one ring about the arc’s axis, and the ring’s plane contains that axis. The polygon tube between two stations is then the revolution crate::revolve bounds, and its triangles are within (R + r)(1 - cos(phi/2)) of it for a step phi: the sagitta of the outer side of the bend, not of the centreline. Its wall quads are planar trapezoids, so their twist adds nothing; the span bound crate::sweep computes, directrix sagitta plus the section’s reach plus crate::loft’s quad bound, is exactly that sum here. The step halves until every span fits c/2; more than 4096 steps on a bend is refused with BudgetExceeded.

Joints

Consecutive pieces share one station, so the walls meet watertight and wound one way: the incoming piece’s end section. The outgoing piece’s own start frame is that frame turned by the least rotation that takes the incoming tangent to the outgoing one, moved to its start point, and its stations follow from it. Where the pieces are tangent continuous and meet, the two frames coincide. Where they turn by theta or leave a gap g, every vertex of the shared station lies at most delta <= g + 2 r sin(theta/2) from the station the outgoing piece’s bound was proved for; moving a triangle’s vertices by at most delta moves each of its points by at most delta (same barycentric weights), so the first span of the outgoing piece adds the measured delta to its bound and is refined until the sum fits.

A swept disk along a directrix that is not tangent continuous is undefined (the section has no single plane at a corner), so a joint turning by more than

theta_max = 2 asin(c / (8 r)) (pi once c >= 8 r)

is a corner and is refused by name: at that turn the shared section leans off the outgoing piece’s own by a quarter of the budget, and any larger turn would be a kink no chord budget can hide. A turn at or below it is tangent continuous at the requested accuracy and costs at most c/4 of the directrix’s half. A gap above c/4 is refused too.

Fillets

A fillet radius f rounds every corner between two straight segments (the swept disk’s fillet_radius): a corner turning by theta is replaced by the arc of radius f tangent to both segments, which cuts f tan(theta/2) off each. Segments shortened to nothing are dropped (two fillets then meet tangentially). Refused by name: a fillet that needs more of a segment than it has, a corner beside an arc (only corners between straight segments are rounded), a disk radius at or above the fillet radius, or any bend radius, whose tube folds through the bend’s axis on its inside (equality is a horn torus, which some formats permit but which bounds no two-manifold solid), and a reversal.

Enums

PathPiece

One smooth piece of a swept disk’s directrix.

Functions

joint_tolerance

The largest turn a joint may make and still be tangent continuous at chord budget chord for a disk of radius radius: 2 asin(c / (8 r)) (see the module notes).

swept_disk_along_pieces

Sweep a disk, optionally hollow, along a chain of segments and arcs so that every point of the exact tube lies within chord of the mesh.