Module pipe 

Swept disks along chains of straight segments and circular arcs (#232, #245).

A pipe run, a rebar or a cable tray edge is a disk swept along legs joined by bends or corners: authoring formats write it as a disk swept along a composite of lines and arcs, or along a polyline whose corners are mitred or rounded to a fillet radius. swept_disk_along_pieces meshes all of them 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 joint turning by at most

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

is tangent continuous at the requested accuracy: its shared section leans off the outgoing piece’s own by at most a quarter of the budget, which that piece’s first span pays as above. A gap above c/4 is refused.

Mitres

A sharper joint between two straight segments is a corner, and a disk swept round a corner is mitred at half angle (#245): both tubes are cut by the plane through the corner whose normal n = (u + v) / |u + v| bisects the incoming and outgoing tangents u and v. Reflection in that plane maps u to -v, so it swaps the two axis lines and with them the two cylinders of radius r (and the two bores); it fixes the plane, so both cylinders cut it in the same ellipse. The outgoing frame is the incoming one turned by the least rotation about u x v, which agrees with that reflection on the plane square to u. So profile point k of the incoming end section and of the outgoing start section, each projected along its own axis onto the mitre plane, land on one point: the two pieces share that mitre ring, and every vertex of it lies on both exact cylinders. Each wall quad of a segment joins two generator lines of the prism over the ring polygon, so it is a planar trapezoid, exactly a face of that prism cut by the end planes: the walls add nothing, and the span bound measures them as zero.

The section pays instead. A point w + s u of the exact cylinder (w square to u, |w| = r; the mitre plane cuts its generator at s_cut(w) = -w.n / u.n, at most r tan(theta/2)) lies within (1 - l) r <= e = r (1 - cos(pi/m)) of the prism point w' + s u at its azimuth, w' = l w on the ring polygon of m points. When that point lies past the plane, the plane crosses the prism’s generator at l s_cut(w), which moves the point by at most (1 - l) s_cut(w) <= (1 - l) r tan(theta/2) along u, square to the first move: the distance is at most e / cos(theta/2), attained where the ellipse’s major axis ends. The rings are therefore chorded to (c/2) cos(theta_m/2) for the sharpest mitre theta_m (only finer elsewhere), and a mitred pipe keeps every point of the exact tube within c of its triangles, as a smooth one does. A hollow disk’s bore is mitred by the same plane and bounded the same way.

Along a segment of length L the tube runs, on the generator at w, from its start plane to its end plane; both cuts are linear in w, so the shortest generator is L - r |g_perp| with g = n1 / u.n1 - n0 / u.n0 (n0 = u at a square end), and it must be positive (one mitre: L > r tan(theta/2)). Refused by name: a segment whose mitres meet inside the tube, which would cut through itself; a reversal (theta = pi, where the mitre plane holds both legs); and a corner beside an arc, because the mitre plane cuts the torus round an arc in a different curve than the tube on its other side, so no ring lies on both exact surfaces and the bound above does not carry over.

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 above the fillet radius, or any bend radius, whose tube folds through the bend’s axis on its inside, and a reversal. A disk radius equal to the fillet (or bend) radius is refused by name too, although the format rule permits it (fillet radius at least the disk radius): the bend is then a horn torus whose inner wall pinches to one point on the bend’s axis, where every section touches. That boundary is not a two-manifold; a closed mesh within the budget would have to either meet itself there (degenerate triangles) or stand off the pinch and misstate it.

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.