Module volume_difference 

The volume between a scanned mesh and the solid it was made to: the material the solid has and the scan lacks, and the material the scan has beyond the solid, as connected regions.

The grid

Space over both bodies’ boxes is cut into cubic cells cell wide, and each cell is read at its centre, its node. A node is in the scan by the parity of the scan’s crossings along the line through it parallel to x (parity), exactly, and on the scan counts as in it. A node is in the solid by the same parity against the solid meshed at a sixteenth of a cell. Where the node lies within that chord, plus the confusion distance, of the solid’s mesh, the band where the mesh and the solid can disagree, it is read again against the solid’s own surfaces by SolidBoundary::holds, and on the solid counts as in it. A node in the solid and out of the scan is missing material; one in the scan and out of the solid is excess material.

Holes in the scan

Vertices of the scan at one position are made one first, so a mesh stored as loose triangles reads as the surface they make. The parity needs every edge used an even number of times; a scan with an edge used an odd number of times, a hole’s rim, is closed first, each rim filled flat with Editable::fill on its own vertices. A differing node whose nearest point on the closed scan lies on a fill was not seen by the scanner: it takes the solid’s side, so a hole reports no material either way.

Such a node is doubtful, and so is every node within a cell of a fill, where the fill may stand where the scan would have, and a node the solid’s surfaces could not classify, every ray meeting a degeneracy, which keeps the reading of the solid’s mesh.

Regions

Nodes that differ the same way are joined across faces, edges and corners into regions. A region’s volume is its nodes’ count times a cell’s volume. A node is counted where its centre lies, so the volume is off by at most the cells the region’s boundary passes through: about the boundary’s area times √3 cells, and less where the boundary’s errors cancel, as they do on a surface not aligned to the grid.

A region’s depth is how far apart the two surfaces stand across it: the greatest distance, from a vertex of either surface within a cell of the region’s cells, to the other surface. A vertex of the scan is measured to the solid’s mesh, within its chord of the solid, and a vertex of the solid’s mesh to the closed scan. A boss standing off a face reads its height, from its top; a pocket its depth, from its floor. A region no vertex comes near reads the greatest distance from its nodes to either surface.

A region shallower than a cell is below the grid’s resolution, which puts at least one node into a slab a cell thick, wherever the grid falls and however the slab is turned, but not into a thinner one: it is not reported. A noisy scan lifts single nodes off its surface all over, and each is that shallow.

Layers and lumps

A region d deep and V in volume is as wide as a square slab of that thickness, w = √(V / d). It is a Form::Layer, a face standing off, where the slab’s two faces have more area than its four sides, 2 w² ≥ 4 w d, so where V ≥ 4 d³; otherwise a Form::Lump.

Unsure

A region is Region::unsure where a doubtful node lies within a cell of one of its nodes, or where it is shallower than two cells. Then its extent rests on nodes nobody measured, or its volume error, about its area times √3 cells against its area times its depth, can reach its whole volume.

Cost

Each row of nodes along x is read in parallel, at a parity line through each body and an exact classification of the nodes in the solid’s band, an eighth of those within a cell of its surface. Rows are read and regions numbered in the grid’s order, so the answer is the same bit for bit on any thread count.

Structs

Region

A connected volume where the scan and the solid differ the same way.

Enums

Form

The shape of a region.

Material

Which way a region differs.

Functions

volume_difference

The regions where mesh, a scan, and the solid shape differ, on a grid of cells cell wide; see the module documentation.