Calculations

Calculations

• Facing faces within a gap

  The pairs of flat faces that face each other, with their gap and overlap.

  n·m ≤ −cos 0.5°; d = (o_B − o_A)·n with −1e-6 ≤ d ≤ gap; overlap = w × h

  Advanced simulation add-on: finding contact pairs addons/advanced-sim/lib/setup.js

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• Triangle areas and centroids

  The area and center of each mesh triangle.

  area = |(b − a) × (c − a)| ÷ 2; centroid = (a + b + c) ÷ 3 × 1000

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Area, mean and peak flux density of each body

  Each body's area in mm², its area-weighted mean |B| and its largest |B|.

  mean B = Σ(Bₑ × areaₑ) ÷ Σ areaₑ

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Result against a hand formula

  The value the result gives where a template's hand formula applies, and its relative difference.

  got = Σ(v × w) ÷ Σ w; diff = got ÷ want − 1

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Flux lines

  Contour segments of the vector potential A, and the flux between neighboring lines.

  levelᵢ = lo + ΔA × (i + 1), ΔA = (hi − lo) ÷ (k + 1); t = (L − aᵢ) ÷ (aⱼ − aᵢ)

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Top of the color scale

  The |B| that the top color stands for.

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Number format of the exports

  How a number is written into the CSV and VTK files.

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Field table (CSV)

  One row per triangle: center, region, area and B.

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• VTK file

  The mesh in mm with B per triangle and A per node.

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Summary table (CSV)

  Peak B, each body's area, mean and peak B, forces, torque and the hand-formula check.

  Magnetics add-on: field readout, flux lines and exports addons/magnetics/lib/results.js

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• Bricks to compute

  Which bricks of the lattice can hold the surface.

  brick range = floor((box ∓ band × h) ÷ h ÷ 8); ball radius = 3.5√3 × h

  Distance grid: baking, sampling and volume field/bake.js

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• Node values of a brick

  The signed distance at each of a brick's 512 nodes.

  value = clamp(f(i·h, j·h, k·h), −band × h, band × h)

  Distance grid: baking, sampling and volume field/bake.js

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• Brick key

  One whole number that names a brick.

  key = ((bx + 65536) × 131072 + (by + 65536)) × 131072 + (bz + 65536)

  Distance grid: baking, sampling and volume field/grid.js

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• Value at a point

  The signed distance at any point, between the lattice nodes.

  trilinear: a + (b − a) × t along x, then y, then z

  Distance grid: baking, sampling and volume field/grid.js

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• Gradient at a point

  The direction and rate of change of the distance.

  g = (f(p + h/2) − f(p − h/2)) ÷ h, per axis

  Distance grid: baking, sampling and volume field/grid.js

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• Box of the stored bricks

  The world box that holds every stored node.

  lo = brick × 8 × h; hi = (brick × 8 + 7) × h

  Distance grid: baking, sampling and volume field/grid.js

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• Enclosed volume

  The volume inside the surface.

  V = h³ × Σ clamp(½ − d ÷ h, 0, 1)

  Distance grid: baking, sampling and volume field/grid.js

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• Welding corners

  Merges mesh corners that sit at the same position.

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Angle-weighted normals

  The pseudo-normal at each corner and edge, used for the sign.

  angle = acos(e₁·e₂ ÷ (|e₁||e₂|)); N_vertex = Σ angle × n_face

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Signed distance at a node

  The distance from a lattice node to the mesh, negative inside.

  d = √(least squared distance); sign = sign((p − closest)·N)

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Closest point on a triangle

  The nearest point of a triangle to a point, and which face, edge or corner it lies on.

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Inside or outside far from the surface

  Marks bricks no triangle comes near as inside or outside.

  x_cross = (w₀a_x + w₁b_x + w₂c_x) ÷ (w₀ + w₁ + w₂); inside when an odd number of crossings lie before the brick

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Density map as stored

  Keeps a density map that steers a wall thickness or cell size.

  Distance grid: baking, sampling and volume field/field.js

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• Number that follows a density map

  The wall thickness or cell size at a point, from the density map.

  v = v₀ + (v₁ − v₀) × density(p), density by trilinear interpolation between cell centers

  Distance grid: baking, sampling and volume field/field.js

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• Edge crossing and its normal

  Where the surface crosses a cell edge, and the surface direction there.

  t = vᵢ ÷ (vᵢ − vₒ); p = p_in + (p_out − p_in) × t

  Surface mesh from a distance grid (dual contouring) field/contour.js

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• Vertex of a surface cell

  The point that best fits the crossing planes of one cell.

  (AᵀA + λI)y = Aᵀb, x = m + y, λ = 0.1

  Surface mesh from a distance grid (dual contouring) field/contour.js

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• Gradient inside a cell

  The gradient of the interpolated distance inside one cell.

  Surface mesh from a distance grid (dual contouring) field/contour.js

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• Which diagonal splits a quad

  Chooses how each four-sided face becomes two triangles.

  use diagonal 0–2 when |p₀ − p₂|² ≤ |p₁ − p₃|²

  Surface mesh from a distance grid (dual contouring) field/contour.js

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• Positions and box as returned

  The mesh's corner positions and its bounding box.

  Surface mesh from a distance grid (dual contouring) field/contour.js

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• Number format of the text exports

  How a number is written into the 3MF and VTI text.

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/export.js

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• Values on the PDF data sheet

  How each value is printed on the data sheet's PDF.

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/export.js

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• Field as VTK ImageData

  The signed distance and an inside mask on the node lattice.

  origin = lowest index × h; spacing = h; inside = value < 0

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/export.js

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• Surface as VTK UnstructuredGrid

  The surface triangles with each triangle's area and unit normal.

  n = (b − a) × (c − a); area = |n| ÷ 2; normal = n ÷ |n|

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/export.js

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  32-bit in part

• 3MF vertices

  The mesh corners as text in the 3MF model, in millimeters.

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/export.js

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• OpenVDB leaves and active voxels

  Maps the grid's bricks onto OpenVDB leaves and marks the narrow band active.

  background = fround(band × h); active when −background < value < background

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/vdb.js

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• OpenVDB value compression

  Chooses how a node's inactive values are stored and writes the values.

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/vdb.js

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• OpenVDB voxel size

  The transform from lattice index to millimeters.

  scale = h; also 1 ÷ h, 1 ÷ h² and 1 ÷ (2h)

  Generated part exports: data sheet, VTK, 3MF and OpenVDB field/vdb.js

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• Distance transform along one row

  The squared distance to the nearest empty voxel along one row.

  s = ((f[q] + q²) − (f[p] + p²)) ÷ (2q − 2p); d[q] = (q − v)² + f[v]

  Printability checks on a voxel solid field/mfg.js

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• Squared distance transform

  For each voxel in a set, the squared distance in voxels to the nearest voxel outside it.

  Printability checks on a voxel solid field/mfg.js

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• Connected pieces

  Labels each connected piece and counts its voxels.

  Printability checks on a voxel solid field/mfg.js

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• Thin regions

  The parts of a solid (or of the empty space) thinner than a limit, with each one's thickness.

  dk = (max(1, t − 1) + 1) ÷ 2; thickness = max(1, 2√(largest d²) − 0.5) voxels

  Printability checks on a voxel solid field/mfg.js

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• Loose pieces and sealed cavities

  The volume and position of each loose piece and each sealed cavity.

  volume = voxel count × h³; position = origin + (mean index − pad + ½) × h

  Printability checks on a voxel solid field/mfg.js

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• Thin wall and gap sizes in millimeters

  The thickness and volume of each thin wall and narrow gap.

  thickness = thickness in voxels × h; volume = voxel count × h³

  Printability checks on a voxel solid field/mfg.js

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• Rounding in the problem list

  Rounds sizes, volumes and positions for the problem list.

  Printability checks on a voxel solid field/mfg.js

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• Overhang area

  The downward-facing area steeper than a process prints without support.

  counts when −n_z ÷ |n| > sin(angle) + 1e-9; share = overhang area ÷ total area

  Printability checks on a voxel solid field/mfg.js

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• File unit to millimeters

  The factor from a file's unit to millimeters.

  μm 0.001; mm 1; cm 10; m 1000; in 25.4; ft 304.8

  Manual: imported product meshes, copies and part volume illustrate/import.js

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• glTF accessor values

  Reads a glTF accessor's numbers, including normalized integers.

  normalized integer: max(x ÷ max, −1)

  Manual: imported product meshes, copies and part volume illustrate/import.js

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• Mesh as stored

  A part's corner positions as the manual keeps them.

  Manual: imported product meshes, copies and part volume illustrate/import.js

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• Weld grid for splitting shells

  Merges corners at the same place so separate pieces of a mesh can be told apart.

  q = max(range, 1e-9) × 1e-7; key = round(x ÷ q), round(y ÷ q), round(z ÷ q)

  Manual: imported product meshes, copies and part volume illustrate/import.js

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• Part signature

  A short key (corner count, triangle count, area, volume) that groups parts that may be copies.

  Manual: imported product meshes, copies and part volume illustrate/import.js

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  32-bit in part