Calculations

Calculations

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  64-bit

• Gradient inside a cell

  The gradient of the interpolated distance inside one cell.

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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• Mesh volume

  The signed volume enclosed by a triangle mesh.

  V = Σ a·(b × c) ÷ 6

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
  64-bit

• Mesh area

  The total area of a triangle mesh.

  A = Σ |(b − a) × (c − a)| ÷ 2

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
  64-bit

• Closed and manifold checks

  Counts open, non-manifold and flipped edges, bad vertices, pieces and the Euler number.

  Euler number = vertices used − edges + triangles

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
  Exact

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• Connected pieces

  Labels each connected piece and counts its voxels.

  Printability checks on a voxel solid field/mfg.js

  • PartCraft
  Exact

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• Rounding in the problem list

  Rounds sizes, volumes and positions for the problem list.

  Printability checks on a voxel solid field/mfg.js

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• Plane of a curve

  The normal of the plane a hull line lies in.

  n = (p_mid − p₀) × (p_end − p₀), made unit

  Hull lines: fairness measures and fairing hull/fair.js

  • PartCraft
  64-bit

• Signed curvature

  The curvature of a curve at a parameter, signed in its plane.

  κ = (C′ × C″)·n ÷ |C′|³

  Hull lines: fairness measures and fairing hull/fair.js

  • PartCraft
  64-bit

• Fairness measures

  Inflection count, largest curvature, roughness, curvature jumps at knots and length.

  R = ∫(dκ/ds)² ds ≈ (du ÷ 3) × Σ wᵢ gᵢ, gᵢ = ((κᵢ₊₁ − κᵢ₋₁) ÷ ds)² × |C′|ᵢ

  Hull lines: fairness measures and fairing hull/fair.js

  • PartCraft
  64-bit

• Fairing a curve

  The rebuild with the fewest control points that stays within tolerance and adds no inflection.

  Hull lines: fairness measures and fairing hull/fair.js

  • PartCraft
  64-bit

• Fairness problems

  Reports curves that have an inflection or a curvature jump.

  Hull lines: fairness measures and fairing hull/fair.js

  • PartCraft
  64-bit

• Sampling a patch

  The parameter values at which each surface is evaluated.

  uᵢ = u₀ + (u₁ − u₀) × i ÷ N, N from 2 to 200

  Hull surfaces to a closed solid and its volume hull/solid.js

  • PartCraft
  64-bit

• Mirror half

  The other half of a hull drawn to the centerline.

  p′ = 2 × at − p on the mirror axis

  Hull surfaces to a closed solid and its volume hull/solid.js

  • PartCraft
  64-bit

• Welding corners

  Merges corners of neighboring patches that lie within the weld distance.

  merge when √(Δx² + Δy² + Δz²) ≤ eps

  Hull surfaces to a closed solid and its volume hull/solid.js

  • PartCraft
  64-bit

• Planar opening test

  Decides whether an opening lies in a plane and can be capped.

  n by Newell's sums; deviation = max |(p − centroid)·n|; cap when deviation ≤ max(4 eps, 2e-4 × size)

  Hull surfaces to a closed solid and its volume hull/solid.js

  • PartCraft
  64-bit

• Volume of the closed mesh

  The volume enclosed by the hull mesh.

  V = Σ a·(b × c) ÷ 6

  Hull surfaces to a closed solid and its volume hull/solid.js

  • PartCraft
  64-bit

• Anchor point as assembled

  Where an annotation's anchor sits when the parts are assembled.

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
  64-bit

• Circle through three points

  The center and radius of the circle through three points.

  n = ab × ac; c = a + ((n × ab)|ac|² + (ac × n)|ab|²) ÷ (2|n|²); r = |c − a|

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
  64-bit

• Measured value

  The length, angle or radius an annotation shows.

  length = |P₁ − P₀|; angle = acos(û·ŵ) × 180 ÷ π; radius from the circle through three points

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
  64-bit

• Unit vector

  A vector scaled to length 1.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Point under a matrix

  A point moved by a 4 × 4 matrix.

  p′ = M₃ₓ₃ p + t

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Matrix product

  The product of two 4 × 4 matrices.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Matrix inverse

  The inverse of a general 4 × 4 matrix.

  M⁻¹ = adj(M) ÷ det(M)

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Box of placed positions

  The axis-aligned box of a mesh's corners under a matrix.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Rigid motion test

  Checks that a matrix is a rotation and a move, with no scale or shear.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Best rigid move between two point sets

  The rotation and move that best take one set of points onto another, and the largest leftover distance.

  R from the unit quaternion of the largest eigenvalue; t = c_B − R c_A; err = max |M pᵢ − qᵢ|

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• Largest eigenvector of a symmetric 4 × 4

  The eigenvector the rigid fit's rotation is read from.

  t = sign(θ) ÷ (|θ| + √(θ² + 1)), θ = (a_qq − a_pp) ÷ (2a_pq)

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

  • PartCraft
  64-bit

• 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

  • PartCraft
  Exact

• 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

  • PartCraft
  64-bit

• Mesh as stored

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

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

  • PartCraft
  32-bit in part