Calculations

Calculations

• Smooth minimum

  Blends two shapes with a radius k.

  h = clamp(½ + (b − a) ÷ 2k, 0, 1); s = b + (a − b)h − k·h(1 − h)

  Field shapes: signed distance functions field/field.js

  • PartCraft
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• Offset and shell

  Grows or shrinks a shape, or hollows it to a wall thickness.

  offset: d − δ; shell inward: max(d, −d − t); shell centered: |d| − t ÷ 2

  Field shapes: signed distance functions field/field.js

  • PartCraft
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• Move, turn and scale

  A shape under a rigid motion and a uniform scale.

  q = Rᵀ(p − move) ÷ s; d = d₀(q) × s

  Field shapes: signed distance functions field/field.js

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• Rotation matrix

  The 3 × 3 matrix of a turn about an axis by an angle.

  R = c·I + s·[u]ₓ + (1 − c)·u uᵀ, c = cos θ, s = sin θ

  Field shapes: signed distance functions field/field.js

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• Beam or round cone

  Signed distance to a beam whose radius changes from one end to the other.

  t = clamp((p − a)·(b − a) ÷ |b − a|², 0, 1); d = |p − a − t(b − a)| − (r₀ + (r₁ − r₀)t)

  Field shapes: signed distance functions field/field.js

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• Lattice of beams

  The least distance to many beams.

  Field shapes: signed distance functions field/field.js

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• Ramped cell size and wall

  A number that changes linearly along an axis or with radius.

  v = v₀ + (v₁ − v₀) × clamp((p − a) ÷ (b − a), 0, 1)

  Field shapes: signed distance functions field/field.js

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• Phase of a graded lattice

  Maps a point to the lattice's own period coordinates when the cell size changes.

  fixed cell: X = 2π·p ÷ cell; ramp: X = ∫k dp with k linear in p

  Field shapes: signed distance functions field/field.js

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• Gyroid, Schwarz P and diamond

  Signed distance to a triply periodic minimal surface, as a sheet or a network.

  d ≈ g ÷ |∇g|; near the surface: two projections q ← q − g∇g ÷ |∇g|², d = ±|p − q|

  Field shapes: signed distance functions field/field.js

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• Brick classification

  Decides whether a ball of space is wholly outside, wholly inside or near the surface.

  outside when f − L·r > band; inside when f + L·r < −band

  Field shapes: signed distance functions field/field.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

  • PartCraft
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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

  • PartCraft
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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

  • PartCraft
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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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• 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

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• 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

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• 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

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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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• 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

  • PartCraft
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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

  • 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

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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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• 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

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• 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
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