Calculations

Calculations

• Brush falloff

  How strongly a vertex follows the brush, from 1 at its middle to 0 at its edge.

  w(x) = 1 − 3x² + 2x³ for 0 < x < 1; 1 for x ≤ 0; 0 for x ≥ 1

  Sculpt brushes mesh/sculpt.js

  • ModelCraft
  64-bit

• Vertex normals for sculpting

  A unit normal per vertex from the triangles round it, weighted by triangle area.

  N(v) = unit(Σ (b − a) × (c − a))

  Sculpt brushes mesh/sculpt.js

  • ModelCraft
  64-bit

• Vertices under the brush

  The vertices within the brush radius of its center, each with its falloff weight.

  d² = |P(v) − p|² ≤ r²; w = falloff(√d² ÷ r)

  Sculpt brushes mesh/sculpt.js

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• Brush plane

  The normal and center of the surface under the brush, and the size of one step.

  n = unit(Σ wᵢ·Nᵢ); c = Σ wᵢ·Pᵢ ÷ Σ wᵢ; step = 0.1·r·k, k = strength × pressure

  Sculpt brushes mesh/sculpt.js

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• Brush moves

  The new position of each vertex under the brush for draw, inflate, flatten, clay, smooth, pinch and crease.

  f = w·(1 − mask); draw: p + n·(±step·f); inflate: p + N(v)·(±step·f); flatten: p − n·dd·min(1, f·max(0.1, k)), dd = (p − plane)·n; smooth: p + (mean of neighbors − p)·min(1, f·max(0.1, k))

  Sculpt brushes mesh/sculpt.js

  • ModelCraft
  32-bit in part

• Grab

  Moves the vertices that were under the brush at the start of the stroke by the pointer's move.

  p = p₀ + δ·w·(1 − mask)

  Sculpt brushes mesh/sculpt.js

  • ModelCraft
  32-bit in part

• Painting the mask

  Raises or lowers the mask of the vertices under the brush, between 0 and 1.

  mask = clamp(mask ± w·k, 0, 1)

  Sculpt brushes mesh/sculpt.js

  • ModelCraft
  32-bit in part

• Split an edge at its middle

  The new vertex when dynamic detail splits a long triangle edge.

  mid = (P_a + P_b) ÷ 2

  Sculpt brushes mesh/sculpt.js

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• Brush radius in mesh units

  Turns the brush's radius in pixels into a length in the mesh's own space at the point under the pointer.

  r = px × pixel size at w ÷ scale, scale = |first column of the placement matrix|

  Sculpt brushes mesh/sculpt-tool.js

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• Dabs along a stroke

  Places dabs a quarter of the brush radius apart between the last dab and the pointer.

  step = 0.25·r; k = min(32, floor(d ÷ step)); pᵢ = last + (hit − last)·i ÷ k

  Sculpt brushes mesh/sculpt-tool.js

  • ModelCraft
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• Storing an unwrap

  Checks which way up the flattened island came out and writes its UVs to the mesh.

  signed = Σ over triangles of (u_b − u_a)(v_c − v_a) − (u_c − u_a)(v_b − v_a); u is negated when signed < 0

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• Are two UVs the same

  Decides whether two face corners share a UV point, which decides where UV islands end.

  |u_a − u_b| ≤ 1e-7 and |v_a − v_b| ≤ 1e-7

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
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• Box projection

  Maps each face onto the side of a box its normal faces most.

  u = (pᵢ − loᵢ) ÷ s, v = (pⱼ − loⱼ) ÷ s; u → 1 − u on the sides seen from behind

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• Cylinder projection

  u is the angle round an axis through the faces' center, v the height along it.

  u = (atan2(p_j − c_j, p_i − c_i) ÷ 2π + 1) mod 1; v = (p_k − lo_k) ÷ height

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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  32-bit in part

• Sphere projection

  u is the angle round the axis, v the latitude, seen from the faces' center.

  d = p − c; u = (atan2(d_j, d_i) ÷ 2π + 1) mod 1; v = 1 − acos(clamp(d_k ÷ |d|, −1, 1)) ÷ π

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• View projection

  UVs from where each corner appears on screen, fitted to the unit square.

  s = max(1e-12, width, height); u = (x − x₀) ÷ s; v = 1 − (y − y₀) ÷ s

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• Smallest bounding rectangle

  The rotation that makes an island's bounding rectangle smallest.

  for each hull edge: θ = −atan2(Δy, Δx); area = (x₁ − x₀)·(y₁ − y₀) of the rotated hull; keep the smallest

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Packing: island scale

  Scales each island so that its UV area matches its area on the mesh.

  A₃ = Σ |(b − a) × (c − a)| ÷ 2; A₂ = Σ |UV triangle area|; k = √(A₃ ÷ A₂)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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  32-bit in part

• Packing: placing the islands

  Places the islands' rectangles on shelves and scales the whole layout into the unit square.

  pad = margin·√(total area)·1.1; widths tried: max(widest, √(sum of areas)·(0.7 + 0.05·k)), k = 0…23; uv = (offset + pad + point) ÷ side

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• Stretch per face

  How much each face's UVs distort it, from 0 for none to 1 for a lot.

  σ₁,₂ = √(tr ÷ 2 ± √(tr² ÷ 4 − det)); angle: 1 − σ₂ ÷ σ₁; area: min(1, |ln(σ₁·σ₂)| ÷ ln 4); per face: area-weighted mean over its triangles

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

  • ModelCraft
  32-bit in part

• Move, rotate and scale UVs

  Moves, turns or scales the selected faces' UVs about their mean as the pointer moves in the UV editor.

  move: (u + Δu, v + Δv); rotate by a = atan2(v₁ − c_v, u₁ − c_u) − atan2(v₀ − c_v, u₀ − c_u); scale by k = |p₁ − c| ÷ max(1e-9, |p₀ − c|)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv-editor.js

  • ModelCraft
  32-bit in part

• UVs and sharp edges of a primitive

  Stores a new primitive's positions, its UVs per face corner and the flags of its sharp edges.

  sharp when n₁·n₂ < cos(sharpDeg·π ÷ 180)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/primitives.js

  • ModelCraft
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• Draft angle

  The angle between a surface normal and the plane square to the pull direction, in degrees.

  angle = asin(clamp(n · pull, −1, 1)) × 180 ÷ π

  Draft and curvature analysis analysis.js

  • PartCraft
  64-bit

• Areas by draft class

  The surface area with enough draft, with too little, and facing away from the pull.

  area = |(B − A) × (C − A)| ÷ 2; each corner adds area ÷ 3 to its class

  Draft and curvature analysis analysis.js

  • PartCraft
  32-bit in part

• Curvature color limit

  The curvature value (1/mm) at which the colors reach full blue or red, stated in the panel with its radius.

  lim = 95th percentile of |mean curvature|; color weight = clamp(mean ÷ lim, −1, 1)

  Draft and curvature analysis analysis.js

  • PartCraft
  32-bit in part

• View scale and position on the sheet

  Picks the drawing scale and maps view coordinates to sheet millimeters.

  scale = first s with w × s ≤ 0.78 × frame width and h × s ≤ 0.78 × frame height; sheet = origin + (u − u₀) × scale

  Assembly drawing sheet and mesh exports asm-drawing.js

  • PartCraft
  64-bit

• Mass text in the bill of materials

  Formats a part's mass for the drawing's parts table.

  g ≥ 1000: kg with 3 decimals; else g with 2 decimals

  Assembly drawing sheet and mesh exports asm-drawing.js

  • PartCraft
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• Assembly as one mesh

  Moves every part's mesh to its place in the assembly and joins them, for STL export and the 3D-printing check.

  p′ = M × p for each vertex

  Assembly drawing sheet and mesh exports asm-drawing.js

  • PartCraft
  32-bit in part

• Numbers written to 3MF

  Turns a coordinate or a matrix entry into the text stored in the 3MF file.

  text = toPrecision(9) of v; |v| < 1e-12 → 0

  Assembly drawing sheet and mesh exports asm-drawing.js

  • PartCraft
  64-bit

• Instance placement in 3MF

  Converts each instance's placement from Y-up to Z-up and writes it as the item transform.

  T = Y_TO_Z × M × Z_TO_Y; 12 entries written row by row

  Assembly drawing sheet and mesh exports asm-drawing.js

  • PartCraft
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• Overlap volume of a pair

  The common volume of two placed parts and whether it counts as a clash.

  v = volume(A ∩ B); clash when v > 1e-6

  Interference between assembly parts assembly.js

  • PartCraft
  32-bit in part

• Thread zones

  Tubes around each cosmetic thread, used to tell a bolt in a tapped hole from a real clash.

  r₀ = 0.97 × minor (default 0.8 × major); r₁ = 1.03 × major + 0.001; length = depth + 2 × max(0.05 × major, 0.001)

  Interference between assembly parts assembly.js

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• Motion check steps

  Steps one joint value from a start to an end and reports interference at each step.

  value_s = from + (to − from) × s ÷ steps, s = 0 … steps

  Interference between assembly parts assembly.js

  • PartCraft
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• Mesh tolerances

  How closely the triangle mesh follows the exact faces.

  linear = 2e-4 × bounding-box diagonal; angular = 2π ÷ max(16, resolution)

  Exact solid: mesh, volume, area and tables brep.js

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• Mesh read from the kernel

  Copies the mesh's vertices, normals and triangles out of the kernel's memory.

  vertices and normals: Float32Array view of the WebAssembly buffer

  Exact solid: mesh, volume, area and tables brep.js

  • PartCraft
  32-bit in part

• Welding the mesh

  Merges vertices that different faces share so the mesh is watertight.

  same vertex when x, y and z are identical

  Exact solid: mesh, volume, area and tables brep.js

  • PartCraft
  32-bit in part

• Volume and surface area

  The solid's volume and total surface area, from the exact faces.

  V = ∭ dV (Gauss–Kronrod, tolerance 1e-12, when a non-line, non-circle edge exists); A = ∬ dA

  Exact solid: mesh, volume, area and tables brep.js

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• Genus

  The number of through holes (handles) of the solid, from the welded mesh.

  χ = V − E + F; genus = round(bodies − χ ÷ 2)

  Exact solid: mesh, volume, area and tables brep.js

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• Face area and center

  Each face's area and center point, listed with the face.

  area = ∬ dA over the face; center = its centroid

  Exact solid: mesh, volume, area and tables brep.js

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• Edge length and points

  Each edge's length, end points and middle point.

  length = ∫ |C′(t)| dt; middle = C((t₀ + t₁) ÷ 2)

  Exact solid: mesh, volume, area and tables brep.js

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• Bounding box

  The smallest axis-aligned box around a shape.

  min, max per axis from the kernel's optimal box

  Exact solid: mesh, volume, area and tables brep.js

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• Facts about an imported STEP file

  The solid count, face count, volume and bounding box shown when a STEP file is imported.

  volume = ∭ dV (default integration)

  Exact solid: mesh, volume, area and tables brep.js

  • PartCraft
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• Stock block

  The size and position of the stock around the part.

  offset mode: min = box min − (side, side, bottom), max = box max + (side, side, top); cylinder: r = ½ × box diagonal in XY + side

  Part and stock in machine coordinates cam.js

  • PartCraft
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• Work origin and frame change

  Moves points from model coordinates to the work frame (G54) and back.

  work = (p · x, p · y, p · z) − origin

  Part and stock in machine coordinates cam.js

  • PartCraft
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• Mesh in the work frame and snap tolerance

  Copies the part's mesh into work coordinates and sets how close a mesh point must be to an exact edge to be taken as that edge.

  tol = 1e-4 + 2e-6 × largest model dimension

  Part and stock in machine coordinates cam.js

  • PartCraft
  32-bit in part

• Where a vertical line meets the part

  The heights at which a vertical line through (x, y) crosses the part's surface.

  weights l₁, l₂, l₃ from the triangle's XY corners; z = l₁ z₁ + l₂ z₂ + l₃ z₃

  Part and stock in machine coordinates cam.js

  • PartCraft
  32-bit in part

• Top facet and its slope

  The height and slope of the highest mesh triangle under a point, used by 3D finishing.

  g_x = ((z₂ − z₁)(y₃ − y₁) − (z₃ − z₁)(y₂ − y₁)) ÷ d₂; g_y likewise

  Part and stock in machine coordinates cam.js

  • PartCraft
  32-bit in part

• Material below a point

  Whether there is part material just under a height at (x, y), used to tell a through hole from a blind one.

  first surface below z faces up

  Part and stock in machine coordinates cam.js

  • PartCraft
  32-bit in part

• Circles of round edges

  The center and radius of each circular edge lying in a plane square to the tool.

  full circle: c = (a + mid) ÷ 2, r = |a − mid| ÷ 2; arc: circle through a, mid, b

  Part and stock in machine coordinates cam.js

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• Circle through three points

  The center and radius of the circle through three 2D points.

  d = 2 (a_x (b_y − c_y) + b_x (c_y − a_y) + c_x (a_y − b_y)); u_x = (|a|² (b_y − c_y) + |b|² (c_y − a_y) + |c|² (a_y − b_y)) ÷ d

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit