Calculations

Calculations

• Surface mesh buffers

  The positions, normals, parameters and colors of a surface's triangles, for drawing.

  Drawing NURBS curves and surfaces nurbs-ui/display.js

  • ModelCraft
  • PartCraft
  32-bit, display only

• Strength of connection between nodes

  Decides which neighboring nodes are strongly coupled, from the Frobenius norms of their stiffness blocks.

  strong when ‖A_IJ‖ > θ × √(‖A_II‖ × ‖A_JJ‖), θ = 0.08

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Tentative prolongator by QR per aggregate

  Orthonormalizes the near-null vectors (the rigid motions) restricted to each aggregate to form the columns of the prolongator.

  q_j ← b_j − Σᵢ (qᵢ·b_j) qᵢ (twice); keep when ‖q_j‖ > 1e-10 × scale × √m

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Largest eigenvalue of D⁻¹A

  Estimates the largest eigenvalue used to set the prolongator smoothing weight.

  λ ≈ 1.05 × ‖D⁻¹A x‖ ÷ ‖x‖ after 15 power iterations

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Prolongator smoothing

  Applies one damped Jacobi step to the tentative prolongator.

  P = P̂ − ω D⁻¹ A P̂, ω = 4 ÷ (3 λmax)

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Sparse matrix product

  Multiplies two sparse matrices, used for A·P and Pᵀ·(A·P).

  C_ij = Σ_r A_ir × B_rj

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Dense Cholesky on the coarsest level

  Factors the coarsest matrix and solves with it by forward and back substitution.

  L Lᵀ = A_c; pivot = √s when s > 1e-12 × max diag, else √(1e-12 × max diag + |s|)

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• Gauss–Seidel sweep

  One forward or backward relaxation sweep, the smoother of the V-cycle.

  x_r = (b_r − Σ_{c≠r} A_rc x_c) ÷ A_rr

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• V-cycle

  Applies the preconditioner: smooth, restrict the residual, solve the coarse problem, add the correction, smooth again.

  x ← S(b); r = b − A x; x ← x + P × cycle(Pᵀ r); x ← Sᵀ(b)

  Multigrid preconditioner for stress analysis amg.js

  • PartCraft
  64-bit

• 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
  64-bit

• 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
  64-bit

• Inverse of a rigid transform

  Inverts a rotation-plus-translation matrix.

  R⁻¹ = Rᵀ; t⁻¹ = −Rᵀ t

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Mate frame on a face

  The frame a joint end uses: the face's axes, with the origin moved to where the chosen axis crosses the face.

  o = a + d × ((f − a) · n) ÷ (d · n)

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Joint value inside its limits

  A joint's angle or distance, held between its minimum and maximum.

  value = min(max, max(min, v))

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Motion of a joint

  The transform of part B relative to part A's mate frame, from the joint's angles and distances.

  revolute, slider, cylindrical, rigid: T(0, 0, offset) × Rz(angle); planar: T(dx, dy, 0) × Rz; ball: Rx(tiltX) × Ry(tiltY) × Rz; pin-slot: T(s cos δ, s sin δ, 0) × Rz

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Placement walk and loop mismatch

  Places each part from one already placed through their joint, and measures how far a loop-closing joint is from closing.

  W_B = W_A × F_A × motion × FLIP × F_B⁻¹; off = max |want − have| (rotation entries × 100)

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Storing solved joint values

  Writes the values that close the loops into the joints, which is what the file saves.

  stored = round(v × 1e9) ÷ 1e9

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Moving a free part by hand

  Applies a typed move or rotation to a part's stored placement.

  M′ = T(move) × R(about, axis, angle) × M

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Rotation matrix to rotation vector

  Turns the rotation part of a mismatch into an axis times an angle in radians.

  θ = acos(clamp((trace − 1) ÷ 2, −1, 1)); w = (R − Rᵀ)ᵛ × θ ÷ (2 sin θ)

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• Loop residuals

  For each loop-closing joint, how far its B end is from where the loop puts it.

  D = have⁻¹ × want; r = (D_x, D_y, D_z, w_x L, w_y L, w_z L)

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• Jacobian by central differences

  The rate of change of every residual with every unknown joint value.

  J_qc = (r_q(x + h e_c) − r_q(x − h e_c)) ÷ 2h, h = 1e-6 × max(1, L)

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• Damped least-norm Newton step

  Moves the joint values toward closing the loops while changing them as little as possible.

  dx = Jᵀ (J Jᵀ + μ D)⁻¹ (−r); accept x + α dx when Σr² falls, α = 1, ½, ¼ …

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• Angle on the nearest turn

  Returns each solved angle on the turn closest to where it started, inside its limits.

  w = s₀ + ((val − s₀ + 180) mod 360) − 180

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• Rank, mobility and conflicts

  Counts the independent conditions at the answer, how many values could still be set, and which loops did not close.

  mobility = unknowns − rank; conflict when max |r| of a loop > max(1000 × tol, 1e-6)

  Closed-loop joint solver assembly.js

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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
  64-bit

• Scale about a point

  The matrix that scales a body by three factors about a fixed point.

  p′ = c + diag(sx, sy, sz) × (p − c); translation = c × (1 − s)

  Body move, scale and path pattern arithmetic bodies.js

  • PartCraft
  64-bit

• Smallest rotation between two directions

  The rotation that turns one unit direction onto another by the smallest angle.

  k = (a × b) ÷ |a × b|; θ = atan2(|a × b|, a · b)

  Body move, scale and path pattern arithmetic bodies.js

  • PartCraft
  64-bit

• Rotation about an axis

  The 3 × 3 rotation about a unit axis by an angle in radians.

  R = cos θ I + sin θ [k]× + (1 − cos θ) k kᵀ

  Body move, scale and path pattern arithmetic bodies.js

  • PartCraft
  64-bit

• Rigid motion about a point

  The 4 × 4 matrix of a rotation about a point followed by a translation.

  p′ = R (p − c) + c + t; offset = c − R c + t

  Body move, scale and path pattern arithmetic bodies.js

  • PartCraft
  64-bit

• Pattern stations along a path

  The arc length at which each copy of a path pattern sits.

  equal: step = L ÷ count (closed) or room ÷ (count − 1) (open); s_i = s₀ ± step × i, wrapped into [0, L) on a closed path

  Body move, scale and path pattern arithmetic bodies.js

  • PartCraft
  64-bit

• Nearest point on an edge

  The point of an edge closest to a given point, with the tangent there.

  circle: p = c + r × unit(radial part of (q − c)); other curves: minimize |C(t) − q|²

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Midpoint of an edge by length

  The point half way along an edge measured along the curve.

  length = Σ over 32 pieces of Σ₈ wᵢ |C′(tᵢ)| × h; bisect for length(t) = total ÷ 2

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Where an edge crosses a plane

  The points at which an edge passes through a plane.

  g(t) = (C(t) − o) · n; bisect each sign change of g between 400 samples

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Arc of a sweep path

  Turns a sketch arc into the three points OpenCascade builds the path arc from.

  r = |s − c|; mid = c + r × (cos, sin)((a₀ + a₁) ÷ 2), a₁ raised by 2π when a₁ ≤ a₀

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Chaining path pieces

  Decides which curve ends meet when pieces are joined into one path.

  same when |p − q| ≤ 1e-6 × max(1, largest |end point|)

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Sweep profile at the path's end

  Checks that the profile's plane passes through one end of the path, and which end.

  |(p − o) · n| ≤ 1e-6 × max(1, |path chord|)

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Extrude up to a face

  How far along the extrude direction each profile point is from the target plane.

  t = ((o − x) · n) ÷ (u · n); parallel target: depth = min t; otherwise depth = 1.02 × max t + 1, then cut back by the plane

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Through, offset and vertex extents

  The extrude depth for through-all cuts, an offset from a face, and up to a vertex.

  through: far + 1 + 0.01 × body range; offset: o = face − n × side × offset; vertex: b = (p − o) · u

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Picking the next face

  Chooses the first face of the body that the extrude meets.

  nearest face by its lowest vertex distance along u, ties (within 1e-7 × scale) by centroid distance

  Reference geometry, sweep paths and extrude extents brep-design.js

  • PartCraft
  64-bit

• Points along a projected edge

  Turns one edge of a view into the points its line segments join.

  circle: n = max(8, ceil(|t₁ − t₀| ÷ (π ÷ 60))); other curves: n = 96; drop a sample within 1e-7 × size of the chord

  Drawing views of the exact solid brep-draw.js

  • PartCraft
  64-bit

• Extent of a view

  The smallest box around a view's segments, used to place and scale it.

  u₀ = min, u₁ = max over segment ends (same for v)

  Drawing views of the exact solid brep-draw.js

  • PartCraft
  64-bit

• Section plane position

  Places the cutting plane square to a view's direction and builds the box that removes the near half.

  o = center − d (center · d) + d × at; box side = 2 × (2 × diagonal + 10)

  Drawing views of the exact solid brep-draw.js

  • PartCraft
  64-bit