Calculations

Calculations

• 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

• Faces in the cutting plane

  Finds the faces created by the cut, whose outlines bound the hatched area.

  ||n · d| − 1| ≤ 1e-9 and |(p − o) · d| ≤ 1e-6 × diagonal

  Drawing views of the exact solid brep-draw.js

  • PartCraft
  64-bit

• Cutting-plane line in the neighboring view

  The point and direction of the line that shows where the section was cut.

  along = unit(d_section × d_view); p = (o · right, o · up); dir = (along · right, along · up)

  Drawing views of the exact solid brep-draw.js

  • PartCraft
  64-bit

• Number of steps round a circle

  How many interpolation points a round direction gets, from its effective radius.

  h = (384 × DEV ÷ R)^¼; n = ceil(2π ÷ h) rounded up to a multiple of 4, between 32 and 720

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Cubic spline through the unit circle

  The poles of a periodic cubic B-spline that passes through the circle at every knot.

  pole i = k × (cos((i − 1)h), sin((i − 1)h)), k = 6 ÷ (4 + 2 cos h), err = h⁴ ÷ 384

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Largest stretch of the map

  The largest factor by which the scale lengthens any direction, used to size the deviation.

  σ = |M x| after 60 power iterations of x ← unit(Mᵀ M x)

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Image of a curve

  A line, circle, ellipse, Bézier or B-spline curve after the scale, as a B-spline in the same parameter.

  line, Bézier, B-spline: poles ↦ T × pole (exact); circle, ellipse: T × (c + r₁ x cos + r₂ y sin) at the spline's poles

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Image of a surface

  A plane, cylinder, cone, sphere, torus, extrusion, revolution or spline surface after the scale.

  plane ↦ plane with frame from T x, T x × T y; cylinder, cone: ring(u) × (R + v sin α) + v cos α Z; sphere, torus: ring(u) × (R₁ + R₂ cos v) + R₂ sin v Z

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Inverse of a placement

  Inverts a 3 × 4 placement matrix found in the shape.

  M⁻¹ = adj(M) ÷ det(M); t⁻¹ = −M⁻¹ t

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Parameter range of straight directions

  How far a line or the straight direction of a surface must extend to cover the body.

  t₀ = ((center − o) · d) ÷ |d|²; range = t₀ ± (1.02 × D ÷ |d| + 1)

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Raised shape tolerances

  The tolerance written on every vertex, edge and face of the scaled body.

  tol = max(old, max(1e-7, 3 × largest deviation + 1e-9))

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Numbers written to the shape text

  Turns each coordinate, weight and knot into the text OpenCascade reads back.

  text = String(v); −0 → 0

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Matching equal placements

  Decides when two uses of the same curve or surface share one mapped copy.

  key = entries rounded to 1e-12, joined

  Non-uniform scale of an exact body brep-gtrsf.js

  • PartCraft
  64-bit

• Rib slab outline

  The flat region the rib is cut from: the sketch chain carried past both ends and offset to one side.

  E = 4 × max(1, diagonal + |box center − sketch origin|); ends ± E along the end directions; far side = + E × D

  Rib feature brep-rib.js

  • PartCraft
  64-bit

• Choosing the rib piece

  Picks the piece of the trimmed slab that lies beside the sketch line and checks it is closed off by the body.

  distance(probe, piece) < 1e-6 × max(1, t); open when reach > 0.5 × E

  Rib feature brep-rib.js

  • PartCraft
  64-bit

• Edge flange frame

  The axes a flange is built in: along the edge, out of the sheet's face, and out of its side.

  e = unit(B − A); square when |n_big · n_side| ≤ 1e-6; P = (A + B) ÷ 2 − v × t ÷ 2 must lie on the side face

  Sheet metal base and edge flanges brep-sheet.js

  • PartCraft
  64-bit

• Flange cross-section

  The corner points of the bend and the leg in the plane square to the edge.

  C = (0, r); point(ρ, φ) = C + ρ (cos φ, sin φ); φ from −π ÷ 2 to −π ÷ 2 + θ; leg end = arc end + length × (−sin φ₁, cos φ₁)

  Sheet metal base and edge flanges brep-sheet.js

  • PartCraft
  64-bit

• Bend relief slot

  The box cut beside an offset flange end so the bend can form.

  depth into the sheet = r + t; box length along u = (r + t) + 1e-3 × max(1, r + t); through 2 × t

  Sheet metal base and edge flanges brep-sheet.js

  • PartCraft
  64-bit

• Side of the first bend

  Which side of an open profile the sheet's thickness goes on.

  turn = sign((p₁ − p₀) × (p₂ − p₁))

  Sheet metal base and edge flanges brep-sheet.js

  • PartCraft
  64-bit

• Cone from its derivatives

  A cone's axis point, axis direction, radius and slope from the surface's derivatives at one point.

  ρ = |P_uu|; e = −P_uu ÷ ρ; z = P_v − (P_v · e) e; axis point = P − ρ e; slope k = (P_v · e) ÷ |z|

  Exact surface records for meshing brep-surf.js

  • PartCraft
  64-bit

• Torus from its derivatives

  A torus's center, axis and two radii from the surface's derivatives at one point.

  ρ = |P_uu|; A = P − ρ e; d = unit(e × P_u); tube center M = P + P_vv; c = A + ((M − A) · d) d; R = |M − c|; r = |P_vv|

  Exact surface records for meshing brep-surf.js

  • PartCraft
  64-bit

• Check against the kernel's face

  Confirms the record describes the same surface as the kernel's face before it is used.

  25 points on a 5 × 5 grid (2 % to 98 % of the parameter box); each within 1e-9 × size of the record's surface

  Exact surface records for meshing brep-surf.js

  • PartCraft
  64-bit

• Closed loops with their curves

  Collects a sketch's curves into closed loops, keeping each curve's exact definition.

  tol = 1e-6 × max(1, extent); ends within 10 × tol share a node; arc mid = c + r (cos, sin)(a₀ + sweep ÷ 2)

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Arc edge through three points

  Builds an arc of the profile in the kernel.

  arc through a (start node), mid, b (end node)

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Loop orientation

  Whether a loop runs counterclockwise, from the sign of its area.

  A = ½ Σ (x_j y_i − x_i y_j)

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Point just inside a loop

  A test point a small step inside the loop's longest edge, used to decide nesting.

  p = midpoint of longest edge + normal × 1e-6 × max(1, edge length)

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Which loops are holes

  Counts how many other loops contain each loop; odd depth is a hole.

  depth_i = number of loops whose polygon contains innerPoint(loop_i); face when depth is even

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Stored edge and face reference points

  The point saved with a picked edge or face so it can be found again after a rebuild.

  at = coordinates rounded to 6 decimals

  Sketch profiles for exact solids brep.js

  • PartCraft
  64-bit

• Extrude depth and start

  Where an extrude starts and how far it runs for blind, mid-plane and through-all ends.

  blind: z₀ = 0 or −h; mid: z₀ = −h ÷ 2; through: h = far side + 1 + 0.01 × body range

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit