Calculations

Calculations

• 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

• 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

• 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

• Revolve axis checks

  Checks that the axis lies in the sketch plane and that the profile stays on one side of it.

  side s = −u_y (p_x − a_x) + u_x (p_y − a_y); crosses when some s > 1e-7 × scale and some s < −1e-7 × scale

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Revolve angle

  The sweep angle given to the kernel.

  full turn when |angle − 360| < 1e-9; else angle × π ÷ 180

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Through-hole depth

  How deep a through hole is drilled.

  depth = distance to the body's far side + 1 + 0.01 × body range

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Countersink depth

  How deep the cone of a countersunk hole goes.

  h = (D_csink − d) ÷ 2 ÷ tan(angle × π ÷ 360)

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Cosmetic thread radii

  The major and minor radius recorded for a tapped hole's thread.

  r₁ = D ÷ 2; r₀ = min(tap drill, D − 1.082532 × P) ÷ 2

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Linear pattern positions

  Where each copy of a linear pattern sits.

  copy i moved by direction × spacing × i

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Circular pattern angles

  The angle of each copy of a circular pattern.

  full circle: step = 360 ÷ count; otherwise step = angle ÷ (count − 1); copy i turned by step × i

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Shell wall offset

  Hollows a body to a wall thickness.

  offset = +t (outward) or −t; kernel tolerance 1e-3

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Fillet and chamfer sizes

  The radius, distances or distance and angle given to the kernel for each picked edge.

  two distances: (d, d₂); distance and angle: (d, angle × π ÷ 180); fillet: r, or (r, r₂) along the edge

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Sewing an imported surface model

  Joins the faces of a STEP file that has no solid into a closed shell.

  sewing tolerance = 1e-6

  Exact-kernel feature dimensions brep.js

  • PartCraft
  64-bit

• Length of a line or arc

  The length of one piece of an outline or toolpath.

  line: |e − s|; arc: |sweep| × r

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Point along a piece

  The point a given fraction of the way along a line or arc.

  line: s + (e − s) t; arc: c + r (cos, sin)(a₀ + sweep × t)

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Is an angle on an arc

  Whether a direction from the center falls within an arc, and how far along.

  u = mod(±(a − a₀), 2π); on the arc when u ≤ |sweep| + eps; t = u ÷ |sweep|

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Distance from a point to a line or arc

  The shortest distance from a point to one piece of an outline.

  line: |q − (s + d × clamp(((q − s) · d) ÷ |d|², 0, 1))|; arc: ||q − c| − r| when the direction is on the arc, else the nearer end

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Signed area of a loop

  The area enclosed by a loop of lines and arcs, positive when counterclockwise.

  A = ½ Σ [line: s × e; arc: r c_x (sin a₁ − sin a₀) − r c_y (cos a₁ − cos a₀) + r² × sweep]

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Winding number

  How many times the loops wind round a point, which says whether the point is inside.

  count signed crossings of a ray to +x; arc crossing at x = c_x ± √(r² − (q_y − c_y)²)

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Line and line

  The point where two line pieces cross, or the ends of their overlap when they lie on one line.

  t_a = ((B.s − A.s) × s) ÷ (r × s); parallel when |r × s| ≤ 1e-12 |r| |s|

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Line and arc

  The points where a line piece meets an arc.

  a t² + b t + c = 0; t = (−b ± √(b² − 4ac)) ÷ 2a

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Arc and arc

  The points where two arcs meet.

  a = (r_A² − r_B² + d²) ÷ 2d; h = √(r_A² − a²); p = c_A + a u ± h n

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Raw offset of a loop

  Moves every piece of a loop sideways by the offset and joins the corners with arcs.

  line: s, e + d × left normal; arc: r ∓ d; corner arc radius d, sweep = atan2(t_a × t_b, t_a · t_b)

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Splitting pieces at intersections

  Cuts every offset piece where it meets another.

  cut at parameter t unless within 1e-5 mm of an end or of the previous cut

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Offset of a region

  The outline at a fixed distance inside or outside a region, which is the path of the tool center.

  keep a piece when dist(midpoint, original outline) ≥ d − 1e-6 and the midpoint is on the offset side

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Chaining pieces into loops

  Joins the kept pieces end to start into closed loops.

  next piece starts within 4e-5 mm of the current end; sharpest left turn wins; then next.start = current.end

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Merging collinear lines and co-circular arcs

  Joins neighboring pieces that lie on the same line or circle into one.

  lines: |u × v| < 1e-10 and end within 1e-9 mm of the line; arcs: centers and radii within 1e-9 mm, same direction

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Splitting arcs to half turns

  Cuts arcs so none sweeps more than 180°, as G-code controls expect.

  n = ceil(|sweep| ÷ π); piece i from a₀ + sweep × i ÷ n

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Points along a piece at a chord tolerance

  Sample points no farther than a tolerance from an arc, for checks and drawing.

  step angle = 2 acos(1 − tol ÷ r)

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Bounding box of a piece

  The extent of a line or arc, including where an arc bulges past its ends.

  ends, plus c ± r along each axis direction the arc passes

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit