Calculations

Calculations

• 64-bit whole numbers

  Reads a signed 64-bit whole number from two 32-bit halves.

  v = hi × 2³² + lo when −2²¹ ≤ hi < 2²¹; otherwise decimal text

  FBX reader: number types as stored io/fbx-parse.js

  • ModelCraft
  Exact

• Arrays by type

  Reads an array of floats, doubles or whole numbers.

  FBX reader: number types as stored io/fbx-parse.js

  • ModelCraft
  64-bit

• Triangles of a referenced object

  The target's triangles in the modified mesh's space, for booleans and similar modifiers.

  q = (toLocal × placement) × p

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Cage point displacements

  How far each cage point has been moved from its place on the even grid.

  rest = min + (max − min) × i ÷ (n − 1); disp = M × now − M × rest

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Cage around a mesh

  A box of grid points around a mesh's bounds.

  d = pad × max side; min′ = min − d; max′ = max + d; p = min′ + (max′ − min′) × i ÷ (n − 1)

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Face points

  The new point in the middle of each face.

  F = Σ corners ÷ n

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Edge points

  The new point on each edge.

  m = (a + b) ÷ 2; e = (a + b + F₁ + F₂) ÷ 4; point = e + (m − e) × s for 0 ≤ s < 1

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Vertex points

  Where each old vertex moves to.

  smooth = (F̄ + 2 R̄ + (n − 3) V) ÷ n; crease = (6 V + a + b) ÷ 8; k = min(1, (sₐ + s_b) ÷ 2); point = smooth + (crease − smooth) × k

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Edge sharpness per level

  The sharpness each new edge inherits.

  s′ = max(0, s − 1); open edge: ∞; crease layer = min(1, s′ ÷ 4)

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Texture coordinates and other corner layers

  Interpolates per-corner values onto the new quads.

  corner: own value; edge: mean of two; face: mean of the face's corners

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Limit positions

  Where each vertex of an all-quad mesh lies on the limit surface.

  smooth: (n² V + 4 Σ E + Σ F) ÷ (n (n + 5)); crease or border: (4 V + a + b) ÷ 6

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Multiplicity of a knot

  How many times a value occurs in a knot vector.

  count of |Uᵢ − u| ≤ tol × (U_last − U_first)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Distinct knots

  The different knot values with their multiplicities.

  a knot joins the previous one when |x − u_prev| ≤ tol × range

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Clamped uniform knot vector

  A knot vector with evenly spaced interior knots.

  uᵢ = a + (b − a) × i ÷ (inner + 1)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Knot vector check

  Checks a knot vector read from a file: count, order, clamping and multiplicities.

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Greville abscissae

  One parameter value per control point.

  gᵢ = (Uᵢ₊₁ + … + Uᵢ₊ₚ) ÷ p

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Binomial coefficients

  The number of ways to choose k of n.

  C(n, k) = C(n − 1, k − 1) + C(n − 1, k)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  Exact

• A number as IGES text

  Writes one real number into the file.

  whole numbers below 10¹⁵: digits and a point; otherwise String(x) with E → D

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Global section

  States the file's number sizes, unit, resolution and largest coordinate.

  unit flag 2 (MM); scale 1; resolution 10⁻⁶; largest coordinate = max |x| over all control points

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• A number from IGES text

  Reads one real number from the file.

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Unit factors

  Millimeters per file unit, by the file's unit flag.

  1: 25.4; 2: 1; 3: 25.4 × 12; 4: 1609344; 5: 10⁶; 6: 1000; 7: 10⁶; 8: 10⁻³; 9: 10⁻³; 10: 10; 11: 25.4 × 10⁻⁶

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Scales a curve's control points by the file's unit as it is read.

  scale = UNITS[flag] (1 when unknown); p′ = p × scale

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• End derivative coefficients

  The factors that link the first control points to the first and second derivative at the start.

  a₁ = p ÷ (U[p+1] − U[1]); b₁ = p ÷ (U[p+2] − U[2]); c₂ = (p − 1) ÷ (U[p+1] − U[2])

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Point, direction and curvature at an end

  What a curve's end looks like to the curve that must meet it.

  t = C′ ÷ |C′|; out = ± t; speed = |C′|

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Curve end matched to a target

  Moves the first one to three control points so the curve meets the target with G0, G1 or G2.

  P₀ = T; P₁ = P₀ + out × speed ÷ a₁; D₂ = out (C″ · out) + κ speed²; P₂ = P₁ + (a₁ (P₁ − P₀) + D₂ ÷ c₂) ÷ b₁

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend curve between two ends

  A Bézier curve joining two curve ends with the asked continuity.

  G1: P₁ = A + out × α ÷ 3; G2: P₁ = A + out × α ÷ 5, P₂ = 2 P₁ − A + κ α² ÷ 20; α = chord × bulge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Frame on the target's edge

  For a point near the target's edge: the nearest edge point, the outward direction in the tangent plane, the normal and the normal curvature.

  d = (n × tₑ) ÷ |n × tₑ|, turned to point away from the surface

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Surface edge matched to a target

  Refits the first one to three rows of control points so the surface meets the target along an edge.

  stations vᵢ = i ÷ M, M = max(samples, 4 × columns); rows fitted by least squares; accepted when gap ≤ g0, angle ≤ g1 and Δκ ≤ g2 (or relative ≤ g2rel)

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend surface between two edges

  A surface joining two surface edges, built from blend sections along them.

  sections at t = i ÷ n, n = 8, 16, … ≤ 128; each a Bézier of degree 1, 3 or 5; interpolated along the edge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• A number as STEP text

  Writes one real number into the file.

  String(x), with a decimal point added when it has none

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Knots and multiplicities

  Turns a full knot vector into STEP's list of distinct knots with their counts.

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Edges that have collapsed to a point

  Leaves a zero-length boundary edge, such as a pole, out of the face loop.

  skipped when arc length ≤ 10⁻⁹ × surface size

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Trim loop written as a curve

  Writes a trim loop as one closed curve on the face.

  3D point = S(u, v) for each loop point; curve = straight segments through them

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Length unit of a file

  Millimeters per file unit.

  SI: prefix × 1000; conversion-based: value × its base unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Reads each control point and scales it by the file's unit.

  p′ = p × unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Trim loops read back

  Turns a face's boundary edges into trim loops in the surface's own parameters.

  33 samples per curved edge; each projected to (u, v); outer when |area| < 0.999 × (u range × v range)

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Volume and box of each body

  The volume and bounding box the body list shows.

  volume = Σ volume of the body's solids; box = min and max over them

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Move matrix

  The matrix a Move body feature applies: by distances, point to point, about an axis, or plane to plane.

  points: T(to − from); align: rotate n_a onto ∓n_b about c, then t = n_b ((o_b − c) · n_b)

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Shape under a move or a uniform scale

  Applies the matrix or the scale to a solid in the kernel.

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Scale center and affine map

  Chooses the center of a scale and builds the map for unequal factors.

  c = picked point, or (box min + box max) × 0.5; A = diag(sₓ, s_y, s_z); t = c (1 − s)

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Quadrature nodes and weights

  The eight points and weights used to integrate the path's speed.

  Pattern along a path brep-bodies.js

  • PartCraft
  64-bit

• Length of the path

  The arc length of the path and the running length at each piece boundary.

  piece length = h Σ wᵢ |C′(m + h xᵢ)|, h = (b − a) ÷ 2, m = (a + b) ÷ 2; L = Σ pieces

  Pattern along a path brep-bodies.js

  • PartCraft
  64-bit

• Parameter at an arc length

  Finds the curve parameter that lies a given length along the path.

  tₙ₊₁ = tₙ − (s(tₙ) − s) ÷ |C′(tₙ)|, kept inside the bracket by bisection

  Pattern along a path brep-bodies.js

  • PartCraft
  64-bit

• Path point nearest the seed

  The arc length of the path point closest to the center of what is copied.

  root of (C(s) − p) · C′(s) = 0 by bisection, beside the nearest of 8 samples per piece

  Pattern along a path brep-bodies.js

  • PartCraft
  64-bit

• Turn of each copy

  Rotates each copy so it follows the path's direction without twisting.

  R ← rot(t_prev → t) × R, in n = ⌈distance ÷ (L ÷ 2000)⌉ steps between stations

  Pattern along a path brep-bodies.js

  • PartCraft
  64-bit

• Modal coordinate word

  Decides whether an address word such as X or Z is written on a line, and writes its value.

  written when round(v × 1000) ≠ round(last × 1000); text = letter + v to 3 decimals

  G-code written by text post definitions cam-postdef.js

  • PartCraft
  64-bit

• Numbers in a line template

  Fills the {variable} holes of a raw, comment or word template with values.

  G-code written by text post definitions cam-postdef.js

  • PartCraft
  64-bit

• Whole-number filter

  The int filter a post can put on a value, for example a spindle speed.

  int(s) = round(number(s))

  G-code written by text post definitions cam-postdef.js

  • PartCraft
  64-bit

• Arc center offsets I and J

  The center of an arc move, written relative to the start point.

  I = round₃(cₓ − round₃(X_last)); J = round₃(c_y − round₃(Y_last))

  G-code written by text post definitions cam-postdef.js

  • PartCraft
  64-bit

• Equality in post conditions

  The == and != tests a post definition can make on two values.

  G-code written by text post definitions cam-postdef.js

  • PartCraft
  64-bit