Calculations

Calculations

• Engagement check

  Re-simulates the passes on a finer grid and reports the largest engagement angle, the equivalent cutting width, uncut area and the smallest wall clearance.

  angle = 360 × (rim points in material) ÷ 360; width = R (1 − cos angle); gouge = min(dist to walls − R − leave)

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Height grid of the part

  Samples the part's top surface on a grid for the first search of the tool-tip height.

  H[i, j] = height at cell center; cell = max(0.02, R ÷ 8)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  32-bit in part

• Tool-tip height

  The lowest height at which the tool at (x, y) touches the part without cutting into it.

  z = max(H − h(ρ)) over grid samples within R; refine at ρ* = A + r_c s ÷ √(1 + s²) uphill; then max over triangle edges by ternary search; result = max(z + leave, floor)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  32-bit in part

• Slope filter

  Whether the surface under a point is within the slope range chosen for the operation.

  s = √(g_x² + g_y²); tan(from) − 1e-9 ≤ s ≤ tan(to) + 1e-9

  3D finishing toolpaths cam-finish.js

  • PartCraft
  32-bit in part

• Finishing one straight line

  The 3D points of a straight pass over the part, dense enough to stay within the tolerance.

  split a span while |z(mid) − (z₀ + z₁) ÷ 2| > tol ÷ 2

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Thinning a pass

  Drops points that lie on a straight run between their neighbors.

  drop p when its distance to the line through the last kept point and the next point ≤ tol ÷ 10

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Parallel pass lines

  The positions of the raster lines across the part at the chosen angle.

  n = ceil((v₁ − v₀) ÷ (stepover × d) − 1e-9); line k at v₀ + (v₁ − v₀) k ÷ n

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Cusp height

  The height of the ridge left between two ball passes a given distance apart on a flat surface.

  h = R − √(R² − (s ÷ 2)²)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Pass spacing for a cusp height

  The distance between passes that leaves ridges of a given height.

  s = 2 √(2 R h − h²); s = 2 R when h ≥ R

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Distance over the surface

  The distance along the part's top surface from the starting cells to every other cell.

  shortest path over grid steps; step length = √((di × c)² + (dj × c)² + ΔH²)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Step directions of the distance search

  The set of grid steps the surface-distance search may take.

  all (di, dj) with |di|, |dj| ≤ 5 and no common factor

  3D finishing toolpaths cam-finish.js

  • PartCraft
  Exact

• Lines of equal surface distance

  Traces the lines at a given surface distance, which become the scallop passes.

  crossing on a cell edge at f = (level − a) ÷ (b − a)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Scallop spacing and levels

  The spacing of scallop passes from the cusp height, and the surface distances they are placed at.

  R_b = ball or corner radius + leave; s = spacing(R_b, cusp); level k = (k + ½) s; grid cell = max(0.02, min(s ÷ 3, d ÷ 8))

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Simplifying a scallop pass

  Reduces the points of a 3D pass while staying within a quarter of the tolerance.

  keep the point farthest from the chord when its distance > tolerance ÷ 4, then repeat on both halves

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Tool grown by the stock to leave

  Models stock to leave by making the cutter larger.

  d′ = d + 2 × leave; corner radius′ = corner radius + leave (bull nose)

  3D finishing toolpaths cam-finish.js

  • PartCraft
  64-bit

• Simulation grid

  Sets the cell size of the stock grid and the tolerances used to call a gouge.

  cell = min(2, max(0.1, max(w, h) ÷ 500)); lateral tolerance = 2e-4 × diagonal + 0.01

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Cutting along a level move

  Lowers the stock cells under one level move of the tool.

  z_cell = min(z_cell, z + h(distance from cell center to the track))

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Cutting along a sloping move

  Cuts a move that changes height as a series of short level moves.

  n = ceil(|Δz| ÷ 0.01); piece i cut at the lower of its two end heights

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Stock volume

  The volume of stock left, from which the volume removed by each operation is found.

  V = cell² × Σ (z − stock bottom) over cells in the stock

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Part height with lateral tolerance

  The part's height at a cell, taken as the lowest of the center and eight points a small distance round it.

  min over the center and 8 points at the lateral tolerance, 45° apart

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Gouge detection

  Finds cells where the stock was cut below the part's surface.

  gouge when z_cell < eroded part height − 0.02

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Stock left and deepest gouge

  The volume of stock left above the part, the thickest stock left and the deepest gouge.

  remaining = cell² × Σ (z − max(part height, stock bottom)) where that exceeds 0.02

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Machining time

  The time a program takes: cutting moves at their feed, rapid moves at the machine's rapid rate, plus tool changes.

  seconds = 60 × (Σ L ÷ feed + Σ L ÷ rapid) + changes × change time

  Stock simulation and machining time cam-sim.js

  • PartCraft
  64-bit

• Arc center words I and J

  The arc center written relative to the arc's start point.

  I = round₃(c_x − round₃(start_x)); J = round₃(c_y − round₃(start_y))

  G-code and toolpath exports cam-post.js

  • PartCraft
  64-bit

• Axis words

  Writes X, Y and Z only when they change at the output resolution.

  write an axis when round₃(new) ≠ round₃(last written)

  G-code and toolpath exports cam-post.js

  • PartCraft
  64-bit

• Toolpath as CSV

  Writes every segment's start, end, arc center, sweep and feed as text.

  value = toFixed(6); sweep in degrees = sweep × 180 ÷ π

  G-code and toolpath exports cam-post.js

  • PartCraft
  64-bit

• Toolpath as JSON

  Writes the setup, tools and every operation's moves.

  JSON.stringify of the moves

  G-code and toolpath exports cam-post.js

  • PartCraft
  64-bit

• Spindle speed and feeds

  The spindle speed, cutting feed, plunge feed and ramp feed for a tool.

  rpm = round(1000 vc ÷ (π d)), limited to the machine's range; feed = rpm × flutes × f_z; plunge = feed ÷ 3; ramp = feed ÷ 2

  Cutting speeds, feeds and tool profiles cam-tools.js

  • PartCraft
  64-bit

• Rounding feeds to one decimal

  Rounds a feed rate to tenths.

  round1(v) = round(v × 10) ÷ 10

  Cutting speeds, feeds and tool profiles cam-tools.js

  • PartCraft
  64-bit

• Tool tip profile

  The height of the cutting surface above the tool's tip at a given distance from its axis.

  ball: R − √(R² − ρ²); bull: r_c − √(r_c² − (ρ − a)²) beyond a = R − r_c; drill and chamfer: (ρ − r_t) ÷ tan(angle ÷ 2); flat: 0

  Cutting speeds, feeds and tool profiles cam-tools.js

  • PartCraft
  64-bit

• Unit factors

  The factor from the file's units to millimeters.

  mm 1; cm 10; m 1000; in 25.4

  Nested sheets from DXF to toolpaths cam-sheets.js

  • PartCraft
  Exact

• Bulged segments to arcs

  Turns a polyline with bulges into lines and true arcs.

  r = (d ÷ 2)(1 + b²) ÷ (2|b|); offset = (d ÷ 2)(1 − b²) ÷ (2b); sweep = 4 atan(b)

  Nested sheets from DXF to toolpaths cam-sheets.js

  • PartCraft
  64-bit

• Extent of an outline

  The box round a sheet or a part outline, used for sheet size and for placing parts on sheets.

  min and max over points sampled along each piece at 0.05 chord tolerance

  Nested sheets from DXF to toolpaths cam-sheets.js

  • PartCraft
  64-bit

• Part coordinates on its sheet

  Moves a part's points so its sheet's lower left corner is the origin, in millimeters.

  x′ = (x − sheet x₀) × k; y′ = (y − sheet y₀) × k

  Nested sheets from DXF to toolpaths cam-sheets.js

  • PartCraft
  64-bit

• Elementary functions

  The one-argument math functions a script can call: roots, exponentials, logarithms, trigonometry, hyperbolic functions, rounding and angle conversion.

  log1p(x) = x − x² ÷ 2 + x³ ÷ 3 for |x| < 1e-4, else log(1 + x); asinh(x) = sign × log(|x| + √(x² + 1)); atanh(x) = ½ log((1 + x) ÷ (1 − x)); sind(x) = sin(x × π ÷ 180)

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Rounding to a number of decimals

  round(x, d): rounds to d decimal places.

  round(x × 10^d) ÷ 10^d

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Gamma function and factorial

  Γ(x), and factorial(n) = Γ(n + 1).

  whole x < 171: (x − 1)! by product; x < 0.5: π ÷ (sin(πx) Γ(1 − x)); else √(2π) t^(x − ½) e^(−t) Σ c_i ÷ (x − 1 + i), t = x + 6.5

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Binomial coefficient

  nchoosek(n, k): the number of ways to choose k of n.

  r ← r × (n − m + i) ÷ i for i = 1 … m, m = min(k, n − k); result = round(r)

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Spacing of numbers

  eps(x): the gap between x and the next larger 64-bit number.

  2^(floor(log₂|x|) − 52); eps(0) = 5e-324

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Evenly spaced values

  linspace(a, b, n): n values from a to b.

  value i = a + (b − a) × i ÷ (n − 1)

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Logarithmically spaced values

  logspace(a, b, n): n values from 10^a to 10^b.

  value i = 10 ^ (a + (b − a) × i ÷ (n − 1))

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Running sum and product

  cumsum and cumprod along a dimension.

  acc ← acc + x_k (or acc × x_k), stored at each step

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Variance and standard deviation

  var and std of a set of values.

  m = Σx ÷ n; var = Σ(x − m)² ÷ (n − 1), or ÷ n with weight 1; std = √var

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Median

  The middle value of a set, or the mean of the two middle values.

  sorted s; n odd: s[(n − 1) ÷ 2]; n even: (s[n ÷ 2 − 1] + s[n ÷ 2]) ÷ 2

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Dot product

  dot(a, b): the sum of products of two vectors.

  Σ a_i b_i

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Trapezoid integration

  trapz(y) or trapz(x, y): the area under sampled data.

  Σ (x_i − x_{i−1}) (y_i + y_{i−1}) ÷ 2

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• One-dimensional interpolation

  interp1(x, y, xi): linear, nearest or previous-value interpolation in a table.

  y = y_lo + (y_hi − y_lo)(q − x_lo) ÷ (x_hi − x_lo)

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Polynomial evaluation

  polyval(p, x): the value of a polynomial.

  s ← s × x + c for each coefficient (Horner)

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Polynomial fit

  polyfit(x, y, n): the least-squares polynomial of degree n through data.

  V_ij = x_i^(n − j); solve V p ≈ y

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit

• Matrix to text

  mat2str(A): text that can be typed back in to recreate a matrix.

  each value written with toPrecision(15) by default

  CraftScript core math functions cs-lib-core.js

  • PartCraft
  64-bit