Calculations

Calculations

• Material below a point

  Whether there is part material just under a height at (x, y), used to tell a through hole from a blind one.

  first surface below z faces up

  Part and stock in machine coordinates cam.js

  • PartCraft
  32-bit in part

• Circles of round edges

  The center and radius of each circular edge lying in a plane square to the tool.

  full circle: c = (a + mid) ÷ 2, r = |a − mid| ÷ 2; arc: circle through a, mid, b

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit

• Circle through three points

  The center and radius of the circle through three 2D points.

  d = 2 (a_x (b_y − c_y) + b_x (c_y − a_y) + c_x (a_y − b_y)); u_x = (|a|² (b_y − c_y) + |b|² (c_y − a_y) + |c|² (a_y − b_y)) ÷ d

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit

• Arc from start, middle and end

  The signed sweep of an arc about a known center that passes from a start through a middle point to an end.

  counterclockwise when mod(a_m − a₀, 2π) < mod(a₁ − a₀, 2π); sweep = ± mod(a₁ − a₀, 2π)

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit

• Exact outline of a flat face

  Turns a face's mesh boundary into lines and arcs whose ends are the exact edge ends.

  mesh corner within tol of an edge end → that end; mesh run within tol of an edge circle → arc on that circle

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit

• Hole position, size and depth

  The center, radius, top and bottom of a round face along the tool axis, and whether it is a hole and goes through.

  c, r from the exact cylinder; top, bottom = mesh z range snapped to an edge end within tol

  Part and stock in machine coordinates cam.js

  • PartCraft
  64-bit

• Chaining picked edges into a loop

  Orders picked edges into one closed outline and makes each start exactly on the previous end.

  ends join when closer than tol; then next.start = previous.end

  Part and stock in machine coordinates cam.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

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  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

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  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

• Splitting a loop at distances along it

  Cuts a loop at given distances from its start, used for tabs and ramps.

  t = (cut − length so far) ÷ piece length

  2D curve offsets for toolpaths cam-offset.js

  • PartCraft
  64-bit

• Depth levels

  The heights of the passes between the top and the bottom of a cut.

  n = ceil((top − bottom) ÷ stepdown − 1e-9); z_i = top − (top − bottom)(i + 1) ÷ n; last = bottom

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Clearance, retract and cut heights

  The heights an operation moves at, from the stock top and the operation's settings.

  clearance = stock top + clearance; retract = min(clearance, stock top + retract); bottom = geometry bottom + offset

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Arc move with changing height

  Writes one line or arc as moves, splitting an arc into half turns and sharing the height change among them.

  z after a piece = z + (z₁ − z₀) × piece length ÷ arc length; last piece ends at z₁

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Ramp along a path

  Lowers the tool linearly while it follows a chain of lines and arcs.

  z = z₀ + (z₁ − z₀) × length so far ÷ total length

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Facing rows

  The positions of the back-and-forth rows that face the stock.

  n = ceil(width ÷ (stepover × d) − 1e-9); row_i = b₀ + (b₁ − b₀) i ÷ n; run from a₀ − R − overhang to a₁ + R + overhang

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Pocket offset levels

  The successive inward offsets a pocket is cleared along.

  level k at distance d₀ + k × stepover, until nothing is left

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Helix entry

  The half-turn arcs that take the tool down into a pocket at a set ramp angle.

  pitch = 2π ρ tan(angle); half turns = ceil((z_from − z) ÷ (pitch ÷ 2) − 1e-9); z after half turn i = z_from − (z_from − z)(i + 1) ÷ n

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Ramp entry along a loop

  How far along the first loop the tool travels while ramping down a level.

  length needed = (z_from − z) ÷ tan(angle); whole loops = floor(needed ÷ L); remainder split off the loop

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Contour offset

  The path of the tool center round an outline.

  d = R + stock to leave; outside: offset by −d (grow); inside: offset by +d

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Lead-in and lead-out arcs

  Quarter-circle arcs that bring the tool onto and off a contour through the air side.

  center = start + ρ × left normal; quarter turn ending (or starting) at the contour's start; ρ halved up to 4 times until the arc is in air

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Holding tabs

  Where the tool lifts over tabs that keep the part attached.

  tab k centered at (k + ½) L ÷ tabs; lifted over tab width + tool diameter; tab top = bottom + tab height

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Drill depth and R plane

  How deep each hole is drilled and the height the drill returns to between holes.

  through: z = bottom − breakthrough − (tip length when full diameter is asked); R = min(retract, highest stock or part near the hole + R plane)

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Highest point between two holes

  The highest part or stock under the drill as it moves from one hole to the next.

  max of heights at 9 points across the tool, every 1 mm along the move

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Chamfer depth and offset

  How deep and how far out a chamfer mill runs to cut a chamfer of a given width.

  β = tool angle ÷ 2; depth = width ÷ tan β; offset = tip radius + tip offset × tan β; z = edge height − depth − tip offset

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Engagement angle of adaptive clearing

  The angle of the cutter's rim in material for a set cutting width, listed with the operation.

  angle = acos(1 − min(1, w ÷ R)) × 180 ÷ π

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Length of a toolpath segment

  The length of a straight or helical move, used for the machining time.

  line: |b − a|; arc: √((|sweep| × r)² + Δz²)

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• Points along a toolpath segment

  Sample points along an arc move for the simulation overlay and checks.

  step angle = 2 acos(1 − tol ÷ r); point i at a₀ + sweep × i ÷ n, z linear

  2.5-axis toolpaths cam-paths.js

  • PartCraft
  64-bit

• One-dimensional squared distance transform

  For each cell of a row, the squared distance to the nearest marked cell, by lower envelopes of parabolas.

  d(q) = min_p ((q − p)² + f(p)); parabolas meet at s = ((f(q) + q²) − (f(p) + p²)) ÷ (2q − 2p)

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Distance to the nearest marked cell

  The straight-line distance, in cells, from every cell to the nearest cell of a mask.

  columns then rows of the 1D transform; distance = √(squared distance)

  Adaptive clearing passes cam-adaptive.js

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  64-bit

• Grid over the region

  Lays a grid over the region and stores, per cell, whether it is inside and its exact distance to the walls.

  cell h = R ÷ 12 (× 1.25 until the grid fits); wall distance from the region's lines and arcs

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Contour of the distance field

  Traces the line where the field crosses zero, which becomes a pass.

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

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Field whose contour is the next pass

  Sets up the field that is zero where the tool center cuts the set width into the remaining material.

  F = (R − w) − distance to material (allowed cells); positive elsewhere

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Keeping passes off the walls

  Moves any pass point that came nearer the walls than the tool radius plus stock to leave.

  p ← nearest wall point + unit(p − nearest) × (R + leave + 1e-6)

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Simplifying a pass

  Drops points of a pass that lie within a tolerance of the line between their neighbors.

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

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit

• Marking what a pass clears

  Marks every grid cell the tool covers along a pass.

  cell cleared when its center is within R of the segment

  Adaptive clearing passes cam-adaptive.js

  • PartCraft
  64-bit