Calculations

Calculations

• Rotation vector from a matrix

  Axis times angle of a rotation, used for orientation error.

  θ = acos((tr R − 1) ÷ 2); v = θ ÷ (2 sin θ) × (R₃₂ − R₂₃, R₁₃ − R₃₁, R₂₁ − R₁₂)

  Rotations, transforms and small linear algebra (motion) math.js Every app
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• Rigid transforms

  Product, inverse and application of a rotation with a translation.

  A·B = (R_A R_B, p_A + R_A p_B); A⁻¹ = (R_Aᵀ, −R_Aᵀ p_A)

  Rotations, transforms and small linear algebra (motion) math.js Every app
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• Dense linear solve

  Solves the small systems of inverse kinematics and linkages.

  Gaussian elimination with partial pivoting

  Rotations, transforms and small linear algebra (motion) math.js Every app
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• Positive-definite test

  Whether a symmetric matrix, such as a mass matrix, is positive definite.

  Cholesky factorization succeeds

  Rotations, transforms and small linear algebra (motion) math.js Every app
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• Singular values

  The singular values of a Jacobian, used to detect singular poses and to count degrees of freedom.

  σᵢ = √(eigenvalues of J Jᵀ or Jᵀ J)

  Rotations, transforms and small linear algebra (motion) math.js Every app
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• Point to segment distance

  The distance from a point to a track center line or a polygon edge.

  t = clamp(((p − a) · (b − a)) ÷ |b − a|², 0, 1); d = |p − (a + t (b − a))|

  Board geometry and clearances geom.js Every app
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• Orientation of three points

  Which side of a line a point lies on.

  o = (b − a) × (c − a)

  Board geometry and clearances geom.js Every app
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• Segment crossing test

  Whether two segments cross or touch.

  cross when the orientation signs differ on both segments; touch when an orientation is 0 and the point lies in the other's box

  Board geometry and clearances geom.js Every app
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• Segment to segment distance

  The distance between two track center lines or edges.

  0 when they cross; else the least of four point-to-segment distances

  Board geometry and clearances geom.js Every app
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• Point in polygon

  Whether a point lies inside a zone, a pad or the board outline.

  even-odd rule along a ray in +x

  Board geometry and clearances geom.js Every app
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• Polygon area

  The signed area of a closed polygon.

  A = ½ Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)

  Board geometry and clearances geom.js Every app
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• Polygon to polygon distance

  The distance between two closed shapes.

  0 when one first vertex is inside the other; else the least edge-to-edge distance

  Board geometry and clearances geom.js Every app
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• Gap between copper shapes

  The edge-to-edge clearance between two tracks, pads or zones.

  gap = max(0, center-line distance − r₁ − r₂)

  Board geometry and clearances geom.js Every app
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• Rotation of a point

  A pad or outline point turned by an angle in degrees.

  (x cos θ − y sin θ, x sin θ + y cos θ), θ = degrees × π ÷ 180

  Board geometry and clearances geom.js Every app
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• Pad position on the board

  Where a footprint pad sits on the board, with its shape for the clearance check.

  world = part position + R(part rotation) × (±x, y)

  Board geometry and clearances board.js Every app
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• Wrap into an integer type

  A number brought into the range of SINT, INT, DINT and the other integer types.

  v mod 2ⁿ, moved into the signed range

  Controller standard functions and integer types std.js Every app
  Exact

• On-delay timer

  The elapsed time and output of a TON block.

  ET = min(now − t₀, PT); Q = ET ≥ PT

  Controller standard functions and integer types std.js Every app
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• Up counter

  The count and output of a CTU block.

  CV ← CV + 1 on a rising edge while CV < 32767; Q = CV ≥ PV

  Controller standard functions and integer types std.js Every app
  Exact

• Round to nearest, ties to even

  The rounding used when a real is converted to an integer.

  nearest whole number; a tie goes to the even neighbor

  Controller standard functions and integer types std.js Every app
  Exact

• Arithmetic functions

  ABS, SQRT, LN, LOG, EXP, the trigonometric functions, MIN, MAX, LIMIT, TRUNC, SHL and SHR.

  SQRT, LN, LOG, EXP, SIN and the rest from Math; SHL: x × 2ⁿ; SHR: ⌊x ÷ 2ⁿ⌋; TRUNC toward zero

  Controller standard functions and integer types std.js Every app
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• Type conversions

  REAL_TO_INT, INT_TO_REAL, BOOL_TO_INT, TIME_TO_REAL and the other A_TO_B functions.

  to an integer type: round to nearest (ties to even), then wrap; to a real type: unchanged

  Controller standard functions and integer types std.js Every app
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• Center, radius and sweep of a bulge arc

  The circle an arc segment lies on, from its two end points and its bulge.

  θ = 4·atan(b); k = (c ÷ 2) ÷ tan(θ ÷ 2); center = midpoint + k × left normal; r = |p − center|

  Arcs from bulges and three points geom.js

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• Arc through three points

  The bulge of the arc that starts at one point, passes through a second and ends at a third.

  center = circumcenter(p, m, q); sweep = atan2(q − center) − atan2(p − center) ± 2π; b = tan(sweep ÷ 4)

  Arcs from bulges and three points geom.js

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• Segment lengths of a run and their sum

  The length of each segment of an open run, and the run's total.

  L = Σ segLength(pᵢ₋₁, pᵢ, bᵢ₋₁)

  Arcs from bulges and three points geom.js

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• Snap to the nearest 45° direction

  Moves a picked point onto the nearest of the eight 45° directions from the point before it, keeping its distance along that direction.

  u = round(cos, sin of k·45° to 12 decimals); p′ = base + u × ((p − base)·u)

  Arcs from bulges and three points geom.js

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• Circle as two half-circle arcs

  Stores a drawn circle as two points on opposite sides of the center, each starting a half-circle arc.

  r = |edge − c|; points (c.x + r, c.y) and (c.x − r, c.y), b = 1 each

  Arcs from bulges and three points takeoffcraft.js

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• Arc reshaped by dragging its middle

  Recomputes a segment's bulge when its arc handle is dragged.

  Arcs from bulges and three points takeoffcraft.js

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• Length and area shown while drawing

  The length of the last segment, the run so far, the area and perimeter so far, or a circle's radius and area, shown in the status line as the pointer moves.

  length = L × k; area = A × k²; circle preview = π·r²

  Arcs from bulges and three points takeoffcraft.js

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• Check distance between two points

  The real distance between two picked points, in feet-inches, decimal feet and meters.

  ft = |b − a| × k; m = ft × 0.3048

  Arcs from bulges and three points takeoffcraft.js

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• Dropping repeated points when a mark is finished

  Removes a point that sits on the one before it, as a double-click adds.

  drop pᵢ when |pᵢ − previous kept point| ≤ 2 ÷ zoom

  Arcs from bulges and three points takeoffcraft.js

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• Arcs broken into chords

  Replaces each arc of a run or outline by points on the arc, at most one step apart.

  k = max(2, ⌈|θ| ÷ step⌉); pⱼ = center + r·(cos tⱼ, sin tⱼ), tⱼ = a₀ + θ·j ÷ k

  Outline checks: inside, crossing, cut-outs geom.js

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• Point inside an outline

  Whether a point lies inside a closed outline.

  Outline checks: inside, crossing, cut-outs geom.js

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• Two segments cross or touch

  Whether two straight segments cross or touch.

  d₁ = (d − c) × (a − c), …; cross when d₁, d₂ lie beyond ±s on opposite sides and d₃, d₄ too

  Outline checks: inside, crossing, cut-outs geom.js

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• Outline crosses itself

  Whether a closed outline crosses itself.

  Outline checks: inside, crossing, cut-outs geom.js

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• Outline that cannot be measured

  Refuses an outline with too few points, one that crosses itself, or one with no area.

  Outline checks: inside, crossing, cut-outs geom.js

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• Cut-out inside its area

  Checks that a cut-out lies fully inside the outline and does not overlap another cut-out.

  Outline checks: inside, crossing, cut-outs geom.js

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• Round half away from zero

  Rounds a number to a given count of decimals.

  r = ⌊|v|·10ᵈ + 0.5 + 1e-9·max(1, |v|·10ᵈ)⌋ ÷ 10ᵈ

  Rounding for display and for money units.js

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• Grouped fixed-point text

  Writes a number with a fixed count of decimals and thousands separators.

  Rounding for display and for money units.js

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• Feet and inches to the nearest fraction

  Writes a length in feet as feet, inches and a fraction of an inch.

  n = round(|ft|·12·denom + 1e-9); feet = ⌊n ÷ (12·denom)⌋; inches = ⌊rem ÷ denom⌋

  Rounding for display and for money units.js

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• Length in a chosen display unit

  Writes a length in feet-inches, decimal feet, inches, meters or millimeters.

  m = ft × 0.3048; mm = ft × 0.3048 × 1000

  Rounding for display and for money units.js

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• Feet per sheet unit for a named scale

  The scale number for a preset such as 1/4 inch = 1 foot or 1:100.

  ftPerUnit = (userUnit ÷ 72) × ratio ÷ 12

  Sheet scale: presets, calibration and drawing units units.js

  • TakeoffCraft
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• Scale from two points and a typed distance

  The scale that makes the distance between two picked points equal a typed real distance.

  ftPerUnit = realFt ÷ |b − a|

  Sheet scale: presets, calibration and drawing units units.js

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• Named scale a number matches

  Finds the preset scale closest to a scale number, if any is close enough.

  err = |ftPerUnit − s| ÷ s ≤ 0.005

  Sheet scale: presets, calibration and drawing units units.js

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• Feet per drawing unit by INSUNITS code

  The table that turns a drawing's unit code into feet per drawing unit.

  code 6: 1 ÷ 0.3048; code 21 as written: 1200 ÷ 3937

  Sheet scale: presets, calibration and drawing units cadcount.js

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• The drawing's length unit

  Reads the INSUNITS code and looks up feet per drawing unit.

  Sheet scale: presets, calibration and drawing units cadcount.js

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• Viewport transform, model to paper

  The matrix that places model space on the paper through a layout's viewport.

  k = h ÷ viewHeight; M = [k·c, k·s, −k·s, k·c, cx − k(c·vx − s·vy), cy − k(s·vx + c·vy)]

  Sheet scale: presets, calibration and drawing units cadset.js

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• Sheet size and the drawing-to-page transform

  Puts a layout on a page in points, or fits model space to a page about 2000 units on its longest side.

  layout: page = (paper − min) × ptU; model: k = 2000 ÷ extent, page = k·x + (24 − k·minX)

  Sheet scale: presets, calibration and drawing units cadset.js

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• Scale a drawing's sheet starts with

  Feet per page unit for a model sheet or a layout.

  model: f = ft ÷ k; layout: f = (1 ÷ ptU) ÷ s × ft, s = h ÷ viewHeight

  Sheet scale: presets, calibration and drawing units cadset.js

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• Drawing scale taken over by the takeoff

  Gives each sheet of an opened drawing the scale the drawing itself states, when the sheet has none yet.

  Sheet scale: presets, calibration and drawing units takeoffcraft.js

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• Calibration stored on the sheet

  Reads the typed distance, computes the scale and stores it with the two points and the text.

  Sheet scale: presets, calibration and drawing units takeoffcraft.js

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