Calculations

Calculations

• Mesh volume

  The volume enclosed by a closed triangle mesh.

  V = Σ p · (q × r) ÷ 6

  Hazardous area extents and zone geometry process/safety/zones.js

  • CadCraft
  • BimCraft
  64-bit

• Release sources from the diagram

  Release sources with position, pressure and temperature taken from the equipment on a process diagram.

  P₀ = 101325 + dp × 10⁵; T = dt + 273.15; position = drawing coordinate × factor, rounded to 6 decimals

  Hazardous area extents and zone geometry process/safety/zones.js

  • CadCraft
  • BimCraft
  64-bit

• Mitigated event frequency

  The frequency of a scenario after its protection layers, against the target.

  f = Σ f_IE × Π p_modifier × Π PFD_IPL; required PFD = target ÷ f

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• Required PFD of a new layer

  The probability of failure a safety function must reach to close the gap.

  required PFD = target ÷ (frequency without that layer)

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• SIL band of a PFD

  The safety integrity level a probability of failure on demand falls in.

  SIL n when 10^−(n+1) ≤ PFD < 10^−n

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• Unavailability of one channel

  The probability that one channel has failed dangerously at a time since new.

  u(t) = 1 − exp(−λ_t·(t mod TI) − λ_n·t)

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• PFDavg by the simplified equations

  The average probability of failure on demand for each voting architecture.

  1oo1: λ·TI ÷ 2; 1oo2: ((1 − β)λ·TI)² ÷ 3 + β·λ·TI ÷ 2; 2oo3: ((1 − β)λ·TI)² + β·λ·TI ÷ 2

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• PFDavg by averaging over the life

  The average probability of failure on demand with imperfect proof tests, by numerical averaging.

  PFDavg = (1 ÷ life) × ∫₀^life [P(at least N − K + 1 channels failed) + common cause] dt

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• Safety function total

  The PFDavg of each subsystem, their total and the SIL of the total.

  PFDavg = Σ subsystem PFDavg

  Layer of protection and SIL arithmetic process/safety/lopa.js

  • CadCraft
  • BimCraft
  64-bit

• Line fit

  The best straight line through a run of center-line pixels, and how far the worst pixel lies from it.

  θ = ½ × atan2(2·Sxy, Sxx − Syy); ends = feet of the first and last points on the line through the centroid

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  • PartCraft
  64-bit

• Circle fit

  The best circle through a run of center-line pixels, its sweep and the angles of an arc.

  Kåsa: solve the 2 × 2 system for the center about the centroid, r = √(uc² + vc² + (Suu + Svv) ÷ n); then Gauss-Newton on d − r

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• Corner detection

  Where a chain of pixels turns sharply, so it can be cut there before fitting.

  turn = |atan2(r − q) − atan2(q − p)| over ± w steps; corner when turn ≥ 0.6 rad and largest nearby

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• Splitting a chain into lines and arcs

  Fits a line, else an arc, else splits the chain at the point farthest from its chord, and joins neighbors that fit as one.

  distance to chord = |(P − A) × (B − A)| ÷ |B − A|

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  • PartCraft
  64-bit

• Joining collinear lines

  Merges two lines that lie on the same straight line and nearly touch.

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
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  64-bit

• Squaring level and plumb lines

  Makes nearly horizontal and vertical lines exactly so, and gives lines of one wall the same coordinate.

  y = (y₁ + y₂) ÷ 2; shared coordinate = Σ(coordinate × length) ÷ Σ length

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• Closing corners

  Moves a line end to the point where its line meets a nearby line.

  s = ((B₁ − E) × v) ÷ (u × v); new end = E + s·u

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• Lines and arcs in image coordinates

  The traced lines and arcs with the window offset added and the arc end points worked out.

  start = (cx + r·cos a₀, cy + r·sin a₀); end = (cx + r·cos a₁, cy + r·sin a₁)

  Tracing a scan into lines and arcs raster/trace.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
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  64-bit

• Crossing of two traced lines

  The point where two traced line segments cross.

  d = (a₂ − a₁) × (b₂ − b₁); t = ((b₁ − a₁) × (b₂ − b₁)) ÷ d; point = a₁ + t·(a₂ − a₁)

  Tracing a scan into lines and arcs raster/snap.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  • PartCraft
  64-bit

• Snap points of a trace

  The ends, middles, arc centers, corners and crossings offered for snapping.

  middle = (p₁ + p₂) ÷ 2; arc middle at angle (a₀ + a₁) ÷ 2; corner = mean of the ends within 3 px

  Tracing a scan into lines and arcs raster/snap.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
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  64-bit

• Black-and-white level (Otsu)

  The grey level that best separates ink from paper for the whole sheet.

  maximize w_B × w_F × (m_B − m_F)² over the 256 levels

  Scan clean-up and JPEG decoding raster/image.js

  • CadCraft
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  • ModelCraft
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  64-bit

• Local threshold (Sauvola)

  Decides ink or paper for each pixel from the mean and spread of the pixels around it.

  ink when v ≤ mean × (1 + k × (sd ÷ 128 − 1)); sd = √(Σv² ÷ n − mean²)

  Scan clean-up and JPEG decoding raster/image.js

  • CadCraft
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  64-bit

• Sheet skew angle

  The angle by which the long lines of a scanned sheet are rotated.

  score(angle) = Σ rows² + Σ columns² of the ink projected at that angle; best angle on a 0.25° then 0.02° grid

  Scan clean-up and JPEG decoding raster/image.js

  • CadCraft
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  • ModelCraft
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  64-bit

• Rotating the image

  Turns a grey image by an angle about its middle.

  source = c + R(angle) × (pixel − c); value = bilinear mix of the 4 nearest source pixels

  Scan clean-up and JPEG decoding raster/image.js

  • CadCraft
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  • ModelCraft
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  64-bit

• JPEG inverse cosine transform

  Turns the stored frequency coefficients of an 8 × 8 block back into pixel values.

  pixel = round(Σ_u Σ_v C(x,u)·C(y,v)·coef(u,v) + 128), clamped to 0–255

  Scan clean-up and JPEG decoding raster/jpeg.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• JPEG color conversion

  Converts the decoded luminance and color-difference values to red, green and blue.

  R = Y + 1.402·Cr; G = Y − 0.344136·Cb − 0.714136·Cr; B = Y + 1.772·Cb

  Scan clean-up and JPEG decoding raster/jpeg.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
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  64-bit

• Table values in millimeters

  Converts each published row to millimeters, mm² and kg/m.

  mm = in × 25.4; mm² = in² × 25.4 × 25.4; kg/m = lb/ft × 1.48816394; r = (k_des − t_f) × 25.4

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
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  • DetailCraft
  • PartCraft
  64-bit

• Exact section area

  The cross-section area worked out from the dimensions, with fillets and rounded corners included.

  I: 2·b_f·t_f + (d − 2t_f)·t_w + (4 − π)·r²; RHS: 2t(h + b − 2t) − (4 − π)(r_o² − r_i²); CHS: π(D − t)·t

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Section outline

  The outline of a section as loops of points, centered on its bounding box.

  arc point = (cx + r·cos a, cy + r·sin a), a = a₀ + (a₁ − a₀)·i ÷ n

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Rounded rectangle loop

  The outer or inner loop of a rectangular hollow section with rounded corners.

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Loop area

  The signed area of an outline loop.

  A = ½ × Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Loop perimeter

  The length around an outline loop.

  L = Σ √(Δx² + Δy²)

  Steel section dimensions, areas and outlines shapes/sections.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Parameter levels

  The evenly spaced values of a parameter between its two bounds.

  vᵢ = from + (to − from) × i ÷ (steps − 1)

  Design study variants and ranking study/study.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Rounding of parameter values

  Rounds a parameter value before it is put in a variant.

  v → v to 12 significant digits

  Design study variants and ranking study/study.js

  • CadCraft
  • BimCraft
  • DetailCraft
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  64-bit

• Variant list

  Every combination of parameter levels, or a seeded random sample between the bounds.

  random: v = from + (to − from) × u, u = 32-bit integer ÷ 2³²

  Design study variants and ranking study/study.js

  • CadCraft
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  64-bit

• Meeting the limits

  Whether a variant's results meet every limit of the study.

  min goal: value ≤ limit; max goal: value ≥ limit

  Design study variants and ranking study/study.js

  • CadCraft
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  64-bit

• Trade-off front

  The variants that no other variant beats on every goal at once.

  a dominates b when a is no worse on every goal and better on at least one

  Design study variants and ranking study/study.js

  • CadCraft
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  64-bit

• Weighted ranking

  A score for each feasible variant from its goals and the weights a person gives.

  x = (value − lo) ÷ (hi − lo); score = Σ w × (x for max goals, 1 − x for min goals)

  Design study variants and ranking study/study.js

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  64-bit

• Load factor to the limit

  The factor by which every load can be multiplied before the design's utilization reaches 1.

  find λ with util(λ) = 1: double λ until util ≥ 1, then bisect with mid = √(lo × hi) until hi − lo ≤ 10⁻⁴ × hi

  Load reserve and escape route checks study/breakit.js

  • CadCraft
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  64-bit

• Escape route widths

  For each room, the route out whose narrowest door is widest, and its margin against the required width.

  width of a route = min of its door widths; margin = width − required

  Load reserve and escape route checks study/breakit.js

  • CadCraft
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  • DetailCraft
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  64-bit

• Part reserve from a stress study

  The load factor to first yield and to buckling of a solved part.

  λ = minimum safety factor ÷ wanted safety factor; utilization = 1 ÷ λ

  Load reserve and escape route checks study/breakit.js

  • CadCraft
  • BimCraft
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  64-bit

• Matrix product

  Combines two transforms into one.

  o[c][r] = Σₖ a[k][r] × b[c][k]

  USD transforms and placement usd/geom.js

  • CadCraft
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  • ModelCraft
  • TakeoffCraft
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  64-bit

• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

  • CadCraft
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  64-bit

• Transforming a point

  Applies a transform to a point.

  p′ = (M × [x, y, z, 1]) ÷ w

  USD transforms and placement usd/geom.js

  • CadCraft
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  64-bit

• Rotation about an axis

  The matrix of a rotation by an angle in degrees about X, Y or Z.

  a = deg × π ÷ 180; entries cos a, ± sin a

  USD transforms and placement usd/geom.js

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  64-bit

• Quaternion to rotation matrix

  The rotation matrix of an orientation stored as a quaternion.

  q ÷ |q|; R₀₀ = 1 − 2(y² + z²), R₀₁ = 2(xy + wz), …

  USD transforms and placement usd/geom.js

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• Value at a time

  An animated value at a given time, between its stored samples.

  v = a + (b − a) × (t − t₀) ÷ (t₁ − t₀)

  USD transforms and placement usd/geom.js

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  64-bit

• An object's own transform

  Builds the transform of one object from its list of translate, rotate, scale and matrix operations.

  M = op₁ × op₂ × … in the listed order; an inverted operation uses its inverse

  USD transforms and placement usd/geom.js

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  64-bit

• World transform

  The transform of an object in the scene, through all its parents.

  W = M_root × … × M_parent × M_own

  USD transforms and placement usd/geom.js

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• Scene units and up axis

  The transform from the scene's own units and up axis to meters with Z up.

  scale by metersPerUnit; Y-up scenes: rotate 90° about X

  USD transforms and placement usd/geom.js

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  64-bit

• Placement of instanced copies

  The transform of each copy of a point instancer from its position, orientation and scale.

  M = translate(p) × rotate(q) × scale(s)

  USD transforms and placement usd/scene.js

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