Calculations

Calculations

• Point inside a zone

  Whether an item's position lies inside a zone volume.

  sphere: |p − c| ≤ r; cylinder: plan distance ≤ r and z₀ ≤ z ≤ z₁

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

  • CadCraft
  • BimCraft
  64-bit

• Circle as a polygon

  The ring of points that stands for a circular zone in exported files.

  (cx + r·cos(2π·k ÷ 48), cy + r·sin(2π·k ÷ 48)), rounded to 6 decimals

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

  • CadCraft
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  64-bit

• Zone footprints as GeoJSON

  The plan footprints of the zones as a GeoJSON file in local meters.

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

  • CadCraft
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  64-bit

• Zone footprints as DXF

  The same footprints as DXF polylines, one layer per level.

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

  • CadCraft
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  64-bit

• Zone volume mesh

  A closed triangle mesh of a sphere, hemisphere, cylinder or extruded polygon zone.

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

  • CadCraft
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  64-bit

• 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
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  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
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• 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

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  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

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• 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
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• 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
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  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

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• 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
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  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
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  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

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  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

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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

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• 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

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  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
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  • 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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  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
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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
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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

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  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
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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

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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

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• 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

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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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  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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  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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• 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

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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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• 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

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  64-bit

• Matrix inverse

  The inverse of the normal matrix, used for the solution and for the cofactors behind the error ellipses.

  [N | I] → [I | N⁻¹] by row operations

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Observation equations

  For each distance, direction, angle, azimuth, vector or control observation: observed minus computed, and its partial derivatives.

  l = observed − computed; ∂d/∂x = Δx ÷ d; ∂az/∂x = Δy ÷ d²; ∂az/∂y = −Δx ÷ d²

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Gauss-Newton iteration

  Builds the weighted normal equations, solves for the corrections and applies them until they are small.

  N = AᵀPA; u = AᵀPl; Δx = N⁻¹u; P = 1 ÷ σ²

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Variance factor, redundancy and standardized residuals

  The a posteriori standard deviation of unit weight, each observation's redundancy number and its standardized residual.

  σ̂₀ = √(vᵀPv ÷ dof); r = 1 − w·aᵀQa; w-test = v ÷ (σ·√r)

  Survey network adjustment adjust.js

  • CivilCraft
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• Error ellipses

  The semi-axes and azimuth of each free point's confidence ellipse, and its standard deviations in x and y.

  λ = (qxx + qyy ± √((qxx − qyy)² + 4qxy²)) ÷ 2; axis = k·σ̂₀·√λ; k = √(−2 ln(1 − confidence))

  Survey network adjustment adjust.js

  • CivilCraft
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• Chi-square limits

  The lower and upper limits the variance test compares vᵀPv with.

  χ²(p, k) ≈ k·(1 − 2/(9k) + z·√(2/(9k)))³

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Critical value for the residual test

  The two-sided normal critical value for the chosen significance level.

  t = √(−2 ln(α/2)); z ≈ t − (c₀ + c₁t + c₂t²) ÷ (1 + d₁t + d₂t² + d₃t³)

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Leveling network

  Adjusted heights of the free points from observed height differences, with residuals and standard deviations.

  weight = 1 ÷ σ², or 1 ÷ length; h = N⁻¹u

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Traverse by the compass rule

  Traverse coordinates, the misclosure, and each point corrected in proportion to the distance traveled.

  x += d·sin(az), y += d·cos(az); correction = misclosure × (length so far ÷ perimeter); precision = perimeter ÷ misclosure

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Adjusted points as CSV

  The export of adjusted coordinates and ellipse values, one point per line.

  value written = round(v × 10⁴) ÷ 10⁴

  Survey network adjustment adjust.js

  • CivilCraft
  64-bit

• Clothoid point

  The offsets along and across the tangent, and the turn so far, at a distance along a clothoid.

  θ = l² ÷ (2·R·Ls); x = l·Σ (−1)ⁿ θ²ⁿ ÷ ((2n)!·(4n+1)); y = l·Σ (−1)ⁿ θ²ⁿ⁺¹ ÷ ((2n+1)!·(4n+3))

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Gauss-Legendre quadrature

  The integral the other transition kinds and the cubic parabola's length are computed with.

  ∫ f ≈ (h ÷ 2)·Σ panels Σ₈ wₖ·f(mid + (h ÷ 2)·xₖ)

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Curvature shapes

  The curvature law of each transition kind and its integral, which gives the turn angle.

  κ = f(u) ÷ R; θ = Ls·F(u) ÷ R; u = l ÷ Ls

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Length along a cubic parabola

  The true length of a cubic parabola transition from its start to a distance along the tangent.

  s(x) = ∫₀ˣ √(1 + (v² ÷ (2·R·Ls))²) dv

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Cubic parabola parameter from a length

  The tangent length Ls of the cubic parabola whose curve has a given length.

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Point on a transition of any kind

  The offsets and turn at a distance along a transition, for all six kinds.

  x = ∫₀ˡ cos θ(s) ds; y = ∫₀ˡ sin θ(s) ds

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
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• Transition constants

  The end offsets Xc and Yc, the shift p and the tangent offset k of a transition, and its true length.

  p = Yc − R·(1 − cos θ); k = Xc − R·sin θ

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit