Calculations

Calculations

• Arrays in the binary file

  Writes an array of numbers into the binary USD file.

  USD mesh coordinates in saved and exported files usd/usdc-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Reading a mesh

  The points, faces and attached values of a mesh read from a USD scene.

  USD mesh coordinates in saved and exported files usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Built-in shapes as meshes

  A cube, sphere, cylinder, cone, capsule or plane of a USD scene as a mesh.

  ring point = (r·cos θ, r·sin θ, z), θ = 2π·j ÷ 24

  USD mesh coordinates in saved and exported files usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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
  64-bit

• 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
  64-bit

• 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

• Angle to (−π, π]

  Brings a direction difference into one turn, so a deflection is read the short way round.

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Deflection, tangent lengths and curve center at a PI

  The deflection angle at each PI, the tangent length of a plain curve, and for a curve with transitions the center and the two tangent lengths.

  Δ = a₂ − a₁; T = R·tan(Δ/2); center from n₁·c = R + p₁, n₂·c = R + p₂; T₁ = k₁ − u₁·c, T₂ = u₂·c + k₂

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Circular curve and its facts

  The arc's center, start angle, length, external, middle ordinate and chord, and the PC and PT stations.

  L = R·Δ; E = R·(1 ÷ cos(Δ/2) − 1); M = R·(1 − cos(Δ/2)); chord = 2R·sin(Δ/2)

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Transition, curve, transition at one PI

  The circular arc left between two transitions and the stations TS, SC, CS and ST.

  Δc = |Δ| − θ₁ − θ₂; Lc = R·Δc

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Point and direction at a station

  The coordinates and forward direction on the alignment at a station.

  line: p₀ + s·(cos dir, sin dir); arc: c + R·(cos a, sin a), a = a₀ ± s ÷ R

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Station and offset of a point

  The nearest station on the alignment to a point, and how far left or right of it the point lies.

  line: s = (p − p₀)·u clamped to the element; arc: s = R × swept angle; offset = (p − foot) × direction

  Alignment lines and circular curves alignment.js

  • CivilCraft
  64-bit

• Station ticks

  The points at every whole multiple of the station interval.

  station = k × step, k whole

  Alignment lines and circular curves alignment.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
  64-bit

• 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
  64-bit

• 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
  64-bit

• 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
  64-bit

• 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
  64-bit

• 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
  64-bit

• 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

• Transition element and its long and short tangents

  Places a transition on the alignment and works out its own point of intersection.

  long tangent = Xc − Yc ÷ tan θ; short tangent = Yc ÷ sin θ

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit

• Point on a placed transition

  The project coordinates and direction at a distance along a transition element.

  p = p₀ + R(dir₀)·(x, ±y)

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit

• Nearest point on a transition

  The station on a transition closest to a given point, for station and offset.

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit

• Alignment as a polyline

  The alignment as vertices for linework and export: arcs and transitions become chords.

  pieces = max(1, ⌈turn ÷ maxTurn⌉, ⌈length ÷ maxSeg⌉); default maxTurn = 0.5°

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit

• Transitions as straight pieces

  Each transition cut into short lines so that crossings with other linework can be found; lines and arcs are kept exact.

  pieces = max(4, ⌈θ ÷ 0.25°⌉)

  Transition curves and sampled alignment geometry alignment.js

  • CivilCraft
  64-bit

• Meters per drawing unit

  The length of each $INSUNITS drawing unit in meters.

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  Exact

• Drawing units to project units

  The factor every imported coordinate is multiplied by.

  inch, yard, mile into feet: meters ÷ 0.3048; metric into feet: meters ÷ (1200 ÷ 3937); into meters: meters ÷ 1

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Object coordinate system

  The rotation from an entity's own coordinate system to world coordinates, from its extrusion direction.

  Az = N ÷ |N|; Ax = (Wy or Wz) × Az, normalized; Ay = Az × Ax

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Composing transforms

  The product of two 3×4 affine transforms, used for nested block references.

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Block reference placement

  The transform of one block reference: base point, scale, array spacing, rotation, insertion point and extrusion.

  M = parent · OCS · T(insert) · R(rotation) · T(column, row) · S(scale) · T(−base)

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Arcs and circles as vertices

  Points along a circular arc, at most 5° apart.

  n = max(2, ⌈sweep ÷ 5°⌉); p = c + r·(cos a, sin a)

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Polyline bulges as vertices

  The arc between two polyline vertices that carry a bulge, as intermediate points.

  θ = 4·atan(bulge); r = chord ÷ (2·sin(θ/2)); center = chord middle + r·cos(θ/2) along the normal

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Splines as vertices

  Points along a B-spline, rational or not, sampled evenly in its parameter.

  samples = min(4000, max(16, 32 × (control points − degree)))

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Splines stored as fit points only

  A smooth curve through the fit points when the file gives no control points.

  centripetal Catmull-Rom: knot step = √(distance between points); 8 vertices per span

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Ellipses as vertices

  Points along an ellipse or elliptical arc.

  minor = ratio × (N × major); p = c + major·cos s + minor·sin s

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• One elevation per object, and flat objects

  The ELEV attribute of an imported object, and whether its vertices keep a z at all.

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Feature lines read back from a drawing

  The vertices of a feature line written earlier by CivilCraft, scaled into project units.

  x, y, z × scale

  DWG and DXF linework into project units cad-linework.js

  • CivilCraft
  64-bit

• Survey point from a block's insertion point

  The world position of each block reference, and of each cell of an arrayed reference, in project units.

  world = OCS · insert; array cell offset rotated by the reference's rotation; × unit scale

  DWG and DXF linework into project units block-points.js

  • CivilCraft
  64-bit

• Number from attribute text

  The elevation or number read from a block attribute such as EL 105.32.

  DWG and DXF linework into project units block-points.js

  • CivilCraft
  64-bit

• Elevation kept for a block point

  Which elevation a point made from a block gets: the insertion z, an attribute, or none.

  DWG and DXF linework into project units block-points.js

  • CivilCraft
  64-bit

• Number from file text

  A coordinate, angle, distance or height read from a field of the file.

  Data collector files and polar reduction collector.js

  • CivilCraft
  64-bit

• Dotted degrees, minutes, seconds to radians

  An angle written as DDD.MMSSss, as many data collectors write it, as radians.

  angle = (D + M ÷ 60 + S ÷ 3600) × π ÷ 180

  Data collector files and polar reduction collector.js

  • CivilCraft
  64-bit

• Degrees or gons to radians

  A file angle in decimal degrees or gons as radians.

  degrees × π ÷ 180; gons × π ÷ 200

  Data collector files and polar reduction collector.js

  • CivilCraft
  64-bit