Calculations

Calculations

• 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

  • CadCraft
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  • CivilCraft
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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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  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

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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

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  64-bit

• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

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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

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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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  64-bit

• 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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  64-bit

• 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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  64-bit

• Writing an object's transform

  Stores an object's placement in the exported scene.

  USD transforms and placement usd/build.js

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  64-bit

• Attribute value to its storage type

  Converts a list of numbers to the number type its USD attribute type calls for.

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

  • CadCraft
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  32-bit in part

• Mesh points and extent

  Stores the points of a mesh and its bounding box in the scene.

  extent = [min x, min y, min z, max x, max y, max z] of the points

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

  • CadCraft
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  32-bit in part

• Lamp intensity

  Converts a lamp's lumens to the intensity a renderer reads.

  intensity = lumens ÷ (683 × area × π)

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

  • CadCraft
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  32-bit, display only

• Single values in the binary file

  Writes one number, tuple or matrix into the binary USD file.

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

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  64-bit

• 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
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  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
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  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

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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
  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