Calculations

Calculations

• 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

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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
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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

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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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  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

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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

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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

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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

• 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

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  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

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  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

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  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

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  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
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  64-bit

• Loop perimeter

  The length around an outline loop.

  L = Σ √(Δx² + Δy²)

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

  • CadCraft
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  64-bit

• Part sizes in millimeters

  The table dimensions of each size of a part family, with inch sizes converted.

  mm = in × 25.4; pitch = 25.4 ÷ threads per inch

  Standard parts and library items shapes/parts.js

  • CadCraft
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  64-bit

• Hexagon area

  The area of a hexagon from its width across flats.

  A = (√3 ÷ 2) × s²

  Standard parts and library items shapes/parts.js

  • CadCraft
  • PartCraft
  64-bit

• Mass from volume

  The mass of a part in kilograms from its volume in mm³.

  kg = mm³ × 10⁻⁹ × 7850, rounded to 4 decimals

  Standard parts and library items shapes/parts.js

  • CadCraft
  • PartCraft
  64-bit

• Inch length as a fraction

  Writes an inch length as a whole number and a fraction for the part's designation.

  n = round(v × 64); reduce n ÷ 64

  Standard parts and library items shapes/parts.js

  • CadCraft
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  64-bit

• Part dimensions and mass

  The full set of dimensions, the bounding box and the mass of one part.

  e = 2s ÷ √3; bolt mass = (hex area × k + π ÷ 4 × d² × l) × ρ; gear: d = m·z, d_a = m(z + 2), d_f = m(z − 2.5), d_b = d·cos 20°

  Standard parts and library items shapes/parts.js

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  64-bit

• Coordinate rounding in library items

  Rounds every coordinate written into a library item's drawing.

  round(v × 1000) ÷ 1000

  Standard parts and library items shapes/items.js

  • CadCraft
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  64-bit

• Steel section as a library item

  The section view or side elevation of a steel section, with its size and weight parameters.

  kg/m = published mass, else area × 10⁻⁶ × 7850; weight = kg/m × length ÷ 1000

  Standard parts and library items shapes/items.js

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  64-bit

• Bolt drawing dimensions

  The head, shank, chamfer and thread lines of a bolt's side view.

  thread root diameter = D − 1.22687 × pitch; chamfer = min(pitch, D ÷ 4)

  Standard parts and library items shapes/items.js

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  64-bit

• Standard part as a library item

  The drawing, size parameters and weight of a standard part as a library item.

  Standard parts and library items shapes/items.js

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  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

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  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

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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

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• 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

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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

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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

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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

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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

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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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• 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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• 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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• 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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• 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

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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

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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

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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