Calculations

Calculations

• Convert between two units

  A value in one unit as a value in another unit of the same kind.

  result = v × (n₁ × d₂) ÷ (d₁ × n₂)

  Unit conversion unit-table.js Every app
  Exact

• Typed text to a unit

  What was typed, with its unit, as a value in the unit the field wants.

  3 yd typed into a feet field = convert(3, yd, ft) = 9

  Unit conversion unit-table.js Every app
  Exact

• A value to the base unit

  A number in any unit as meters, square meters, cubic meters, radians or rise over run.

  base = v × n ÷ d (v × π × n ÷ d for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• A base value to a unit

  A stored base value as a number in the unit shown.

  value = base × d ÷ n (base ÷ π × d ÷ n for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• Member stiffness matrix

  The 12 by 12 stiffness of one beam-column in its own axes.

  EA ÷ L, GJ ÷ L, 12EI ÷ L³, 6EI ÷ L², 4EI ÷ L, 2EI ÷ L

  3D frame analysis frame/frame-solve.js

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• Equivalent nodal loads

  A load along a member as forces and moments at its two ends.

  p = ∫ N(ξ) · w(ξ) dx, by 4-point Gauss-Legendre quadrature

  3D frame analysis frame/frame-solve.js

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• Banded Cholesky factorisation

  Factorises the stiffness matrix once, for every load set.

  K = L · Lᵀ; pivot s = k_jj − Σ l²; mechanism when s ≤ 1e-11 × k_jj

  3D frame analysis frame/frame-solve.js

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• Forward and back substitution

  Displacements for one load set from the factorised matrix.

  L · y = f, then Lᵀ · u = y

  3D frame analysis frame/frame-solve.js

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  64-bit

• Add, subtract, multiply, divide

  The four operations on values that carry units.

  a + b and a − b need the same dimensions; a × b and a ÷ b combine them

  Calc sheets calc.js Every app
  64-bit

• Comparisons

  Pass or fail checks such as M <= 40 kN m.

  a ≤ b is true when a ≤ b + 1e-12 × max(1, |a|, |b|)

  Calc sheets calc.js Every app
  64-bit

• Number shown

  A result as text in the sheet.

  6 significant digits; exponent form from 1e9 up and below 1e-4

  Calc sheets calc.js Every app
  64-bit

• A piece as stored

  One stud, plate, header, joist or rafter with its length and its place in the panel.

  stored value = round(v × 10) ÷ 10

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Stud length of a wall

  The length of a full-height stud: the wall height less the plates.

  stud length = H − (top plates + 1) × plate thickness

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Studs on the spacing grid

  Where the regular studs stand, left out where an opening is.

  center c = spacing, 2 × spacing, …; stud from c − face ÷ 2 to c + face ÷ 2

  Framing panels and cut lists framing.js

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• Framing round an opening

  The jack studs, header, cripples and sill of one opening.

  jack = sill + opening height − bottom plate; header = opening width + 2 × stud face; cripple above = H − top plates − header top

  Framing panels and cut lists framing.js

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  64-bit

• Floor panel

  The joists, rim joists and blocking of a floor panel.

  joist length = span − 2 × rim thickness; blocking = clear gap between neighboring joists

  Framing panels and cut lists framing.js

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  64-bit

• Roof panel

  The rafters and ridge board of a gable roof panel, and its pitch.

  slope = √(run² + rise²); rafter = slope + overhang × slope ÷ run; pitch = atan2(rise, run) × 180 ÷ π

  Framing panels and cut lists framing.js

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  64-bit

• Piece count and total length

  How many pieces a panel has and their total length in meters.

  meters = Σ piece lengths ÷ 1000, rounded to 0.01

  Framing panels and cut lists framing.js

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  64-bit

• Cut list

  One row per mark: size, length, count and total length.

  total (m) = round(length × count) ÷ 1000

  Framing panels and cut lists framing.js

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• Panel placed in the model

  A point of the panel turned by the panel's angle and moved to its place in the model.

  P = O + u·(cos a, sin a) + n·(−sin a, cos a) + (0, 0, z), each coordinate rounded to 0.1 mm

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Timber profile and weight

  The profile a placed timber piece gets: its dressed size and a density for its weight.

  b × t = dressed size, rounded to 0.1 mm; density = 500 kg/m³

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Scale of the shop sheet

  The scale the panel is drawn at on its sheet, and the scale printed in the title block.

  k = min(drawing width ÷ panel width, drawing height ÷ panel height); printed scale = 1 : round(1 ÷ k)

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Scale text in the title block

  The line of the title block that states the scale and the piece count.

  text = "1:" + max(1, round(1 ÷ k))

  Framing panels and cut lists framing.js

  • BimCraft
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  64-bit

• Number format in the file

  Turns a number into the text written in the PDF.

  toFixed(4), trailing zeros removed; |v| < 5e-5 → 0

  Vector PDF output pdf.js

  • BimCraft
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  64-bit

• Millimeters to points

  The factor between millimeters and PDF points.

  K = 72 ÷ 25.4

  Vector PDF output pdf.js

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  Exact

• Page scale matrix

  The matrix that lets the rest of the page be written in millimeters.

  [K 0 0 K 0 0] with K written as num(K) = 2.8346

  Vector PDF output pdf.js

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  64-bit

• Page size in points

  The page box from the sheet size in millimeters.

  points = num(mm × K)

  Vector PDF output pdf.js

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  64-bit

• Circle as four curves

  A circle written as four cubic Bézier curves.

  control distance k = 0.5522847498 × r

  Vector PDF output pdf.js

  • BimCraft
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  64-bit

• Text position and matrix

  Where a text starts and how it is turned and sized.

  x = x₀ + dx·cos θ, y = y₀ + dx·sin θ; matrix = [h·cos θ, h·sin θ, −h·sin θ, h·cos θ, x, y]

  Vector PDF output pdf.js

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  64-bit

• SI value of a file unit

  Meters per file length unit and radians per file angle unit.

  SI unit: 10^prefix; conversion-based unit: factor × factor of its base unit; by name: degree π ÷ 180, inch 0.0254, foot 0.3048

  IFC units and placements ifc/geo-ctx.js

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• Axis placement as a matrix

  The matrix of an IfcAxis2Placement3D, 2D or IfcAxis1Placement.

  columns = (x, z × x, z, origin), x = reference direction projected square to z

  IFC units and placements ifc/geo-ctx.js

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• First projection axis

  The X axis of a placement: the reference direction projected into the plane normal to Z.

  x = normalize(v − z (v · z))

  IFC units and placements ifc/geo-ctx.js

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• Cartesian transformation operator

  The matrix of an IfcCartesianTransformationOperator, uniform or not, in 2D or 3D.

  columns = (x s₁, y s₂, z s₃, origin)

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• World matrix of a placement

  The position of an element in the model from its chain of relative placements.

  world = parent world × local

  IFC units and placements ifc/geo-ctx.js

  • BimCraft
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  64-bit

• Grid placement

  The point where two grid axes meet, with their offsets, as a placement.

  t = ((pB − pA) × dB) ÷ (dA × dB); point = pA + dA t

  IFC units and placements ifc/geo-ctx.js

  • BimCraft
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  64-bit

• Map conversion

  The matrix that takes model coordinates to map coordinates, with its rotation and scale.

  map = (E, N, H) + R(θ) · S · model; θ = atan2(ordinate, abscissa)

  IFC units and placements ifc/geo-ctx.js

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  64-bit

• Matrix product

  The product of two 4 × 4 matrices.

  IFC units and placements ifc/geo-math.js

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• Point through a matrix

  Transforms a point by a matrix.

  p' = M · (x, y, z, 1)

  IFC units and placements ifc/geo-math.js

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• Matrix inverse

  The inverse of a 4 × 4 matrix, or null when it has none.

  M⁻¹ = adjugate ÷ determinant

  IFC units and placements ifc/geo-math.js

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• Rotation about an axis

  The matrix that turns by an angle about an axis through the origin.

  R = c I + s [a]× + (1 − c) a aᵀ

  IFC units and placements ifc/geo-math.js

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• Polygon normal

  The normal of a 3D polygon, with a length of twice its area.

  n = Σ ((yₐ − y_b)(zₐ + z_b), (zₐ − z_b)(xₐ + x_b), (xₐ − x_b)(yₐ + y_b))

  IFC units and placements ifc/geo-math.js

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  64-bit

• Signed area of a 2D polygon

  The area of a profile outline, positive when counter-clockwise; used to orient outlines and holes.

  A = Σ (xₐ y_b − x_b yₐ) ÷ 2

  IFC units and placements ifc/geo-math.js

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• Positions and normals handed to the app

  Packs the mesh into the vertex and index arrays the apps read.

  vertices = Float32Array of (x, y, z, nx, ny, nz) per vertex

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

  • BimCraft
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  32-bit in part

• Planar face

  Adds one planar face with holes as triangles, all with the face's normal.

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

  • BimCraft
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  64-bit

• Mesh moved by a matrix

  Adds another mesh, transformed, with its normals turned to match.

  p' = M p; n' = normalize((M⁻¹)ᵀ n); a mirror swaps the winding

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

  • BimCraft
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  64-bit

• Signed volume

  The volume enclosed by the mesh, positive when the triangles face outward.

  V = Σ a · (b × c) ÷ 6

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

  • BimCraft
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  64-bit

• Shared vertices

  Merges vertices that fall in the same grid cell, for boolean operations.

  key = (round(x ÷ eps), round(y ÷ eps), round(z ÷ eps))

  Triangle mesh built from IFC geometry ifc/geo-mesh.js

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• Segments of an arc

  How many chords replace an arc of a given sweep.

  n = max(2, ceil(|sweep| ÷ 2π × 36))

  IFC curves and profiles as point lists ifc/geo-curve.js

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  Exact

• Arc through three points

  The circular arc of an indexed poly-curve from its start, middle and end points.

  center = a + ((|ab|² (ac × n)) + (|ac|² (n × ab))) ÷ (2 |n|²), n = ab × ac

  IFC curves and profiles as point lists ifc/geo-curve.js

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• Circle and ellipse points

  Points on a circle or ellipse between two parameters.

  p = M · (r₁ cos θ, r₂ sin θ, 0), θ = a₀ + sweep × i ÷ n

  IFC curves and profiles as point lists ifc/geo-curve.js

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  64-bit