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

• Which side of an edge

  Whether three points turn counter-clockwise, clockwise, or lie on one line.

  sign of (b − a) × (c − a)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Is an edge Delaunay

  Whether the point across an edge lies inside the circle through the triangle on this side.

  sign of the in-circle determinant of a, b, c, d

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Triangle under a point

  The triangle a point falls in and its three barycentric weights.

  w₁ = ((y_b − y_c)(x − x_c) + (x_c − x_b)(y − y_c)) ÷ d, w₃ = 1 − w₁ − w₂

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Elevation at a point

  The surface elevation at an easting and northing.

  z = w₁·z₁ + w₂·z₂ + w₃·z₃

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Slope of a triangle

  Rise over run and the downhill direction of one triangle.

  n = u × v; slope = √((n_x ÷ n_z)² + (n_y ÷ n_z)²)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Plan and surface area

  The surface's area in plan and along the slope, with its elevation range.

  area in plan = Σ |n_z| ÷ 2; surface area = Σ |u × v| ÷ 2

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Length of a segment or arc

  The true length of one straight or curved piece of a run.

  θ = 4 · atan(b); r = c ÷ (2 · sin(θ ÷ 2)); length = r × |θ|

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  64-bit

• Area under an arc

  The area between an arc and its chord.

  area = r² × (θ − sin θ) ÷ 2

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  64-bit

• Area of an outline

  The area inside a closed outline by the shoelace formula, arcs included.

  A = ½ Σ (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ) + Σ arc areas

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  64-bit

• Perimeter

  The length around a closed outline.

  P = Σ segment lengths, the closing segment included

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  64-bit

• Net area with cut-outs

  Gross area, cut-outs, net area and perimeter of an area mark.

  net = gross − Σ cut-outs

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  64-bit

• Centroid

  The center of area of an outline, where its label goes.

  C_x = Σ (xᵢ + xᵢ₊₁) · fᵢ ÷ 6A, fᵢ = xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ

  Takeoff lengths and areas geom.js

  • TakeoffCraft
  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

• Knot span

  Which span of the knot vector a parameter falls in.

  i such that U[i] ≤ u < U[i + 1], by bisection

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N(i, p)(u) by the Cox-de Boor recurrence (A2.2)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Basis function derivatives

  The basis functions and their derivatives up to a chosen order.

  N⁽ᵏ⁾(i, p)(u) for k = 0 … d (A2.3)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Reproject points

  Every point of a file from one coordinate system to another.

  (x, y) in the source system → (x, y) in the target system, point by point

  Map coordinate reprojection reproject-batch.js

  • BimCraft
  • CivilCraft
  64-bit

• Map origin from latitude and longitude

  Eastings and northings of a typed latitude and longitude.

  (lon, lat) → (E, N), rounded to 0.000001 of the map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Model to map conversion

  Where a model sits on the map: origin, rotation and scale.

  X axis = (cos a, sin a) rounded to 1e-12; scale = model unit ÷ map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Line and line

  The point where two lines cross.

  t = ((b₁ − a₁) × d₂) ÷ (d₁ × d₂); parallel when |d₁ × d₂| < 1e-12 × |d₁| × |d₂|

  Line, arc and circle intersections geom.js

  • CadCraft
  64-bit

• Line and circle

  The points where a line meets a circle or arc.

  A·t² + B·t + C = 0; t = (−B ± √(B² − 4AC)) ÷ 2A

  Line, arc and circle intersections geom.js

  • CadCraft
  64-bit

• Circle and circle

  The points where two circles or arcs meet.

  x = (d² + r₁² − r₂²) ÷ 2d; h = √(r₁² − x²)

  Line, arc and circle intersections geom.js

  • CadCraft
  64-bit

• Is the point on the object

  Whether a computed crossing lies on the drawn part of each object.

  −tol ≤ t·L ≤ L + tol, with tol = 1e-7; arcs: the angle within 1e-7 of the sweep

  Line, arc and circle intersections geom.js

  • CadCraft
  64-bit

• Conjugate gradient solve

  The displacement of every node under the loads.

  repeat until ‖r‖ ÷ ‖b‖ < 1e-8

  Stress analysis fea.js

  • PartCraft
  64-bit

• Von Mises stress

  One equivalent stress from the six stress components.

  σᵥ = √(½[(σ_x − σ_y)² + (σ_y − σ_z)² + (σ_z − σ_x)²] + 3(τ_xy² + τ_yz² + τ_zx²))

  Stress analysis fea.js

  • PartCraft
  64-bit

• Stress and displacement at each node, and their maximum

  The values the plot colors by, and the maximum the report states.

  σ at a node = mean of its corners; |u| = √(u_x² + u_y² + u_z²); maximum over the nodes

  Stress analysis fea.js

  • PartCraft
  32-bit in part

• Volume, surface area and bounding box

  The measured properties of the finished body.

  volume, area, genus and bounding box, asked of the kernel

  Solid modeling kernel.js

  • PartCraft
  64-bit

• Vertex positions of the body

  The mesh that faces, edges and the display are built from.

  positions = 32-bit copy of the kernel's vertices

  Solid modeling kernel.js

  • PartCraft
  32-bit in part

• Sabine constant

  The constant both reverberation formulas multiply the volume by.

  C = 0.161 s/m

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Sound absorbed by the air

  The attenuation of a pure tone in air, in dB per meter, at the room's temperature, humidity and pressure.

  α = 8.686 f² [1.84e-11 (p_r ÷ p) √(T ÷ T₀) + (T ÷ T₀)^−2.5 (0.01275 e^(−2239.1 ÷ T) ÷ (f_rO + f² ÷ f_rO) + 0.1068 e^(−3352 ÷ T) ÷ (f_rN + f² ÷ f_rN))]

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Air absorption per meter

  Turns the dB figure into the energy absorption coefficient the formulas use.

  m = α ÷ 4.343

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Reverberation time per band

  The Sabine and Eyring times of a room in each of six octave bands, and the mid-frequency mean.

  A = Σ Sᵢαᵢ + extra; ᾱ = min(0.999, A ÷ S); Sabine = C V ÷ (A + 4mV); Eyring = C V ÷ (−S ln(1 − ᾱ) + 4mV)

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Reverberation over the target

  The finding raised when a room's mid-frequency time is over the person's target, with the absorption that would meet it.

  T = round(T_mid × 100) ÷ 100; need = round(max(0, V C ÷ target − (A₅₀₀ + A₁₀₀₀) ÷ 2))

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Distance from a point to a room outline

  How far an opening's middle is from the room's outline, to decide which room the opening belongs to.

  d = min over edges of |P − (A + t (B − A))|, t clamped to 0…1

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room perimeter

  The length of the room's outline, used for the wall area when the walls' finishes are not read from the model.

  P = Σ |pᵢ₊₁ − pᵢ|

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Is a point inside a room

  Decides whether a ceiling, a floor or a piece of furniture lies in a room.

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room height and opening areas

  The height the room is given and the area of the doors and windows in its walls.

  H = height ÷ 1000; opening area = (width ÷ 1000) × (height ÷ 1000)

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Wall area per finish

  The area of wall each finish covers in a room, less its doors and windows.

  wall area = Σ edge length × H; net = max(0, gross − glass − doors); each finish × net ÷ gross

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room volume and the figures reported

  The volume passed to the formulas and the area, height and volume listed for each room.

  V = floor area × H; aspect = longest ÷ max(shortest, 0.1)

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Fast Fourier transform

  The transform behind the band filters and the convolution.

  X_k = Σ x_n e^(∓2πi kn ÷ N), N a power of two

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit