Calculations

Calculations

• 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

  • CadCraft
  • PartCraft
  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

  • CadCraft
  • PartCraft
  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

  • CadCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  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

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

• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

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

• Writing an object's transform

  Stores an object's placement in the exported scene.

  USD transforms and placement usd/build.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  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

• Systems the coordinates fit

  The catalog systems in which a project point falls inside the system's area of use.

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Latitude and longitude as degrees, minutes, seconds

  The status bar text for a latitude or longitude.

  seconds = round(|v| × 360000) ÷ 100, then split into D, M, S

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Latitude and longitude as decimal degrees

  The status bar text in decimal degrees.

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Typed latitude or longitude

  A typed angle, decimal or in degrees, minutes and seconds, with a sign or a hemisphere letter.

  value = ±(D + M ÷ 60 + S ÷ 3600)

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Local fit: project to grid

  Grid coordinates of a project point through a scale, rotation and shift.

  grid = to + scale · R(rotation) · (p − from)

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Misclosure of a third control point

  How far a two-point fit puts a third known point from its grid coordinates, as a distance and in parts per million.

  ppm = distance ÷ span × 10⁶

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Converting a project's coordinates

  Every stored x and y of the project (points, breaklines, boundary, alignment PIs, pads, structures) passed through a conversion.

  Coordinate system helpers crs-pick.js

  • CivilCraft
  64-bit

• Arc from two points and a bulge

  The center, radius, start angle, sweep and length of the arc between two vertices.

  k = (1 − b²) ÷ (4b); center = chord middle + k·(−Δy, Δx); r = c·(1 + b²) ÷ (4|b|); sweep = 4·atan(b); L = r·|sweep|

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Bulge of an arc through three points

  The bulge of the arc drawn from a start, through a middle point, to an end.

  sweep = 2·(π − angle at the middle point); bulge = sign × tan(sweep ÷ 4)

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Bulge for a typed radius

  The bulge of the shorter arc of a given radius between two vertices.

  Δ = 2·asin(c ÷ 2r); bulge = ±tan(Δ ÷ 4)

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Point and direction along a segment

  The plan point and travel direction at a fraction of a straight or arc segment.

  arc: p = c + r·(cos θ, sin θ), θ = a₀ + sweep × u

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Segment lengths and grades

  Each segment's plan length, chord, 3D length, rise and grade.

  length = chord or r·|sweep|; 3D length = √(L² + Δz²); grade = Δz ÷ L

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Stations along a feature line

  The plan distance from the start to each vertex.

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Elevation from a grade

  The far vertex of a segment moved to the elevation a typed grade gives.

  z₂ = z₁ + grade × length

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• One grade between two vertices

  The vertices between two chosen ones placed on a single grade.

  z = z₀ + (z₁ − z₀) × (s − s₀) ÷ L

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Stepped offset

  A copy of the line moved sideways with mitred corners and raised by a height or a cross slope.

  corner shift = bisector × d ÷ cos(half angle); z + Δz, or z + |d| × slope

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Straight segment against an alignment element

  Where a straight feature line segment crosses a line or arc of the alignment, with the station.

  line: 2×2 solve by cross products; arc: A·t² + B·t + C = 0, t = (−B ± √(B² − 4AC)) ÷ 2A

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Arc segment against an alignment element

  Where an arc of a feature line crosses a line or arc of the alignment.

  two circles: along = (r₁² − r₂² + d²) ÷ 2d; h = √(r₁² − along²)

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Feature line read from a file

  The vertices kept when a feature line is read from a project file or a drawing.

  Feature line grades, arcs, offsets and crossings featureline.js

  • CivilCraft
  64-bit

• Elevations of a newly drawn line

  The elevation each vertex of a new feature line gets.

  Feature line grades, arcs, offsets and crossings features-ui.js

  • CivilCraft
  64-bit

• Elevation and grade boxes

  A typed vertex elevation or segment grade applied to the line.

  grade = typed percent ÷ 100

  Feature line grades, arcs, offsets and crossings features-ui.js

  • CivilCraft
  64-bit

• Offset distance and amount

  The typed offset distance and height or slope passed to the stepped offset.

  slope = typed percent ÷ 100

  Feature line grades, arcs, offsets and crossings features-ui.js

  • CivilCraft
  64-bit

• Layers made end to end

  The depth limits of a boring's layers after gaps and overlaps are removed.

  Boring logs and the soil profile between borings geotech.js

  • CivilCraft
  64-bit