Calculations

Calculations

• Turn angle of a fitting

  The angle a two-port fitting turns the pipe through.

  turn = 180 − acos(p · q) × 180 ÷ π, rounded to 2 decimals

  Piping in an IFC file as PCF components ifc/pipes.js

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  64-bit

• Center point of a bend

  The corner where the axes of a bend's two ports meet.

  s = (b e − c d) ÷ (a c − b²), t = (a e − b d) ÷ (a c − b²); center = midpoint of the two closest points

  Piping in an IFC file as PCF components ifc/pipes.js

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  64-bit

• A group value by its code

  Turns one group's text into a real number, a whole number, a true or false, or text.

  codes 10–59, 110–149, 210–239, 460–469, 1010–1059: parseFloat; 60–99, 160–179, 270–289, 370–389, 400–409, 420–429, 440–459, 1060–1071: parseInt

  DXF reading into the shared drawing model dxf-model.js

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  64-bit

• Arc angles

  An arc's start and end angles in radians.

  angle = degrees × (π ÷ 180)

  DXF reading into the shared drawing model dxf-model.js

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  64-bit

• Multi-line text direction

  The direction vector of multi-line text, from the file's vector or from its rotation angle.

  x axis = group 11 when present; otherwise (cos(a), sin(a)), a = degrees × (π ÷ 180)

  DXF reading into the shared drawing model dxf-model.js

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  64-bit

• Hatch arc and ellipse edges

  The center, radius or axes, and start and end angles of the curved edges of a hatch boundary.

  angle = degrees × (π ÷ 180); missing end angle = 360 degrees

  DXF reading into the shared drawing model dxf-model.js

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  64-bit

• abs, round, ceil and floor

  The four functions an expression may call.

  round(x) = floor(x + 0.5) for x ≥ 0, −floor(−x + 0.5) for x < 0; ceil(x) = ⌈x − 1e-9·max(1, |x|)⌉; floor(x) = ⌊x + 1e-9·max(1, |x|)⌋

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Adding and subtracting

  Sums and differences of lengths, left to right.

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Multiplying and dividing

  Products and quotients of a length and plain numbers, left to right.

  Arithmetic on typed lengths lencalc.js

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  64-bit

• The answer

  The final value and whether it is a length or a plain number.

  Arithmetic on typed lengths lencalc.js

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• Unit sizes

  Builds each unit's row: its fraction of the base unit and the factor used for quick conversion.

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

  Unit table: reading and writing values unit-table.js Every app
  Exact

• Numbers and fractions

  Reads 6, 6.5, 6 1/2, 6-1/2 or 1/2 as a number.

  whole + numerator ÷ denominator

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Feet and inches

  Reads 12'6", 12'-6 1/2" or 6" as inches.

  inches = feet × 12 + inches

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Degrees, minutes and seconds

  Reads 45°30'15" or 45d30m15s as degrees.

  degrees + minutes ÷ 60 + seconds ÷ 3600

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Slopes

  Reads 2%, 3H:1V, 1:3 or 6/12 as a grade (rise over run).

  percent ÷ 100; H:V → V ÷ H; rise:run → rise ÷ run

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing feet and inches

  Writes a length as feet, inches and a fraction of an inch.

  n = round(|inches| × den); feet = ⌊n ÷ 12den⌋; inches = ⌊rest ÷ den⌋; fraction = rest ÷ den, reduced

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing degrees, minutes and seconds

  Writes an angle as degrees, minutes and seconds.

  t = round(|degrees| × 3600 × 10^dec); split by 3600 and 60

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing decimals

  Writes a number with a fixed count of decimals.

  text = v.toFixed(dec)

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Guard on every written measure

  Decides what is written when a value is missing or not a number.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Arc angles brought into one turn

  Brings an angle into the range from 0 up to 2π before it is written.

  r = a mod 2π; r + 2π when r < 0

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Arc

  Writes an arc's center, radius and two angles.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Line

  Writes a line's two end points.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Polyline vertices

  Writes the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Vertex widths

  Writes the start and end width at a polyline vertex.

  equal widths: one value, −|width|

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Linetype pattern length

  The total length of a linetype's dash pattern.

  length = Σ |element length|

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Text direction

  The x-axis direction written for a block of text.

  x axis = (cos θ, sin θ, 0)

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Block scale

  Writes the three scale factors of a block reference in the format's compact form.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Single-line text

  Writes a text's insertion point, height, rotation, width factor and oblique angle.

  DWG object values written and read dwg/objects-write.js

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• Extension record values

  Writes the numbers kept in an extension record.

  DWG object values written and read dwg/objects-write.js

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• Block content matrix of a multileader

  The 4 × 4 matrix written for a multileader's block content when the model gives none.

  M = translate(l) · rotate_z(θ) · scale(s)

  DWG object values written and read dwg/mleader-write.js

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  64-bit

• Line read

  Reads a line's two end points.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Polyline vertices read

  Reads the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• North direction of older geolocation data

  Turns the north angle of a version 1 geolocation object into a direction vector.

  north = (sin a, cos a)

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Fit point parameters and knots

  The parameter of each fit point along the spline, and the knot vector built from them.

  u₀ = 0; uᵢ = uᵢ₋₁ + |Qᵢ − Qᵢ₋₁|; knots = [0,0,0,0, u₁ … uₙ₋₂, L,L,L,L]

  Spline through fit points dwg/spline-fit.js

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  64-bit

• Cubic basis functions at a parameter

  The four B-spline basis values that weigh the control points at one fit point.

  Cox–de Boor recurrence, degree 3

  Spline through fit points dwg/spline-fit.js

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• Start condition

  The rows that pin the first control point and set the spline's start tangent or a natural start.

  P₀ = Q₀; P₁ − P₀ = T₀ × u₄ ÷ 3 or P₀ ÷ u₄ − P₁ (1 ÷ u₄ + 1 ÷ u₅) + P₂ ÷ u₅ = 0

  Spline through fit points dwg/spline-fit.js

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• End condition

  The rows that pin the last control point and set the spline's end tangent or a natural end.

  Pₘ₋₁ = Qₙ₋₁; Pₘ₋₁ − Pₘ₋₂ = T₁ × (L − uₘ₋₁) ÷ 3 or the zero second derivative row

  Spline through fit points dwg/spline-fit.js

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• Solve for the control points

  Solves the banded system for the x, y and z of every control point.

  A · P = B

  Spline through fit points dwg/spline-fit.js

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• Unit vector

  A vector scaled to length 1.

  n = a ÷ |a|; [0, 0, 0] when |a| ≤ 1e-15

  3D vectors, planes and transforms m3d/vec.js

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• Polygon normal

  The unit normal of a polygon given by its corner points.

  N = Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁) taken per axis, then normalized

  3D vectors, planes and transforms m3d/vec.js

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• Polygon area in 3D

  The area of a flat polygon in space.

  area = |Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁)| ÷ 2

  3D vectors, planes and transforms m3d/vec.js

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• Plane through a point

  A plane as a unit normal and an offset, with the normal's sign made canonical.

  n·p = d; first component of n with |n| > 1e-9 is made positive

  3D vectors, planes and transforms m3d/vec.js

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• Same plane test

  Whether two planes are the same plane.

  |n₁·n₂| > 1 − 1e-9 and |d₁ × sign(n₁·n₂) − d₂| ≤ 1e-6

  3D vectors, planes and transforms m3d/vec.js

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• Closest points of a line and a segment

  The nearest points between a ray's line and an edge.

  den = (d·d)(u·u) − (d·u)²; t = (B·E − C·D) ÷ den; s = (A·E − B·D) ÷ den, clamped to [0, 1]

  3D vectors, planes and transforms m3d/vec.js

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• Closest points of two lines

  The nearest points between two infinite lines, or nothing when they are parallel.

  den = A·C − B²; parallel when |den| < 1e-14 × A × C

  3D vectors, planes and transforms m3d/vec.js

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• Ray and plane

  The point where a ray's line meets a plane.

  t = (d_plane − n·o) ÷ (n·dir); p = o + t × dir

  3D vectors, planes and transforms m3d/vec.js

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• Ray and triangle

  The distance along a ray to a triangle, or nothing when it misses.

  Möller–Trumbore: det = e₁·(d × e₂); u, v, t from three scalar triple products ÷ det

  3D vectors, planes and transforms m3d/vec.js

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• Rotation about an axis through a point

  The 4×4 matrix that turns geometry by an angle about an axis through a given point.

  M = T(about) × R(axis, θ) × T(−about), R from Rodrigues' formula

  3D vectors, planes and transforms m3d/vec.js

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• Transform a point

  A point moved by a 4×4 matrix.

  p′ = (M × [x, y, z, 1]) ÷ w

  3D vectors, planes and transforms m3d/vec.js

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• Inverse of a 4×4 matrix

  The matrix that undoes a transform, used to move points into a group's own coordinates.

  M⁻¹ = adj(M) ÷ det(M), with det from twelve 2×2 minors

  3D vectors, planes and transforms m3d/vec.js

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