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
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• 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
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• 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

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• Area under an arc

  The area between an arc and its chord.

  area = r² × (θ − sin θ) ÷ 2

  Takeoff lengths and areas geom.js

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• 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

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• Perimeter

  The length around a closed outline.

  P = Σ segment lengths, the closing segment included

  Takeoff lengths and areas geom.js

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• 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

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• 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

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• 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
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• 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
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• 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
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• Combine two transforms

  The product of two plan transforms, each six numbers.

  [a b c d e f]: x′ = a·x + c·y + e, y′ = b·x + d·y + f

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Block insert transform

  Where the content of a block lands: insertion point, rotation, scale, base point and its place in a row and column array.

  M = T(insert) · R(rotation) · S(sx, sy) · T(−base + (col × colSpacing ÷ sx, row × rowSpacing ÷ sy))

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Tilted block seen from above

  The plan transform of an insert whose extrusion direction is neither up nor down.

  Ax = unit(Wz × N) (or Wy × N when |Nx| and |Ny| < 1/64); Ay = unit(N × Ax); offset = N·z

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Downward extrusion

  Mirrors an entity in X when its extrusion direction points down.

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Transform points

  A list of points moved through a transform into the space's coordinates.

  x′ = a·x + c·y + e; y′ = b·x + d·y + f

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Segments in an arc

  How many straight segments stand in for an arc.

  n = max(8, min(128, ⌈|sweep| ÷ 2π × 72⌉))

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Points along an arc or circle

  An arc or circle as points on it.

  pᵢ = (cx + r·cos aᵢ, cy + r·sin aᵢ), aᵢ = a₀ ± sweep × i ÷ n

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Points along an ellipse

  An ellipse or elliptical arc as points on it.

  p(t) = c + R(rot) · (L·cos t, L·ratio·sin t), L = |major axis|, rot = atan2(my, mx)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Polyline with bulge arcs

  A polyline whose segments may be arcs, as points.

  θ = 4·atan(bulge); r = chord ÷ (2·sin(θ ÷ 2)); center = chord middle + normal × r·cos(θ ÷ 2)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Points along a spline

  A spline as points: its fit points, or samples of the B-spline from its control points.

  de Boor: dⱼ = (1 − a)·dⱼ₋₁ + a·dⱼ, a = (u − kᵢ) ÷ (kᵢ₊ₚ₋ᵣ₊₁ − kᵢ); point = (Σ w·P) ÷ Σ w

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• What a sheet viewport looks at

  The model point at the middle of a viewport on a sheet.

  vx = target_x + cos(tw)·cx + sin(tw)·cy; vy = target_y − sin(tw)·cx + cos(tw)·cy

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• A viewport's window on the model

  The box in model space that a viewport shows, used to say when it looks at nothing.

  half width = w ÷ 2 × viewHeight ÷ h; half height = viewHeight ÷ 2; corners turned by the twist

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Picture placement

  The transform from a picture's pixels to drawing coordinates.

  world = P + u·px − v·py + v·h (pixel y runs down; P is the lower left corner)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Lines of a paragraph of text

  Where each line of a multi-line text starts.

  pitch = h × 1.667 × line spacing; line i at insert + (top − pitch × i) along the up direction

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Text position, height and angle

  One line of text carried through the entity's transform.

  H = h × |second column of M|; angle = atan2(m₁, m₀) ± rotation (− when M mirrors)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Paper unit of a sheet

  Whether a sheet is in inches or millimeters.

  unit = mm when the sheet is wider than 100 units, else inches

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Line weight in millimeters

  A stored line weight, in hundredths of a millimeter, as millimeters.

  mm = v ÷ 100 for 0 ≤ v ≤ 211, else 0.25

  DWG and DXF drawings compiled for the plan view dwg-compile.js

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• Pointer position in drawing coordinates

  The drawing coordinates under the pointer, for the coordinate readout.

  x = cx + (px − W ÷ 2) ÷ zoom; y = cy − (py − H ÷ 2) ÷ zoom

  DWG and DXF drawings compiled for the plan view dwg-view.js

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• Model to sheet through a viewport

  The transform that carries a model point onto the sheet through one viewport.

  k = h ÷ viewHeight; paper = center + k · R(twist) · (model − view point)

  DWG and DXF drawings compiled for the plan view dwg-view.js

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• Millimeters per paper unit

  The size of one paper unit of a sheet in millimeters, for line weights.

  1 for millimeters, 25.4 for inches

  DWG and DXF drawings compiled for the plan view dwg-view.js

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  Exact

• 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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• 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

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• 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

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• 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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• 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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• 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

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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

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• 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

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• 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

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