Calculations

Calculations

• Collada positions and normals

  The vertex positions and face normals of each definition in the Collada file.

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada placement matrix

  The 4×4 matrix of each placement, written row by row.

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Is a face convex

  Decides whether a face is written as one polygon or as triangles in Collada.

  every turn (b − a) × (c − b) ≥ −1e-12 × size²

  Export to OBJ and Collada m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• IFC number format

  How every coordinate, direction component and elevation is written.

  text = x rounded to 9 decimals

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Shared points

  The corner points of the faces, each written once.

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Is a placement rigid

  Decides whether a group is exported in its own coordinates with a placement, or in world coordinates.

  |column| = 1 ± 1e-9; columns square within 1e-9; determinant > 0

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Placement of an element

  Where each exported element sits relative to its level.

  location = (t_x, t_y, t_z − level elevation); axis = third column; ref direction = first column

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Level of an element

  Which level an element is listed under.

  the highest level with elevation ≤ z_min + 1e-6

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Precision stated in the file

  The precision value the IFC file declares for its geometry.

  Export to IFC m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Drawing unit factors

  The factor from the drawing's unit ($INSUNITS) to meters.

  in 0.0254; ft 0.3048; mi 1609.344; mm 0.001; cm 0.01; m 1; km 1000; yd 0.9144

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  Exact

• Which unit factor is used

  Picks the factor for a drawing.

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Mirrored entities

  How an entity drawn in a flipped coordinate system is read.

  x → −x when the extrusion direction's z is negative

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Arc angles

  The start and end angles of a DXF arc.

  radians = degrees × π ÷ 180; mirrored: (a₀, a₁) → (π − a₁, π − a₀)

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Placing and scaling a segment

  Moves each line segment by the block placements above it and converts it to meters.

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

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Arcs, circles and bulges as segments

  The points that stand for an arc, a circle or a polyline bulge.

  n = max(2, ⌈|sweep| ÷ (2π ÷ 48)⌉); pⱼ = c + r (cos aⱼ, sin aⱼ), aⱼ = a₀ + sweep × j ÷ n; bulge: sweep = 4 atan b

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Ellipse as segments

  The points that stand for an ellipse or an elliptical arc.

  p = c + major × cos t + minor × sin t, minor = major turned 90° × ratio

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline (NURBS) as segments

  The points that stand for a spline with control points.

  de Boor on homogeneous points (x·w, y·w, w); 12 samples per knot span

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Spline stored with fit points only

  The points that stand for a spline that carries fit points and no control points.

  centripetal Catmull–Rom, 12 samples per span

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Bulge arcs in hatch boundaries

  The points of a curved side of a hatch boundary given as a polyline bulge.

  center = chord midpoint + ((1 ÷ b − b) ÷ 2) × (chord turned 90°) ÷ 2; sweep = 4 atan b

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Block insert transform

  The transform that places a block, with its scale, rotation and array spacing.

  M = T(x, y) · R(rot) · S(s_x, s_y) · T(−base); array offset = R(rot) · (col × d_c, row × d_r)

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Hatch holes

  Which loops of a hatch are outlines and which are holes.

  area = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|; a loop inside an odd number of larger loops is a hole

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• 3D face corners

  The corners of 3D faces and solids, which keep their heights.

  (x, y) placed by the block transform; all three × k

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Text line pitch

  The distance between lines of imported multi-line text, which is drawn as edges.

  pitch = height × 1.667 × spacing factor

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Edges of imported solids

  How the edges of an imported solid are matched to decide which ones show.

  key = round(x × 1e5), round(y × 1e5), round(z × 1e5)

  Import of DXF and DWG drawings into the 3D model m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• OBJ vertex

  A vertex line of an OBJ file.

  (x, y, z) → (x, −z, y)

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF accessor values

  The numbers of a glTF buffer, such as vertex positions.

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF node transform

  The matrix of a glTF node from its matrix, or from its translation, rotation and scale.

  M = T · R(q) · S

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• glTF vertex in the model

  A glTF vertex moved by its node transforms and turned to Z up.

  p′ = M · p, then (x, y, z) → (x, −z, y)

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada numbers

  How lists of numbers in a Collada file are read.

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada unit and up axis

  The change from the file's unit and up axis to Z-up meters.

  C = R_up · S(meter); node matrix → C · M · C⁻¹; point → C · p

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada node transforms

  The matrix of a Collada node from its matrix, translate, rotate and scale elements.

  rotate: angle in radians = degrees × π ÷ 180

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Collada vertex in the model

  A Collada vertex turned to Z-up meters.

  p′ = C · p

  Import of OBJ, glTF and Collada models m3d/io.js

  • CadCraft
  • ModelCraft
  64-bit

• Arc of a glyph

  The points of the round parts of letters.

  n = max(2, ⌈|a₁ − a₀| ÷ 15⌉); p = (c_x + r_x cos a, c_y + r_y sin a)

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Width of a string

  The advance of a string in drawing units, used to align and wrap text.

  width = (Σ (glyph width + 2) − 2) × h ÷ 9 × width factor

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Strokes of a string

  The polylines of a string at a given height, width factor and slant.

  x = x₀ + p_x × s_x + p_y × s × tan(oblique); y = p_y × s; s = h ÷ 9

  Text from drawings as stroked edges m3d/strokefont.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Day of the year

  The day number of a date.

  N = days before the month + day (+1 after February in a leap year)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  Exact

• Equation of time and declination

  How far sun time runs ahead of clock time, and the sun's angle above the equator, for the date.

  γ = 2π ÷ (365 or 366) × (N − 1 + (hour − 12) ÷ 24); eqtime = 229.18 × (0.000075 + 0.001868 cos γ − 0.032077 sin γ − 0.014615 cos 2γ − 0.040849 sin 2γ)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Hour angle and altitude

  The sun's height above the horizon.

  tst = hour × 60 + eqtime + 4 × lon − 60 × tz; H = tst ÷ 4 − 180°; cos Z = sin φ sin δ + cos φ cos δ cos H; altitude = 90° − Z

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Azimuth

  The sun's compass direction, in degrees clockwise from north.

  azimuth = atan2(sin H, cos H sin φ − tan δ cos φ) + 180°, wrapped to 0…360

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Direction toward the sun

  The unit vector shadows are cast from, in model axes (north is +Y, east is +X).

  dir = (sin a cos alt, cos a cos alt, sin alt)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Unit constants

  The Boltzmann constant, Coulomb constant, acceleration factor and pressure factor for reduced and real units.

  real: kB = 0.0019872041 kcal/mol/K; ke = 332.0637; accel = 418.4; pressure = 68568.4 atm per kcal/mol/ų

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Complementary error function

  erfc(x), used for the real-space part of the Ewald sum.

  erfc(x) ≈ t × exp(−x² + P₉(t)), t = 1 ÷ (1 + |x| ÷ 2); 2 − that for x < 0

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Pair parameters and the cutoff shift

  σ and ε for each pair of atom types and the energy subtracted so the pair energy is zero at the cutoff.

  σᵢⱼ = (σᵢ + σⱼ) ÷ 2; εᵢⱼ = √(εᵢ εⱼ); shift = 4 εᵢⱼ (s² − s), s = (σᵢⱼ ÷ r_c)⁶

  Molecular dynamics molecular/md.js

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  • BimCraft
  64-bit

• Kinetic energy

  The kinetic energy of the system in its energy unit.

  KE = ½ Σ mᵢ (vx² + vy² + vz²) ÷ accel

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Temperature

  The instantaneous temperature from the kinetic energy.

  T = 2 KE ÷ (dof × kB), dof = max(1, 3N − 3 − constraints)

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Pair forces and energies

  The Lennard-Jones and real-space Coulomb force, energy and virial of one pair of atoms.

  U = 4ε(s⁶² − s⁶) − shift, s⁶ = (σ² ÷ r²)³; Coulomb: ke qᵢqⱼ erfc(αr) ÷ r; excluded pairs: −ke qᵢqⱼ erf(αr) ÷ r

  Molecular dynamics molecular/md.js

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  • BimCraft
  64-bit

• Bond stretch

  The energy, force and virial of each harmonic bond.

  U = ½ k (r − r₀)²; F = −k (r − r₀) ÷ r along the bond

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Bond angles

  The energy, forces and virial of each harmonic angle.

  θ = acos(a·b ÷ (|a||b|)); U = ½ k (θ − θ₀)²; dU/dcos = −k (θ − θ₀) ÷ sin θ

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Dihedral angles

  The energy and forces of each periodic dihedral.

  φ = atan2(|b₂| b₁·n₂, n₁·n₂); U = k_d (1 + cos(nφ − δ))

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit

• Ewald reciprocal sum and self term

  The long-range part of the Coulomb energy and force.

  E = (2π ke ÷ V) Σ_k exp(−k² ÷ 4α²) ÷ k² × |S(k)|², |n|² ≤ kmax²; self = −ke α ÷ √π × Σ q²

  Molecular dynamics molecular/md.js

  • CadCraft
  • BimCraft
  64-bit