Calculations

Calculations

• Torus

  The point at two angles on a torus, and the inverse.

  c = R + r cos v; S(u, v) = o + c cos u · x + c sin u · y + r sin v · z

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
  • ModelCraft
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  64-bit

• Surface of revolution

  The point of a profile curve turned about an axis by an angle, and the inverse.

  S(u, v) = o + h z + rr cos u + (z × rr) sin u, h = (C(v) − o)·z, rr = (C(v) − o) − h z

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Angle of a point on an ellipse

  The ellipse parameter of the point on the ellipse nearest a given point.

  f(t) = (E(t) − q)·E′(t) = 0, solved by t ← t − f ÷ f′

  STEP curves and surfaces: parameters from points step/geom.js

  • BimCraft
  • ModelCraft
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  64-bit

• Parameter of a point on a B-spline curve

  The curve parameter nearest a given point.

  STEP curves and surfaces: parameters from points step/geom.js

  • BimCraft
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  64-bit

• Parameters of a point on a B-spline surface

  The (u, v) of the surface point nearest a given point.

  [S_u·S_u S_u·S_v; S_u·S_v S_v·S_v] d = −[r·S_u; r·S_v], r = S(u, v) − q

  STEP curves and surfaces: parameters from points step/geom.js

  • BimCraft
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  64-bit

• Surface of linear extrusion

  The point of a curve swept along a vector, and the (u, v) of a point.

  S(u, v) = C(u) + v · vec; v = (q − C(u))·d ÷ |vec|, u = project(q − v · vec)

  STEP curves and surfaces: parameters from points step/geom.js

  • BimCraft
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  64-bit

• Offset surface

  The point of a surface moved a distance along its normal.

  S_off(u, v) = S(u, v) + d · n, n = (ΔS_u × ΔS_v) ÷ |…|

  STEP curves and surfaces: parameters from points step/geom.js

  • BimCraft
  • ModelCraft
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  64-bit

• Length and angle units

  The file's length unit in millimeters and its plane-angle unit in radians.

  mm = prefix × 1000 for a meter; a conversion-based unit = its value × its base unit

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Point coordinates

  A CARTESIAN_POINT in millimeters.

  (x, y, z) × unit in mm

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Radii and cone angles

  The radius of a cylinder, cone, sphere or torus in millimeters and a cone's half angle in radians.

  r × unit in mm; α × angle unit, accepted when 0 < |α| < π ÷ 2

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
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  64-bit

• Knot vectors

  The full knot vector of a B-spline from the file's knots and multiplicities, or the one a uniform, quasi-uniform or Bézier form implies.

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  Exact

• Model size and chord tolerance

  The size of the model and the distance a triangle may stray from a curved face.

  size = diagonal of the box around every point; tol = max(4e-4 × size, 1e-4 mm)

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Closing a loop of edges

  Drops the last point of an edge loop when it repeats the first.

  |first − last| ≤ 0.01 × tol

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Placement matrices

  A placement as a 4 × 4 matrix, the product of two and the inverse of one.

  M⁻¹ = [Rᵀ, −Rᵀ t]

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• A component placed in its assembly

  The transform that carries a component's coordinates into the assembly that uses it.

  T = M(A₂) × M(A₁)⁻¹

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Placed vertex positions

  Every vertex of a solid moved to its place and stored for the app.

  p′ = R p + t

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Faceted faces

  Triangles for a flat face given as loops of points.

  n = Σ ((yᵢ − yⱼ)(zᵢ + zⱼ), (zᵢ − zⱼ)(xᵢ + xⱼ), (xᵢ − xⱼ)(yᵢ + yⱼ))

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
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  64-bit

• Points along an edge

  The points of an edge from its start vertex to its end vertex along its curve.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• How finely a curve is divided

  The parameters at which an edge curve is sampled.

  circle step = min(π ÷ 24, 2 acos(1 − tol ÷ r)); else halve while the middle is more than tol from the chord or the turn exceeds π ÷ 24

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
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  64-bit

• Turn and crossing tests

  Whether three points turn left, right or run straight, and whether two segments cross.

  x = (a − o) × (b − o); left when x > 1e-9 × |ab| × |bc|, right when x < −1e-9 × |ab| × |bc|

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
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  64-bit

• Holes joined to the outer ring

  One ring made from a face's outer loop and its holes by bridges, ready for ear clipping.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Ear clipping

  Triangles of a simple polygon in the plane.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
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  64-bit

• A loop in the surface's (u, v)

  The parameters of each boundary point, with periodic directions unwrapped and poles split.

  uᵢ ← uᵢ + round((uⱼ − uᵢ) ÷ period) × period

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
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  64-bit

• Scaled layout and first triangles

  The (u, v) points scaled to lengths, the loops given their turning sense, and the first triangulation.

  M = (u × s_u, v × s_v); loop sense = sign of Σ (xᵢ yⱼ − xⱼ yᵢ)

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
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  64-bit

• Apex of a cone

  The parameter at which a cone's radius reaches zero, for a face that closes at the apex.

  R = |S(0, 0) − S(π, 0)| ÷ 2; k = |S(0, 1) − S(π, 1)| ÷ 2 − R; v_apex = −R ÷ k

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Interior grid spacing

  How far apart interior points are placed on each kind of surface.

  angle step = min(π ÷ 24, 2 acos(1 − tol ÷ r))

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Interior points inserted

  Grid points added inside a face and joined into the triangulation.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
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  64-bit

• In-circle test

  Whether a fourth point lies inside the circle through three others.

  det of rows (a − d, |a − d|²), (b − d, |b − d|²), (c − d, |c − d|²)

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Edge flips

  Inside edges flipped toward a Delaunay triangulation in the scaled (u, v).

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Refinement to the chord tolerance

  Inside edges split where the surface is farther than the tolerance from the triangles.

  split when |S(mid uv) − (P_a + P_b) ÷ 2| > tol, or when a triangle's center is more than tol from its plane

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Facing and self-check

  The 3D position of every point, the triangles turned to face outward, and a check of how far the mesh is from the surface.

  worst = max |S(center uv) − center of triangle| over up to 64 triangles; noted when worst > 25 × tol

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Unit factors

  What a number with a unit is multiplied by to give millimeters or degrees.

  mm 1; cm 10; m 1000; in 25.4; ft 304.8; deg 1; rad 180 ÷ π

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• A number with a unit

  Reads a number and converts it by the unit written right after it.

  v = Number(text) × factor

  Parameter expressions expr.js

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  64-bit

• Sine, cosine and tangent in degrees

  sin, cos and tan of an angle given in degrees.

  sin(x) = clean(Math.sin(x × π ÷ 180)); tan refused when |cos| < 1e-12

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Trig result rounded to 15 digits

  Rounds a sine, cosine or tangent to 15 significant digits and turns a value below 1e-15 into 0.

  |x| < 1e-15 → 0; otherwise Number(x.toPrecision(15))

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Inverse trigonometry and square root

  asin, acos, atan, atan2 in degrees, and sqrt.

  asin(x) = Math.asin(clamp(x, −1, 1)) ÷ (π ÷ 180); sqrt(x) = √max(0, x)

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Power, exponential and logarithms

  pow, exp, ln and log.

  pow(a, b) = aᵇ; ln and log need x > 0

  Parameter expressions expr.js

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  64-bit

• ceil, floor and int

  Rounds up, down or toward zero to a whole number.

  ceil(x) = ⌈x − ε⌉; floor(x) = ⌊x + ε⌋; int(x) = trunc(x + sign(x) × ε); ε = 1e-9 × max(1, |x|)

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• round(x, digits)

  Rounds to a number of decimal places from 0 to 10, halves away from zero.

  y = |x| × 10ᵈ; r = ⌊y + 0.5 + 1e-9 × max(1, y)⌋; result = ± r ÷ 10ᵈ

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Comparisons

  The six comparison operators, giving 1 or 0.

  tol = 1e-9 × max(1, |a|, |b|); a == b when |a − b| ≤ tol; a < b when a < b − tol

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Arithmetic operators

  Addition, subtraction, multiplication, division, remainder and power.

  a % b = a − b × ⌊a ÷ b⌋; division and remainder refuse b = 0

  Parameter expressions expr.js

  • CadCraft
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  64-bit

• Number as text

  The text a number is written as when it is put back into an expression or a message.

  Number(v.toPrecision(12)) as text

  Parameter expressions expr.js

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  64-bit

• Solve for a parameter

  The value of the named parameter for which the expression equals the target.

  secant: c = b − f(b) × (b − a) ÷ (f(b) − f(a)); accept when |f| ≤ 1e-9 × max(1, |target|, |x₀|)

  Solving an expression for one parameter expr.js

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  64-bit

• Rounding of the answer

  Replaces the answer by its 13-digit rounding when the two are practically equal.

  r = x to 13 significant digits; use r when |r − x| ≤ 1e-12 × max(1, |x|)

  Solving an expression for one parameter expr.js

  • CadCraft
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  64-bit

• 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