Calculations

Calculations

• Thread diameters

  The major, minor and pitch diameter and the thread depth of a size.

  ISO: depth = 5√3 ÷ 16 × P, d₁ = d − 2 × depth, d₂ = d − 0.649519 P; trapezoidal: depth = P ÷ 2, d₂ = d − P ÷ 2

  Thread dimensions and modeled threads threads.js

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• Size that fits a round face

  Chooses the thread size nearest a shaft's or a bore's diameter.

  shaft: nearest d; bore: nearest (d₁ + d₂) ÷ 2; accepted when the difference ≤ 0.12 × the face diameter

  Thread dimensions and modeled threads threads.js

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• Profile corners over one pitch

  The crest, flank and root corners of the thread profile as axial position and radius.

  ISO: crest P ÷ 8, root P ÷ 4; trapezoidal: crest = root = P (0.5 − tan(π ÷ 12) ÷ 2); flank = (P − crest − root) ÷ 2

  Thread dimensions and modeled threads threads.js

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• Profile radius at an axial position

  The radius of the thread profile at any position along the axis.

  r = r₀ + (r₁ − r₀)(w − u₀) ÷ (u₁ − u₀), w = u mod P

  Thread dimensions and modeled threads threads.js

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• Cross-section of a modeled thread

  The polygon cut square to the axis whose twisted extrusion is the thread.

  point = (r cos θ, r sin θ), r = profile(∓P θ ÷ 2π)

  Thread dimensions and modeled threads threads.js

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• Helical solid

  The thread's solid shape from the cross-section.

  turns = h ÷ P; slices = ceil(turns × max(24, segments)); twist = 360° × turns

  Thread dimensions and modeled threads threads.js

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• Cutting tool of a modeled thread

  The solid that is cut from a shaft or a bore to leave the thread, with its clearance.

  profile shifted by ∓clearance; start moved back by ceil(ext₀ ÷ P) × P

  Thread dimensions and modeled threads threads.js

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• Numbers in the thread callout

  Formats the pitch and the thread length written in a drawing's thread note.

  text = value rounded to 3 decimals, trailing zeros dropped

  Thread dimensions and modeled threads threads.js

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• Rounding to 3 decimals

  Rounds a number to a multiple of 0.001 before it is stored.

  round(v × 1000) ÷ 1000

  Add-on display bodies: stored coordinates and placement addons/addon-bodies.js

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• Body coordinates as stored

  The box corners or mesh positions, offset and turn of one body as they go into the part file.

  Add-on display bodies: stored coordinates and placement addons/addon-bodies.js

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• Body placed in the world

  Turns a body's points about the Y axis and moves them by its offset.

  x′ = x cos a + z sin a + t_x; y′ = y + t_y; z′ = −x sin a + z cos a + t_z; a = rotY × π ÷ 180

  Add-on display bodies: stored coordinates and placement addons/addon-bodies.js

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• Box of all add-on bodies

  The least and largest coordinates of every placed add-on body.

  Add-on display bodies: stored coordinates and placement addons/addon-bodies.js

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• Power and logarithm used

  Which power and logarithm functions the curves are evaluated with.

  Material packs: property curves by temperature engineer/material-pack.js

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• One curve segment

  The fitted curve's value at a temperature.

  y = 10^(Σ cᵢ Lⁱ), L = log10 T; or y = c₀ + T(c₁ + T(c₂ + T(c₃ + T c₄)))

  Material packs: property curves by temperature engineer/material-pack.js

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• Property at a temperature

  The property value with its unit, range, fit error and source.

  value = curve(T) × scale

  Material packs: property curves by temperature engineer/material-pack.js

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• Series coefficients

  The factors 1/n! of the sine and cosine series.

  cₙ = ±1 ÷ n!

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Angle reduction

  Brings an angle into [−π, π].

  n = floor(x ÷ 2π + ½); r = (x − n × 2π_hi) − n × 2π_lo

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine series

  The sine of a reduced angle.

  sin r = r(c₀ + c₁r² + … + c₁₂r²⁴)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine series

  The cosine of a reduced angle.

  cos r = c₀ + c₁r² + … + c₁₂r²⁴

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine

  sin x for any finite x.

  sin r = sin(π − r) for r > π/2; sin r = sin(−π − r) for r < −π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine

  cos x for any finite x.

  cos r = −cos(π − |r|) for |r| > π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Exponential

  The exponential eˣ of a number.

  k = floor(x ÷ ln 2 + ½); r = (x − k × ln2_hi) − k × ln2_lo; eˣ = 2ᵏ × Σ rⁿ ÷ n! (n = 0 … 18)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Natural logarithm

  The natural logarithm ln x of a number.

  x = m × 2ᵉ, m in [√2/2, √2]; ln m = 2(s + s³/3 + … + s²⁹/29), s = (m − 1) ÷ (m + 1); ln x = e × ln2_hi + (ln m + e × ln2_lo)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-2 logarithm

  The base-2 logarithm log₂ x of a number.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-10 logarithm

  The base-10 logarithm log₁₀ x of a number.

  log₁₀ x = ln x ÷ 2.302585092994046

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Whole-number power

  xⁿ for a whole n ≥ 0.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Power

  A number x raised to a power y.

  xʸ = exp(y × ln x) for a fractional y

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arctangent

  The arctangent atan x of a number.

  atan z = 2·atan(z ÷ (1 + √(1 + z²))), twice; then 4(z − z³/3 + … + z⁴¹/41)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Two-argument arctangent

  The angle of the point (x, y).

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arcsine

  The arcsine asin x of a number in [−1, 1].

  asin x = atan2(x, √((1 − x)(1 + x)))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arccosine

  The arccosine acos x of a number in [−1, 1].

  acos x = 2·atan2(√(1 − x), √(1 + x))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Tangent

  The tangent tan x of an angle.

  tan x = sin x ÷ cos x

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic sine

  The hyperbolic sine sinh x of a number.

  sinh x = (eˣ − e⁻ˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic cosine

  The hyperbolic cosine cosh x of a number.

  cosh x = (eˣ + 1 ÷ eˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic tangent

  The hyperbolic tangent tanh x of a number.

  tanh x = (e²ˣ − 1) ÷ (e²ˣ + 1)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cube root

  The cube root ∛x of a number.

  r₀ = exp(ln|x| ÷ 3); r ← r − (r³ − |x|) ÷ (3r²), twice

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sphere

  Signed distance to a sphere.

  d = √((x − cx)² + (y − cy)² + (z − cz)²) − r

  Field shapes: signed distance functions field/field.js

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• Box with rounded edges

  Signed distance to a box, optionally with rounded edges and corners.

  q = |p − c| − (half − round); d = |max(q, 0)| + min(max(q_x, q_y, q_z), 0) − round

  Field shapes: signed distance functions field/field.js

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• Cylinder

  Signed distance to a cylinder between two points.

  t = (p − m)·u; d_r = |p − m − t·u| − r; d_t = |t| − L ÷ 2; d = |max(d_r, d_t, 0)| + min(max(d_r, d_t), 0)

  Field shapes: signed distance functions field/field.js

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• Torus

  Signed distance to a torus about the z axis.

  d = √((√(x² + y²) − R)² + z²) − r

  Field shapes: signed distance functions field/field.js

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• Half space

  Signed distance to a plane.

  d = (n·p − d₀) ÷ |n|

  Field shapes: signed distance functions field/field.js

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• Solid of revolution

  Signed distance to the solid under a profile r(x) turned about the x axis.

  ρ = √(y² + z²); d = least distance from (x, ρ) to the profile's segments; inside when ρ ≤ r(x)

  Field shapes: signed distance functions field/field.js

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• Union, intersection and subtraction

  Combines shapes by union, intersection or subtraction.

  union = min(a, b); intersection = max(a, b); subtraction = max(a, −b)

  Field shapes: signed distance functions field/field.js

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• Smooth minimum

  Blends two shapes with a radius k.

  h = clamp(½ + (b − a) ÷ 2k, 0, 1); s = b + (a − b)h − k·h(1 − h)

  Field shapes: signed distance functions field/field.js

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• Offset and shell

  Grows or shrinks a shape, or hollows it to a wall thickness.

  offset: d − δ; shell inward: max(d, −d − t); shell centered: |d| − t ÷ 2

  Field shapes: signed distance functions field/field.js

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• Move, turn and scale

  A shape under a rigid motion and a uniform scale.

  q = Rᵀ(p − move) ÷ s; d = d₀(q) × s

  Field shapes: signed distance functions field/field.js

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• Rotation matrix

  The 3 × 3 matrix of a turn about an axis by an angle.

  R = c·I + s·[u]ₓ + (1 − c)·u uᵀ, c = cos θ, s = sin θ

  Field shapes: signed distance functions field/field.js

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• Beam or round cone

  Signed distance to a beam whose radius changes from one end to the other.

  t = clamp((p − a)·(b − a) ÷ |b − a|², 0, 1); d = |p − a − t(b − a)| − (r₀ + (r₁ − r₀)t)

  Field shapes: signed distance functions field/field.js

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• Lattice of beams

  The least distance to many beams.

  Field shapes: signed distance functions field/field.js

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• Ramped cell size and wall

  A number that changes linearly along an axis or with radius.

  v = v₀ + (v₁ − v₀) × clamp((p − a) ÷ (b − a), 0, 1)

  Field shapes: signed distance functions field/field.js

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