Calculations

Calculations

• Rotation about an axis

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

  R = c I + s [d]× + (1 − c) d dᵀ, c = cos a, s = sin a, a = deg × π ÷ 180; translation = p − R p

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Reflection in a plane

  The matrix that mirrors geometry in a plane.

  M = I − 2 n nᵀ; translation = 2 (o·n) n

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Matrix applied to a point

  Transforms a point by a 4 × 4 matrix.

  p′ = M₃ₓ₃ p + t

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Matrix product

  Multiplies two 4 × 4 matrices.

  C = A B

  Vectors, plane frames and transforms vec.js

  • PartCraft
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• Small linear system

  Solves A x = b for a small dense system.

  Gaussian elimination with partial pivoting, then back substitution

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Number shown with up to four decimals

  Formats a length or number for display.

  text = value rounded to dp decimals (4 by default); |value| < 10⁻⁹ shown as 0

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Cage move from the gizmo

  The transform applied to the selected cage vertices while the gizmo is dragged.

  M = W_now × W_start⁻¹

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Point for a new face

  The position stored when a face is drawn by clicks.

  existing vertex within 9 px, else ray ∩ plane y = 0 rounded to 0.001

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Direction of an edge extrusion

  The direction and distance open edges are extruded by.

  d = Σ (edge middle − face center); offset = d ÷ |d| × distance

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Tolerance for joining a new face

  The distance within which a new face's points join existing cage vertices.

  tol = 10⁻⁶ × max(1, size setting)

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Picked point as an anchor

  The anchor position in the part's own axes and the face normal at the pick.

  p = M⁻¹ p_pick, each coordinate rounded to 6 decimals; n = (b − a) × (c − a) ÷ length

  Anchor points of manual annotations ws-illustrate.js

  • PartCraft
  64-bit

• Is a face square to the tool axis

  Decides whether a picked flat or round face can be machined from the setup's tool axis.

  accepted when ||n·axis| − 1| ≤ 10⁻⁹

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Matching a pick to one already stored

  Decides whether a clicked face is already in the operation's geometry, to add or remove it.

  same when names agree and |at − ref| < 10⁻³

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Numbers shown in the panel

  Formats stock sizes, feeds, gouge depth and remaining stock for the panel.

  text = value rounded to dp decimals (3 by default), trailing zeros dropped

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Rounding for the panel

  Rounds a result to a few significant digits before the panel words it.

  Advanced simulation add-on: result summary and color bands addons/advanced-sim/lib/results.js

  • PartCraft
  64-bit

• Summary rows

  The load reached, largest displacement and stress, plastic strain, contact force and penetration of a result.

  load = √(F_x² + F_y² + F_z²) × λ; penetration % = 100 × penetration ÷ h

  Advanced simulation add-on: result summary and color bands addons/advanced-sim/lib/results.js

  • PartCraft
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• Color bands and legend

  Splits the plot mesh into bands of equal value range and gives each band's limits.

  band = floor((v − lo) ÷ (hi − lo) × n), clamped to 0 … n − 1; limits: lo + (hi − lo) × i ÷ n

  Advanced simulation add-on: result summary and color bands addons/advanced-sim/lib/results.js

  • PartCraft
  64-bit

• Permeability of free space

  The constant μ0 used in every hand formula.

  μ0 = 4 × 10⁻⁷ × π

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• Round conductor as a polygon

  The corners of the polygon that stands for a round wire.

  pᵢ = (cx + r cos(2πi ÷ n), cy + r sin(2πi ÷ n)), n = 48

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• C-core gap field

  The flux density in the air gap of a C-core electromagnet by the magnetic circuit formula.

  B = μ0 × NI ÷ ((g + l ÷ μr) × 10⁻³), l = 2(w − tk) + 2(h − tk) − g

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
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• Magnet in a closed iron frame

  The flux density in a permanent magnet set in one leg of a closed frame.

  B = Br × (lm ÷ μrm) ÷ (lm ÷ μrm + L ÷ μr), L = 2(w − tk) + 2(h − tk) − lm

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• Field round one wire

  The flux density at two radii from the center of a round conductor.

  B = μ0 × I ÷ (2π × r × 10⁻³), r = 2a

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• Force between two wires

  The force per meter between two parallel conductors.

  F = μ0 × |I₁ × I₂| ÷ (2π × d × 10⁻³)

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• Template sizes within their ranges

  Clamps each size a person types to the template's allowed range.

  Magnetics add-on: template shapes and hand formulas addons/magnetics/lib/templates.js

  • PartCraft
  64-bit

• Mesh volume

  The signed volume enclosed by a triangle mesh.

  V = Σ a·(b × c) ÷ 6

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
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• Mesh area

  The total area of a triangle mesh.

  A = Σ |(b − a) × (c − a)| ÷ 2

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
  64-bit

• Closed and manifold checks

  Counts open, non-manifold and flipped edges, bad vertices, pieces and the Euler number.

  Euler number = vertices used − edges + triangles

  Mesh volume, area and watertightness checks field/measure.js

  • PartCraft
  Exact

• Anchor point as assembled

  Where an annotation's anchor sits when the parts are assembled.

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
  64-bit

• Circle through three points

  The center and radius of the circle through three points.

  n = ab × ac; c = a + ((n × ab)|ac|² + (ac × n)|ab|²) ÷ (2|n|²); r = |c − a|

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
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• Measured value

  The length, angle or radius an annotation shows.

  length = |P₁ − P₀|; angle = acos(û·ŵ) × 180 ÷ π; radius from the circle through three points

  Manual annotations: measured lengths, angles and radii illustrate/annotate.js

  • PartCraft
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• Significand width of a number

  How many bits of a number this device keeps.

  bits = the first n where 1 + 2⁻ⁿ = 1

  Device number check accuracy.js Every app
  Exact

• Digits of a graphics float

  The number of significant digits stated for the graphics processor's float type.

  bits < 23: digits = max(1, ⌊(bits + 1) × log₁₀ 2⌋); bits ≥ 23: the fixed text "about 7"

  Device number check accuracy.js Every app
  64-bit

• Size and position text

  A dimension, port size or port position as text in millimeters.

  text = round(x × 10⁴) ÷ 10⁴

  Library configurator sizes library-configurator.js Every app
  64-bit

• Option value used for matching

  The text an option value is compared by when the page picks the variant that matches the selected options.

  numbers: round(v × 10⁴) ÷ 10⁴ as text; other values: trimmed text

  Library configurator sizes library-configurator.js Every app
  64-bit

• Required and found values as text

  A limit or a measured value in words, such as ≤ 196.85 mm or 863.6–965.2 mm.

  text = round(x × 100) ÷ 100, then the unit

  Check report values problems.js Every app
  64-bit

• Built-in unit factors

  The factor from each built-in unit of mass, time, force, pressure, energy, power, current, voltage and volume to SI.

  lb = 0.45359237 kg; lbf = 4.4482216152605 N; psi = 6894.757293168 Pa; psf = 47.88025898 Pa

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Constants and fallback lengths

  pi, e, standard gravity g0, and the length units used when no unit table is given.

  g0 = 9.80665 m/s²; in = 0.0254 m; ft = 0.3048 m; yd = 0.9144 m

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Unit lookup

  The SI factor and dimension of a unit name.

  deg = π ÷ 180; rad = 1

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Dimension comparison

  Whether two values have the same dimensions, so they can be added, compared or converted.

  |aᵢ − bᵢ| < 1e-9 for each of the five exponents

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Power

  a ^ b for a value with units.

  value = aᵇ; dimensions × b

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Conversion with to

  M to kN m: checks the dimension and records the unit to show the answer in.

  shown = value ÷ factor of the target unit

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Shown amount

  The number and unit a row displays, and the amount passed to other modules.

  amount = value ÷ factor

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Square root and absolute value

  sqrt(a) and abs(a) for a value with units.

  √a halves the dimensions

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Rounding to a step

  round, floor and ceil, with an optional step such as round(x, 5 mm).

  round(a ÷ s) × s

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Minimum, maximum, sum and average

  min, max, sum and avg of values of one kind.

  sum = Σ aᵢ in order; avg = sum ÷ n

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Trigonometry, logarithms and exponent

  sin, cos, tan, their inverses, atan2, ln, log10, exp and pow.

  sin, cos, tan of radians; asin, acos need |x| ≤ 1; ln, log10 need x > 0

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Reading an app parameter

  @Name: the app's parameter as a value in SI.

  value = parameter value × factor of its unit

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Values written to driven parameters

  What each drive would set in the app: the row's answer in the parameter's unit.

  value = row value ÷ factor of the unit; change when |current − value| > 1e-9 × max(1, |value|)

  Calc sheet units, functions and driven parameters calc.js Every app
  64-bit

• Rotation about an axis

  The rotation matrix for an angle about an axis.

  R = cos θ I + sin θ [a]× + (1 − cos θ) a aᵀ

  Rotations, transforms and small linear algebra (motion) math.js Every app
  64-bit

• Roll, pitch and yaw from a matrix

  The three angles of a rotation matrix.

  pitch = asin(−R₃₁); roll = atan2(R₃₂, R₃₃); yaw = atan2(R₂₁, R₁₁)

  Rotations, transforms and small linear algebra (motion) math.js Every app
  64-bit