Calculations

Calculations

• Check before a call

  Refuses a call when the calls per hour, the day's or month's spending, or the day's tokens could pass a limit.

  refuse when spent + worst-case cost > limit; worst case = cost(estimated input, largest allowed output)

  Spending on a person's own AI model ai-byok.js Every app
  64-bit

• Recorded cost and the token estimate

  Stores each call's cost and estimates the input tokens of the next request.

  estimated tokens = ⌈JSON length ÷ 3⌉ + 500

  Spending on a person's own AI model ai-byok.js Every app
  64-bit

• Flow through a passage

  Persons per second through a door or corridor of a given width.

  rate = max(0, Fsm × (width − 2 × boundary))

  People flow and evacuation time people.js Every app
  64-bit

• Speed and specific flow at a density

  Walking speed and flow per meter of width at a given density of people.

  S = k × (1 − 0.266 × D); Fs = S × D; k = 1.4 m/s

  People flow and evacuation time people.js Every app
  64-bit

• Rate and speed of each passage

  The flow rate and walking speed each passage gets, from its own values or the defaults for its kind.

  People flow and evacuation time people.js Every app
  64-bit

• Door service time and walking time

  How long a door takes per person, how long the passage takes to walk, and each room's distance to an exit in time.

  service = 1 ÷ rate; walk = length ÷ speed; to exit = least total walking time

  People flow and evacuation time people.js Every app
  64-bit

• Which way a room empties

  The next passage for people in a room: the one with the least remaining walking time to an exit.

  keep a hop when toExit(next) < toExit(room) − 1e-9; choose the least walk + toExit(next)

  People flow and evacuation time people.js Every app
  64-bit

• Evacuation results

  People evacuated, the time the last person reaches an exit, peak occupancy and density per room, and queues per door.

  time = latest arrival at any exit; density = peak people in a room ÷ its area

  People flow and evacuation time people.js Every app
  64-bit

• Joint frames at a pose

  Each joint's rotation and offset from its parent at the given joint values.

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Joint torques for a motion (inverse dynamics)

  The torque or force each joint needs for given positions, velocities and accelerations, with gravity, external loads and damping.

  outward: ω, α, a of each link; F = m × a_com; N = I α + ω × (I ω); inward: τᵢ = axis · n (revolute) or axis · f (prismatic) + damping × q̇ᵢ

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Mass matrix

  The joint-space mass matrix at a pose.

  column j = rnea(q, q̇ = 0, q̈ = unit vector j) with no gravity, damping or external loads

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Joint accelerations for given torques (forward dynamics)

  The acceleration of each joint under applied torques.

  M q̈ = τ − h

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Kinetic and potential energy

  The mechanism's kinetic, potential and total energy at a pose and velocity.

  KE = ½ q̇ᵀ M q̇; PE = −Σ m × (g · p_com)

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Simulated motion over time

  Joint positions and velocities at each time step under gravity and applied torques.

  RK4: y₊ = y + dt ÷ 6 × (k₁ + 2k₂ + 2k₃ + k₄); steps = round(T ÷ dt); t = step × dt

  Rigid-body dynamics of a mechanism dynamics.js Every app
  64-bit

• Courtyard of a footprint

  The rectangle around all pads, grown by a margin.

  courtyard = (min(x − w ÷ 2) − m, min(y − h ÷ 2) − m, max(x + w ÷ 2) + m, max(y + h ÷ 2) + m)

  Parametric PCB footprints footlib.js Every app
  64-bit

• Two-terminal chip

  Two pads at half the pitch either side of the center.

  x = ∓ pitch ÷ 2

  Parametric PCB footprints footlib.js Every app
  64-bit

• Dual-row surface-mount package

  Pads in two rows, numbered counter-clockwise from the top left.

  y = ((n ÷ 2 − 1) ÷ 2 − i) × pitch; x = ∓ row spacing ÷ 2

  Parametric PCB footprints footlib.js Every app
  64-bit

• Dual in-line through-hole package

  Drilled pads in two rows.

  y = ((n ÷ 2 − 1) ÷ 2 − i) × pitch; x = ∓ row spacing ÷ 2

  Parametric PCB footprints footlib.js Every app
  64-bit

• Pin header

  Drilled pads in one or two rows at a pitch.

  x = (r − (rows − 1) ÷ 2) × pitch; y = ((n − 1) ÷ 2 − i) × pitch

  Parametric PCB footprints footlib.js Every app
  64-bit

• Quad flat package

  Pads on four sides, numbered counter-clockwise from the top left.

  along a side: ((n − 1) ÷ 2 − i) × pitch; across: ± span ÷ 2

  Parametric PCB footprints footlib.js Every app
  64-bit

• Axial through-hole part

  Two drilled pads at half the lead pitch either side of the center.

  x = ∓ pitch ÷ 2

  Parametric PCB footprints footlib.js Every app
  64-bit

• Meets the limit

  True, false or null for a found value against a rule.

  min − e ≤ found ≤ max + e

  Pass or fail against a regulation limit addons/regs/lib/rows.js

  • CadCraft
  • BimCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Relative difference

  The fraction by which a coarse result differs from the refined result.

  |a − b| ÷ |b|

  Spread between coarse and refined results study/live.js

  • CadCraft
  • BimCraft
  • DetailCraft
  • PartCraft
  64-bit

• Ring orientation

  The signed area that decides if a ring runs counter-clockwise.

  A = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Polygon triangulation by ear clipping ifc/geo-tri.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Cross product and inside test

  Which side of a line a point lies on and whether a point is inside a triangle.

  cross = (bx−ax)(cy−ay) − (by−ay)(cx−ax)

  Polygon triangulation by ear clipping ifc/geo-tri.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Hole bridge

  The vertex of the outer ring a hole is joined to.

  Polygon triangulation by ear clipping ifc/geo-tri.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Ear test

  Whether a corner can be cut off as a triangle.

  Polygon triangulation by ear clipping ifc/geo-tri.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Triangulate rings on a plane

  The triangles of 3D rings that lie on one plane.

  Polygon triangulation by ear clipping ifc/geo-tri.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Vertices and loose edges

  The point positions of a model definition and its loose edges.

  Model points written to USD usd/m3d.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Mesh arrays

  The point, normal, UV and color arrays of a mesh.

  Model points written to USD usd/m3d.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Number to text

  The text a number is written as, by type.

  f32: text = shortest s with fround(+s) = fround(x)

  Number text in USD files usd/usda-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Shortest digits

  A number to p significant digits without trailing zeros.

  Number text in USD files usd/usda-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Type name to number type

  Which number type a USD type name uses (32-bit float, half or 64-bit double).

  Number text in USD files usd/sdf.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  Exact

• Inline values

  A value that the file stores inside its 32-bit payload word.

  Floats read from USD files usd/usdc-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Raw arrays

  A packed array of numbers of one type.

  Floats read from USD files usd/usdc-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Compressed float arrays

  An array stored as whole numbers or as a lookup table and indexes.

  Floats read from USD files usd/usdc-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Float text rounding

  A float or half value read from .usda text.

  Floats read from USD files usd/usda-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Arrays from text

  A list of parsed numbers as a typed array of the right type.

  Floats read from USD files usd/usda-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Half to float

  The value of a 16-bit half float.

  v = ± (1 + m ÷ 1024) × 2^(e − 15)

  Half floats in USD usd/util.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Float to half

  The 16-bit half closest to a number.

  Half floats in USD usd/util.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Measure to SI

  A property value in meters, square meters or cubic meters.

  area = v × L²; volume = v × L³

  IFC measures in SI for USD usd/ifc.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• 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

• Mesh buffers

  The item's box or mesh as geometry for drawing, turned from z up to y up.

  Library 3D preview library-3d.js Every app
  32-bit, display only

• Camera position

  Where the camera sits as the view is turned and zoomed.

  position = target + dist × (sin φ sin θ, cos φ, sin φ cos θ)

  Library 3D preview library-3d.js Every app
  32-bit, display only

• Variant body buffers

  The chosen variant's box or mesh as geometry for drawing.

  Library 3D preview library-configurator.js Every app
  32-bit, display only

• 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