Calculations

Calculations

• Reading a 64-bit integer

  Reads an unsigned or signed 64-bit integer.

  Numbers sent between people editing together live-client.js Every app
  Exact

• Cost of one call

  The cost of a call from its token counts and the model's prices.

  cost = (in × price in + out × price out) ÷ 1,000,000

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

• Spending today and this month

  The cost, tokens and number of calls for the current day and month, and the calls in the last hour.

  amount = Σ cost of the calls in the period

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

• 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

• Content stream number writer

  Formats one number for a PDF content stream.

  text = toFixed(v, 4) without trailing zeros

  Numbers written into edited PDF page content base-lex.js

  • MarkupCraft
  64-bit

• Round to 3 decimals

  The rounding applied to every coordinate in the package.

  round(v × 1000) ÷ 1000

  Redline package coordinates redline.js

  • MarkupCraft
  64-bit

• Position as a fraction of the sheet

  A markup's box and points as fractions of the sheet's width and height.

  fx = (x − x₀) ÷ w; fy = (y − y₀) ÷ h

  Redline package coordinates redline.js

  • MarkupCraft
  64-bit

• Ring area

  The area inside a closed outline.

  A = |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| ÷ 2

  Area markup with cut-outs, and clipboard placement holes.js

  • MarkupCraft
  64-bit

• Ring perimeter

  The closed length of an outline.

  P = Σ hypot(Δx, Δy)

  Area markup with cut-outs, and clipboard placement holes.js

  • MarkupCraft
  64-bit

• Segment crossing test

  Whether two segments properly cross, used to reject cut-outs that cross each other or the outline.

  cross = (b − a) × (c − a); crossing if signs differ on both pairs

  Area markup with cut-outs, and clipboard placement holes.js

  • MarkupCraft
  64-bit

• Point in outline

  Whether a point is inside an outline.

  x < (xb − xa)(py − ya) ÷ (yb − ya) + xa

  Area markup with cut-outs, and clipboard placement holes.js

  • MarkupCraft
  64-bit

• Box round copied markups

  The bounding box of all points and rectangles in a clip.

  min and max of all x and y

  Area markup with cut-outs, and clipboard placement clip.js

  • MarkupCraft
  64-bit

• Paste offset

  Moves pasted markups by an offset, or centers them on a point.

  dx = cx − (x₀ + x₁) ÷ 2; x′ = x + dx

  Area markup with cut-outs, and clipboard placement clip.js

  • MarkupCraft
  64-bit

• Character advance

  The advance width of one character in 1/1000 em.

  Text width from built-in font metrics font.js

  • MarkupCraft
  64-bit

• String width

  The width of a string at a font size.

  width = (Σ advance ÷ 1000) × size

  Text width from built-in font metrics font.js

  • MarkupCraft
  64-bit

• Millimeters per PDF point

  The size of one PDF point in millimeters, the base of every preset scale.

  MM_PER_PT = 25.4 ÷ 72

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Calibration from a preset scale

  Real millimeters per point for a scale chosen from the list, such as 1/4 inch = 1 foot or 1:100.

  mmPerPt = (25.4 ÷ 72) × ratio

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Calibration from two picked points

  Real millimeters per point from a distance picked on the sheet and the real length typed for it.

  mmPerPt = real length (mm) ÷ picked distance (pt)

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Fractions and mixed numbers

  Reads 1/2, 6 1/2 or 12.5 as a number.

  a/b → a ÷ b; n a/b → n + a ÷ b

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• A typed length in millimeters

  Reads what a person types for a length (24'-0", 12' 6 1/2", 3.5 m, 900 mm, or a bare number in the chosen unit) as millimeters.

  mm = feet × 304.8 + inches × 25.4; metric: value × 1, 10 or 1000

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• A typed length with arithmetic

  Reads entries such as 12'6" + 3/4" or 2*450mm as one length in millimeters.

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Length as text

  Writes a length in millimeters in the sheet's unit: feet and inches, feet, inches, meters or millimeters.

  ft-in: round(mm ÷ 25.4 × 16) sixteenths; ft: mm ÷ 304.8; in: mm ÷ 25.4; m: mm ÷ 1000

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Area as text

  Writes an area in square millimeters in the area unit that goes with the sheet's length unit.

  sq ft = mm² ÷ 92903.04; sq in = mm² ÷ 645.16; m² = mm² ÷ 1e6

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Area and perimeter of an outline

  The area and the perimeter of a closed outline, in points² and points.

  area = |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| ÷ 2; perimeter = Σ hypot(Δx, Δy)

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Area measurement with cut-outs

  The area and perimeter text of an area markup: the outline less its cut-outs, in real units.

  area = (A_outline − Σ A_cut-out) × mmPerPt²; perimeter = (P_outline + Σ P_cut-out) × mmPerPt

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Length under a calibration

  A distance in points as a real length in the sheet's unit.

  length = points × mmPerPt

  Sheet scale, typed lengths and measured values units.js

  • MarkupCraft
  64-bit

• Measured values of a markup

  The length, area and perimeter, count, angle or arc length listed for a markup in the Properties panel and, through valueOf, in lists, CSV and BCF.

  length = hypot(Δx, Δy) × mmPerPt; arc = r × |sweep| × mmPerPt; count = number of marks

  Sheet scale, typed lengths and measured values markupcraft.js

  • MarkupCraft
  64-bit

• Pointer position on the paper

  The pointer's position on the sheet in the status line, in millimeters or inches of paper.

  mm = pt ÷ 72 × 25.4; in = pt ÷ 72

  Sheet scale, typed lengths and measured values markupcraft.js

  • MarkupCraft
  64-bit

• Scale carried to a replacement sheet

  When markups are carried to a new revision of a sheet, the old sheet's calibration is given to the new one if it has none.

  |w_old − w_new| ≤ 1 pt and |h_old − h_new| ≤ 1 pt

  Sheet scale, typed lengths and measured values markupcraft.js

  • MarkupCraft
  64-bit

• Size of a pasted area

  The size on the sheet of a copied area pasted as a picture: the same size on paper, or the same real size when both sheets have a scale.

  k = mmPerPx ÷ mmPerPt (both scaled), else ptPerPx; w = px_w × k; h = px_h × k

  Sheet scale, typed lengths and measured values markupcraft.js

  • MarkupCraft
  64-bit