Calculations

Calculations

• Grid lines and dots

  How many grid steps fit inside the margins and where each line or dot is drawn.

  n = floor((size − 2 × margin) ÷ pitch + 1e-6); start = (size − n × pitch) ÷ 2; line i = start + i × pitch

  Blank pages: paper sizes and grid blank.js

  • MarkupCraft
  64-bit

• Old and new sheet mixed per pixel

  The color of each pixel of the compare picture, and counts of removed, added and unchanged pixels.

  Sheet compare picture compare.js

  • MarkupCraft
  32-bit, display only

• Value of a number row

  The number handed to the app when a person changes a number box.

  v = min(max, max(min, typed value))

  Numbers typed into the property grid app-propgrid.js Every app
  64-bit

• Rounding of a slider value

  Removes binary noise such as 0.30000000000000004 from a slider's value.

  v = round to (decimals of the step + 2) decimals, at most 8

  Numbers typed into the property grid app-propgrid.js Every app
  64-bit

• Value of a slider row

  The number handed to the app while a slider is dragged or its box is changed.

  v = fix(clamp(value, min, max), step)

  Numbers typed into the property grid app-propgrid.js Every app
  64-bit

• Integer or float

  Decides how a number is written.

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

• Writing a float

  Writes a number as 8 bytes.

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

• Writing an integer

  Writes a whole number in the shortest integer form that holds it.

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

• Reading a float

  Reads a 4-byte or an 8-byte float from a message.

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

• 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

• Positions stored as offsets from the origin

  The kept points as 32-bit offsets from a whole-unit origin.

  pos = fround(x − floor(min))

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  32-bit in part

• A point back in file coordinates

  The file-unit position of one kept point.

  p = pos + origin

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  32-bit in part

• Points in a slice

  The points that lie within a thickness of a plane.

  |pos − (at − origin)| ≤ thick ÷ 2

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  32-bit in part

• Intensities scaled to 0 to 65535

  The intensity of each kept point on a common 16-bit scale.

  out = round((v − lo) ÷ (hi − lo) × 65535)

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  32-bit in part

• LAS record decoder

  The x, y, z of each LAS point from its stored integers.

  x = Int32 × scale + offset

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  64-bit

• Size of the cloud

  The extent of the cloud in each axis shown in the summary.

  size = max − min

  Point cloud positions and read-in values pointcloud.js

  • CadCraft
  • BimCraft
  64-bit

• Round a box edge outward to 32-bit

  A 32-bit number that is never above (down) or never below (up) the 64-bit value.

  Ray hits against 32-bit triangle records compute/bvh-core.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Ray and triangle

  Whether a ray meets a triangle and at what distance t and barycentric u, v.

  t = (e2 · (s × e1)) ÷ (e1 · (d × e2))

  Ray hits against 32-bit triangle records compute/bvh-core.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Ray and box slab test

  The distance at which a ray enters a box, or Infinity.

  t = (box − o) ÷ d, max of the near, min of the far

  Ray hits against 32-bit triangle records compute/bvh-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Instance inverse matrix

  The world-to-object matrix of an instance as 12 numbers.

  M⁻¹ = adj(M) ÷ det(M), translation = −M⁻¹ t

  Ray hits against 32-bit triangle records compute/bvh-scene.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Ray into object space of an instance

  The ray origin and direction in an instance frame, and the triangle test that follows.

  Ray hits against 32-bit triangle records compute/bvh-scene.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Triangle records from world triangles

  The 12-number triangle records the scatter ray tests use.

  e1 = v1 − v0; e2 = v2 − v0

  Ray hits against 32-bit triangle records rt/rt-scatter-ui.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Ray against boxes

  Calls back for every item whose box a ray passes through.

  t₀ = max(nearₐ), t₁ = min(farₐ); hit if t₀ ≤ t₁

  Box tree for picking and collisions m3d/bvh.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Boxes of point lists

  The box of each item from its points, grown by a pad.

  Box tree for picking and collisions m3d/bvh.js

  • CadCraft
  • ModelCraft
  • PartCraft
  64-bit

• Accumulation buffer read back

  The per-pixel sums of red, green, blue and sample count.

  Path-traced picture sums and the saved file render/pt-gpu.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Sum of an exact range of samples

  The sum of samples s0 to s0 + spp − 1 for every pixel.

  Path-traced picture sums and the saved file render/pt-gpu.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Render elements read back

  The per-pixel render elements and light-group sums.

  Path-traced picture sums and the saved file render/pt-gpu.js

  • BimCraft
  • ModelCraft
  32-bit in part