Calculations

Calculations

• Polygon split by a plane

  Cuts a convex polygon into the part in front of a plane and the part behind it.

  t = dₐ ÷ (dₐ − d_b); x = a + (b − a) t

  IFC openings, clippings and half-space cuts ifc/geo-bool.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Mesh clipped by a plane

  Keeps the part of a mesh on one side of a plane and closes the cut.

  dᵢ = (pᵢ − p) · n; ε = 10⁻⁹ × (d_max − d_min); cut point = a + (b − a) dₐ ÷ (dₐ − d_b)

  IFC openings, clippings and half-space cuts ifc/geo-bool.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Cap of a cut

  Joins the cut segments into loops, sorts outer loops from holes and fills them.

  key = round(x ÷ tol), round(y ÷ tol), round(z ÷ tol); area = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|

  IFC openings, clippings and half-space cuts ifc/geo-bool.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Half-space cut

  Removes the part of an element behind a plane, optionally only inside a polygon boundary.

  L = 4 × max(1, box diagonal) + |P₀ − box min|; prism height = 2 L

  IFC openings, clippings and half-space cuts ifc/geo-bool.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  32-bit in part

• Vertex data of one geometry

  Returns a geometry's positions and normals for drawing and for measuring.

  IFC geometry handed to the apps ifc/api.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  32-bit in part

• Placement matrix of each piece

  The matrix that takes a geometry's vertices to model space in meters, Y up.

  M = C × Y-up × scale(u) × placement

  IFC geometry handed to the apps ifc/api.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Shift to the origin

  Finds the translation that brings the model near the origin.

  c = (box min + box max) ÷ 2 of the first element; C = translate(−(Y-up × scale(u) × placement × c))

  IFC geometry handed to the apps ifc/api.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• File units to meters, Z up to Y up

  Scales a placement by the file's length unit and turns its axes.

  M′ = Y-up × scale(u) × M

  IFC geometry handed to the apps ifc/api.js

  • BimCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  64-bit

• Rounding to one and two decimals

  Rounds a value for a message or for a warning's stored fields.

  f1(x) = round(10 x) ÷ 10; f2(x) = round(100 x) ÷ 100

  MEP check messages: units and rounding mep/problems-mep.js

  • BimCraft
  64-bit

• Pipe flow and velocity in the message

  Shows a flow in gallons per minute or liters per second and a velocity in feet or meters per second.

  gpm = q ÷ 0.003785411784 × 60; L/s = 1000 q; ft/s = v ÷ 0.3048

  MEP check messages: units and rounding mep/problems-mep.js

  • BimCraft
  64-bit

• Stored velocity and pressure values

  Writes the required and found velocity and the pressure left at a fixture into each pipe warning.

  found = f2(V) m/s; required = f2(v_max) m/s; pressure = f1(kPa)

  MEP check messages: units and rounding mep/problems-mep.js

  • BimCraft
  64-bit

• Duct velocity, friction rate and aspect ratio

  Writes a duct's flow, velocity, needed round size and aspect ratio into its warning.

  fpm = round(V ÷ 0.00508); L/s = f1(1000 q); needed size = round(mm)

  MEP check messages: units and rounding mep/problems-mep.js

  • BimCraft
  64-bit

• Tube dimensions for a size

  The outside, wall and inside dimensions of one tube, in inches and millimeters.

  id = od − 2 × wall; odMm = round(od × 25.4 × 1000) ÷ 1000; wallMm = wall × 25.4

  Refrigeration copper tube sizes mep/tables-acr.js

  • BimCraft
  64-bit

• Filmic tone curve

  Maps scene brightness to screen brightness for the filmic view transform.

  h(x) = (x (0.15 x + 0.05) + 0.004) ÷ (x (0.15 x + 0.5) + 0.06) − 0.02 ÷ 0.3; output = h(2 c) ÷ h(11.2)

  Real-time view and render bench rt/rt-post.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Baked probe light

  Collects the bounce light at each probe and removes the sky's own part.

  shᵢ,ₖ = bakedᵢ,ₖ − skyₖ (k = 0…11)

  Real-time view and render bench rt/rt-gi.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Mean radiance per pixel

  Averages the accumulated samples of each pixel for the bench and the parity comparison.

  mean = Σ samples ÷ n

  Real-time view and render bench render/bench-kernels.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Scene for the standalone viewer

  Packs the drawn scene into the viewer's file.

  Real-time view and render bench rt/rt-export.js

  • BimCraft
  • ModelCraft
  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