Calculations

Calculations

• Global section

  States the file's number sizes, unit, resolution and largest coordinate.

  unit flag 2 (MM); scale 1; resolution 10⁻⁶; largest coordinate = max |x| over all control points

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• A number from IGES text

  Reads one real number from the file.

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Unit factors

  Millimeters per file unit, by the file's unit flag.

  1: 25.4; 2: 1; 3: 25.4 × 12; 4: 1609344; 5: 10⁶; 6: 1000; 7: 10⁶; 8: 10⁻³; 9: 10⁻³; 10: 10; 11: 25.4 × 10⁻⁶

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Scales a curve's control points by the file's unit as it is read.

  scale = UNITS[flag] (1 when unknown); p′ = p × scale

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• End derivative coefficients

  The factors that link the first control points to the first and second derivative at the start.

  a₁ = p ÷ (U[p+1] − U[1]); b₁ = p ÷ (U[p+2] − U[2]); c₂ = (p − 1) ÷ (U[p+1] − U[2])

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Point, direction and curvature at an end

  What a curve's end looks like to the curve that must meet it.

  t = C′ ÷ |C′|; out = ± t; speed = |C′|

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Curve end matched to a target

  Moves the first one to three control points so the curve meets the target with G0, G1 or G2.

  P₀ = T; P₁ = P₀ + out × speed ÷ a₁; D₂ = out (C″ · out) + κ speed²; P₂ = P₁ + (a₁ (P₁ − P₀) + D₂ ÷ c₂) ÷ b₁

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend curve between two ends

  A Bézier curve joining two curve ends with the asked continuity.

  G1: P₁ = A + out × α ÷ 3; G2: P₁ = A + out × α ÷ 5, P₂ = 2 P₁ − A + κ α² ÷ 20; α = chord × bulge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Frame on the target's edge

  For a point near the target's edge: the nearest edge point, the outward direction in the tangent plane, the normal and the normal curvature.

  d = (n × tₑ) ÷ |n × tₑ|, turned to point away from the surface

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Surface edge matched to a target

  Refits the first one to three rows of control points so the surface meets the target along an edge.

  stations vᵢ = i ÷ M, M = max(samples, 4 × columns); rows fitted by least squares; accepted when gap ≤ g0, angle ≤ g1 and Δκ ≤ g2 (or relative ≤ g2rel)

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend surface between two edges

  A surface joining two surface edges, built from blend sections along them.

  sections at t = i ÷ n, n = 8, 16, … ≤ 128; each a Bézier of degree 1, 3 or 5; interpolated along the edge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• A number as STEP text

  Writes one real number into the file.

  String(x), with a decimal point added when it has none

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Knots and multiplicities

  Turns a full knot vector into STEP's list of distinct knots with their counts.

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Edges that have collapsed to a point

  Leaves a zero-length boundary edge, such as a pole, out of the face loop.

  skipped when arc length ≤ 10⁻⁹ × surface size

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Trim loop written as a curve

  Writes a trim loop as one closed curve on the face.

  3D point = S(u, v) for each loop point; curve = straight segments through them

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Length unit of a file

  Millimeters per file unit.

  SI: prefix × 1000; conversion-based: value × its base unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Reads each control point and scales it by the file's unit.

  p′ = p × unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Trim loops read back

  Turns a face's boundary edges into trim loops in the surface's own parameters.

  33 samples per curved edge; each projected to (u, v); outer when |area| < 0.999 × (u range × v range)

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• 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

• 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