Calculations

Calculations

• Phase of a graded lattice

  Maps a point to the lattice's own period coordinates when the cell size changes.

  fixed cell: X = 2π·p ÷ cell; ramp: X = ∫k dp with k linear in p

  Field shapes: signed distance functions field/field.js

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• Gyroid, Schwarz P and diamond

  Signed distance to a triply periodic minimal surface, as a sheet or a network.

  d ≈ g ÷ |∇g|; near the surface: two projections q ← q − g∇g ÷ |∇g|², d = ±|p − q|

  Field shapes: signed distance functions field/field.js

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• Brick classification

  Decides whether a ball of space is wholly outside, wholly inside or near the surface.

  outside when f − L·r > band; inside when f + L·r < −band

  Field shapes: signed distance functions field/field.js

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• Plane of a curve

  The normal of the plane a hull line lies in.

  n = (p_mid − p₀) × (p_end − p₀), made unit

  Hull lines: fairness measures and fairing hull/fair.js

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• Signed curvature

  The curvature of a curve at a parameter, signed in its plane.

  κ = (C′ × C″)·n ÷ |C′|³

  Hull lines: fairness measures and fairing hull/fair.js

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• Fairness measures

  Inflection count, largest curvature, roughness, curvature jumps at knots and length.

  R = ∫(dκ/ds)² ds ≈ (du ÷ 3) × Σ wᵢ gᵢ, gᵢ = ((κᵢ₊₁ − κᵢ₋₁) ÷ ds)² × |C′|ᵢ

  Hull lines: fairness measures and fairing hull/fair.js

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• Fairing a curve

  The rebuild with the fewest control points that stays within tolerance and adds no inflection.

  Hull lines: fairness measures and fairing hull/fair.js

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• Fairness problems

  Reports curves that have an inflection or a curvature jump.

  Hull lines: fairness measures and fairing hull/fair.js

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• Sampling a patch

  The parameter values at which each surface is evaluated.

  uᵢ = u₀ + (u₁ − u₀) × i ÷ N, N from 2 to 200

  Hull surfaces to a closed solid and its volume hull/solid.js

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• Mirror half

  The other half of a hull drawn to the centerline.

  p′ = 2 × at − p on the mirror axis

  Hull surfaces to a closed solid and its volume hull/solid.js

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• Welding corners

  Merges corners of neighboring patches that lie within the weld distance.

  merge when √(Δx² + Δy² + Δz²) ≤ eps

  Hull surfaces to a closed solid and its volume hull/solid.js

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• Planar opening test

  Decides whether an opening lies in a plane and can be capped.

  n by Newell's sums; deviation = max |(p − centroid)·n|; cap when deviation ≤ max(4 eps, 2e-4 × size)

  Hull surfaces to a closed solid and its volume hull/solid.js

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• Volume of the closed mesh

  The volume enclosed by the hull mesh.

  V = Σ a·(b × c) ÷ 6

  Hull surfaces to a closed solid and its volume hull/solid.js

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• Unit vector

  A vector scaled to length 1.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Point under a matrix

  A point moved by a 4 × 4 matrix.

  p′ = M₃ₓ₃ p + t

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Matrix product

  The product of two 4 × 4 matrices.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Matrix inverse

  The inverse of a general 4 × 4 matrix.

  M⁻¹ = adj(M) ÷ det(M)

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Box of placed positions

  The axis-aligned box of a mesh's corners under a matrix.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Rigid motion test

  Checks that a matrix is a rotation and a move, with no scale or shear.

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Best rigid move between two point sets

  The rotation and move that best take one set of points onto another, and the largest leftover distance.

  R from the unit quaternion of the largest eigenvalue; t = c_B − R c_A; err = max |M pᵢ − qᵢ|

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Largest eigenvector of a symmetric 4 × 4

  The eigenvector the rigid fit's rotation is read from.

  t = sign(θ) ÷ (|θ| + √(θ² + 1)), θ = (a_qq − a_pp) ÷ (2a_pq)

  Illustrate: vector, matrix and rigid-fit math illustrate/m4.js

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• Time-weighted average

  The average of a value that changes at events: queue length, units busy, work in process.

  average = (Σ vᵢ × Δtᵢ) ÷ (t − t₀)

  Flow simulation (discrete events) engine.js Every app
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• Mean, deviation and percentiles of observations

  Waiting time, service time and time in system: mean, standard deviation, minimum, maximum, P50 and P95.

  m ← m + (x − m) ÷ n; s ← s + (x − m_old)(x − m_new); sd = √(s ÷ (n − 1)); P = sorted[⌊p × k⌋]

  Flow simulation (discrete events) engine.js Every app
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• Event time

  When a scheduled event happens.

  t_event = t_now + delay

  Flow simulation (discrete events) engine.js Every app
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• Conveyor slots and timing

  How many items fit on a belt, the gap between items and the travel time.

  slots = max(1, ⌊length ÷ pitch + 1e-9⌋); headway = pitch ÷ speed; travel = length ÷ speed

  Flow simulation (discrete events) engine.js Every app
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• Shift edges

  When a server goes on and off shift.

  x = t − ⌊t ÷ period⌋ × period; on shift when a ≤ x < b

  Flow simulation (discrete events) engine.js Every app
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• Server utilization

  The share of time a server's units were busy, blocked, down and idle.

  utilization = average busy units ÷ units; idle = max(0, 1 − busy − blocked − down)

  Flow simulation (discrete events) engine.js Every app
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• Throughput

  Finished entities per unit of time at a sink.

  throughput = count ÷ (t − t₀)

  Flow simulation (discrete events) engine.js Every app
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• Resource utilization

  The share of a shared resource's capacity that was in use.

  utilization = average in use ÷ average capacity

  Flow simulation (discrete events) engine.js Every app
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• Vehicle trip times

  How long a vehicle takes to reach the pick station and to carry the load to the drop station.

  to pick = distance ÷ speed + load time; carry = distance ÷ speed + unload time

  Flow simulation (discrete events) engine.js Every app
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• Student t factor

  The factor for a 95 % confidence range.

  df ≤ 30: table value; df ≤ 60: 2.0; df ≤ 120: 1.98; above: 1.96

  Flow simulation (discrete events) engine.js Every app
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• Replication mean and confidence half-width

  The mean of each result over independent runs and its 95 % range.

  mean = Σx ÷ n; sd = √(Σ(x − mean)² ÷ (n − 1)); half = t × sd ÷ √n

  Flow simulation (discrete events) engine.js Every app
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• Uniform and normal draws

  The random numbers every arrival, service time and route choice is drawn from.

  u = (k + 0.5) ÷ 2³², k a 32-bit integer; z = √(−2 ln u) × cos(2π v)

  Flow simulation (discrete events) rng.js Every app
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• Draw from a distribution

  One value from an exponential, uniform, triangular, normal, lognormal, Erlang, Weibull or empirical distribution.

  exp: −mean × ln u; tri: a + √(u (b − a)(c − a)) or b − √((1 − u)(b − a)(b − c)); weibull: scale × (−ln u)^(1 ÷ shape)

  Flow simulation (discrete events) rng.js Every app
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• Mean of a distribution

  The mean used for load estimates and for routing to the least loaded target.

  tri: (a + m + b) ÷ 3; weibull: scale × Γ(1 + 1 ÷ shape)

  Flow simulation (discrete events) rng.js Every app
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• Gamma function

  Γ(x), used for the mean of a Weibull distribution.

  Lanczos approximation, g = 7, 9 coefficients

  Flow simulation (discrete events) rng.js Every app
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• Thermal voltage

  kT/q, the voltage scale of every diode and transistor junction.

  VT = 0.025852 V

  Circuit solver circuit.js Every app
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• Value with an engineering suffix

  4.7k, 100n, 1meg: a component value read from text.

  value = number × factor of the suffix (t, g, meg, k, m, u, n, p, f); mil = 25.4e-6

  Circuit solver circuit.js Every app
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• Source waveform value

  The value of a DC, sine or pulse source at a time.

  sin: off + amp × e^(−damp (t − delay)) × sin(2π f (t − delay) + phase); pulse: linear edges over tr and tf

  Circuit solver circuit.js Every app
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• Diode current and conductance

  The current through a diode and its slope at a junction voltage.

  i = Is (e^(v ÷ (n VT)) − 1); g = Is e^(v ÷ (n VT)) ÷ (n VT)

  Circuit solver circuit.js Every app
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• Transistor currents

  Collector and base current of a bipolar transistor.

  ic = If − Ir − Ir ÷ βr; ib = If ÷ βf + Ir ÷ βr; If = Is (e^(vbe ÷ VT) − 1)

  Circuit solver circuit.js Every app
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• MOSFET level 1 current

  Drain current of a MOSFET in the triode and saturation regions.

  triode: id = kp (vov vds − vds² ÷ 2)(1 + λ vds); saturation: id = kp vov² (1 + λ vds) ÷ 2

  Circuit solver circuit.js Every app
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• Switch conductance

  The conductance of a voltage-controlled switch.

  ln g = ln(1 ÷ roff) + a (ln(1 ÷ ron) − ln(1 ÷ roff)); a = 1 ÷ (1 + e^(−(v − vt) ÷ vh))

  Circuit solver circuit.js Every app
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• Circuit equations

  The matrix and right side for one Newton step, at DC or at one time step.

  capacitor: geq = C ÷ h (backward Euler) or 2C ÷ h (trapezoid); inductor: r = L ÷ h or 2L ÷ h

  Circuit solver circuit.js Every app
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• Newton iteration

  Repeats the linear solve until the voltages and currents stop changing.

  stop when |Δxᵢ| ≤ 1e-3 × max(|xᵢ new|, |xᵢ old|) + 1e-6 V (or 1e-12 A) for every unknown

  Circuit solver circuit.js Every app
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• Operating point

  The DC voltages and currents of the circuit.

  direct Newton, then gmin stepping, then source stepping

  Circuit solver circuit.js Every app
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• DC sweep

  The circuit solved for each value of a swept source.

  Circuit solver circuit.js Every app
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• AC small-signal response

  Complex node voltages and branch currents at each frequency.

  capacitor: jωC; inductor: −jωL on its branch row; ω = 2π f

  Circuit solver circuit.js Every app
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• Transient

  Voltages and currents over time.

  trapezoidal rule after a first backward-Euler step; output at t = k × tstep

  Circuit solver circuit.js Every app
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• Junction voltage limiting

  Keeps a diode or transistor voltage from jumping too far in one iteration.

  v = v_old + vt ln(1 + (v_new − v_old) ÷ vt) when the step exceeds 2 vt above vcrit

  Circuit solver circuit.js Every app
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