Calculations

Calculations

• Placing people

  Places figures over the region; along a line they face along it, half each way.

  heading = atan2(Δy, Δx) of the line; yaw = heading + π ÷ 2 (+ π for half of them) + (rand − ½) × 0.3; scale = 0.97 + 0.06 × rand

  Placing scene trees and people scene-entourage.js

  • ModelCraft
  64-bit

• Scattering points with a minimum spacing

  Seeded points inside a rectangle or along a line, none closer than the minimum distance to another.

  rectangle: x = x₀ + margin + rand × w, y likewise; line: p = a + (b − a) × i ÷ (count − 1) + normal × jitter; kept x, y = round to 0.01

  Placing scene trees and people scene-say.js

  • ModelCraft
  64-bit

• Placement matrix

  The 4×4 matrix for a turn about the vertical axis, a uniform scale and a position.

  c = cos(yaw)·scale; s = sin(yaw)·scale; columns [c, s, 0], [−s, c, 0], [0, 0, scale], [x, y, z]

  Placing scene trees and people scene-say.js

  • ModelCraft
  64-bit

• Arc and circle as straight segments

  The ground segments for one traced line, arc or circle.

  aₖ = a₀ + sweep × k ÷ n; pₖ = [cx + r·cos aₖ, cy + r·sin aₖ, 0]

  Traced sketch into model edges sketch-photo-mc.js

  • ModelCraft
  64-bit

• Did another program change the points

  Decides whether a definition's native USD points still match the model's own.

  changed if any fround(own coordinate) ≠ file coordinate, or the counts differ

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  Exact

• Mesh from the native points

  Rebuilds a definition that was changed elsewhere from the file's native points and faces.

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit

• Was a placement moved elsewhere

  Decides whether a placement's matrix in the file differs from the model's and takes the file's if so.

  moved if any |mₖ(file) − mₖ(own)| > 10⁻⁷ × max(1, |mₖ(own)|)

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit

• Placements through the stage's frame

  Brings a placement matrix from the file's units and up axis into meters with Z up.

  M′ = F · M · F⁻¹

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit

• Imported mesh points

  The points of an imported mesh in meters with Z up.

  p′ = F · p (rotation, scale and translation of the frame)

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit

• Lamp output from a USD light

  The lumens of an imported lamp when the file does not carry ModelCraft's own value.

  area = width × height × scale² or 4π·(radius × scale)²; lumens = intensity × 2^exposure × 683 × max(area, 10⁻⁶) × π

  USD save, open and import usd-modelcraft.js

  • ModelCraft
  64-bit

• Event order

  Decides which of two queued events runs first.

  earlier time first; equal times: lower priority number first; then lower sequence number

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Time-weighted mean and maximum

  The mean and maximum over time of a queue length, a count of busy servers or an on/off state.

  area = Σ value × (time it held); mean = (area + value × (now − last change)) ÷ (now − start)

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Rounding of reported statistics

  Rounds a statistic to 6 decimals before it goes into the results.

  r6(x) = round(x × 10⁶) ÷ 10⁶

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Load type by weight

  Picks the type of a new load in proportion to the types' weights.

  u = random × Σ weights; subtract each weight in turn until u < 0

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Throughput and lead time at a sink

  Counts finished loads and works out loads per hour and the mean and maximum time from creation to the sink.

  lead = now − born; per hour = n ÷ span × 3600; mean lead = Σ lead ÷ n

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Station utilization, blocking and waiting

  The share of server time spent busy, blocked and idle, the mean queue and the mean wait at a station.

  utilization = mean busy ÷ servers; blocked = mean blocked ÷ servers; idle = 1 − (mean busy + mean blocked) ÷ servers; wait = Σ wait ÷ starts

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Gate cycle phase

  Whether a cycling gate starts open and when it first switches.

  period = open + closed; phase = ((0 − offset) mod period + period) mod period; open when phase < open

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Position of a load on a belt

  Where the tail of a load is now, on an accumulating belt or on a belt that stops as a whole.

  accumulating: p = min(v × t + c, S); stopping belt: p = D(t) − D at entry, D(t) = D₀ + v × (t − t₀)

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• When the next load can get on a belt

  The time at which the last load has moved far enough for the next one to enter.

  pitch = load length + gap (or the fixed zone pitch); accumulating: t = (pitch − c_last) ÷ v; stopping belt: t = now + (pitch − gap so far) ÷ v

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• A load gets on a belt

  Sets the two numbers that describe the new load's motion: where it is held back and where it stops.

  empty belt: c = −v × now (or less, to keep the pitch behind the load that just left), S = length − L; else c = min(−v × now, c_last − pitch), S = S_last − pitch

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• When the head load reaches the end

  The time the first load's head arrives at the end of the belt.

  accumulating: t = (length − L − c) ÷ v; stopping belt: t = t₀ + (D at entry + length − L − D₀) ÷ v

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Loads close up when the head leaves

  Works out again, for every load left on the belt, where it is held back and where it stops.

  first: c = min(−v × entry, length − L_gone − v × now − pitch), S = length − L; others: c = min(−v × entry, c_front − pitch), S = S_front − pitch

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• A belt starts again after a breakdown

  Shifts every load's timeline by the time the belt stood still.

  dur = now − time it stopped; c = c − v × dur; entry = entry + dur

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Photo-eye covered by a load

  Whether a load is over a sensor placed a given distance along a belt.

  covered when p − 1e-7 ≤ at ≤ p + L + 1e-7

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Shortest vehicle routes

  The shortest path distance from one path node to every other, used to pick the nearest idle vehicle and its route.

  Dijkstra: dist(to) = min over edges of dist(from) + edge length

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Vehicle travel time on a segment

  When a vehicle that enters a path segment reaches its end.

  t₁ = now + length ÷ speed

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Vehicle time shares and distances

  For each vehicle: the share of time in each state, the distance driven loaded and empty, and how busy it was.

  share = time in state ÷ span; busy = 1 − idle share − down share

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Run results

  Throughput per hour and the lead time count, mean, minimum, median, 95th percentile and maximum for the whole run.

  per hour = done ÷ span × 3600; mean = Σ lead ÷ n; percentile p = sorted[min(n − 1, floor(p × n))]

  Material flow simulation addons/flow/lib/sim.js

  • ModelCraft
  64-bit

• Seeds from a string

  Turns the seed and a stream key into four 32-bit seeds.

  Random numbers and time distributions for material flow addons/flow/lib/rng.js

  • ModelCraft
  Exact

• Uniform random number

  One number from 0 up to, but not including, 1.

  next = (32-bit state output) ÷ 4294967296

  Random numbers and time distributions for material flow addons/flow/lib/rng.js

  • ModelCraft
  64-bit

• One time from a distribution

  A time in seconds drawn from a constant, uniform, exponential, Erlang, normal or triangular distribution.

  exponential: −mean × ln(1 − u); Erlang: −(mean ÷ k) × ln(Π (1 − uᵢ)); normal: mean + sd × √(−2 ln u₁) × cos(2π u₂); triangular: a + √(u (b − a)(m − a)) or b − √((1 − u)(b − a)(b − m))

  Random numbers and time distributions for material flow addons/flow/lib/rng.js

  • ModelCraft
  64-bit

• Mean of a distribution

  The mean time of a distribution, as the panel shows it.

  uniform: (a + b) ÷ 2; triangular: (a + m + b) ÷ 3; others: the stated mean

  Random numbers and time distributions for material flow addons/flow/lib/rng.js

  • ModelCraft
  64-bit

• Shortest turn between two directions

  The quaternion that turns one unit vector onto another.

  q = (u × v, 1 + u · v), made unit

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Two-bone reach

  Where the elbow and the end of a two-bone chain go so the end reaches a target.

  cos A = (L₁² + d² − L₂²) ÷ (2 L₁ d); sin A = √(1 − cos² A); elbow = a + dir × L₁ cos A + n × L₁ sin A

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Chain of three or more bones (FABRIK)

  Joint positions of a longer chain whose tip reaches for a target.

  backward sweep from the target, forward sweep from the root, each joint kept its segment length from its neighbor

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Point along a path by length

  The point and direction a given fraction of the way along a polyline.

  s = k × Σ segment lengths; walk the segments subtracting each length until s falls inside one

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Point an axis at a direction

  A matrix whose chosen axis points along a direction, with its up axis leaning to world +Z.

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Blend by influence

  Mixes two matrices by a factor from 0 to 1: place and size linearly, the turn along the shortest arc.

  t = t_a + (t_b − t_a) k; s = s_a + (s_b − s_a) k; q = slerp(q_a, q_b, k)

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• IK on a rig

  Builds the chain for an IK constraint, solves it and swings each bone so it runs to the next joint.

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• World matrix back to a local pose

  Works a bone's local matrix back from a wanted world matrix through its parent.

  local = (parent world × parent rest⁻¹ × rest)⁻¹ × wanted world

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• Constraints on placements

  Applies look-at, copy, limit, child-of and follow-path constraints to placements in world coordinates and returns their new local matrices.

  new local = blend(local, parent world⁻¹ × constrained world, influence)

  Inverse kinematics and constraints anim/constraints.js

  • ModelCraft
  64-bit

• World matrix of a placement

  The product of the matrices of a placement and everything it sits in.

  W = M₁ × M₂ × … × M_own

  Inverse kinematics and constraints anim/cns-model.js

  • ModelCraft
  64-bit

• A curve object's line in world coordinates

  The polyline a follow-path constraint runs along.

  Inverse kinematics and constraints anim/cns-model.js

  • ModelCraft
  64-bit

• A body from a placement

  The center, half sizes and turn of the box or sphere that stands for a placement.

  scale = lengths of the matrix columns; half = max(1e-4, half box × scale); sphere radius = largest half

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Rotation matrix to quaternion

  The unit quaternion of a rotation given by its three axes.

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Placement matrix from a body state

  The placement's world matrix for a simulated position and turn.

  axes = q applied to X, Y, Z, times the placement's scale; origin = body center − rotated, scaled center offset

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Mass and inertia

  The inverse mass and inverse moments of inertia of a box or a sphere.

  sphere: I = 0.4 m r²; box: I_x = m (b² + c²) ÷ 12, and likewise for y and z

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Depth of a point inside a box

  How far a point is inside a box and the outward normal of the nearest face.

  local = q⁻¹ (p − center); depth = min over axes of (half + margin − |local|) − margin

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Positional correction

  Pushes two bodies apart along a normal by a given distance, sharing it by their mass and inertia.

  w = Σ (1 ÷ m + (r × n) · I⁻¹ (r × n)); λ = c ÷ w; x += n λ ÷ m; q += ½ (Δω, 0) q, made unit

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Step size and frame count

  How many frames are recorded and how long one substep is.

  frames = max(1, round((to − from) × fps) + 1); h = 1 ÷ fps ÷ substeps

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit