Calculations

Calculations

• Mean and variance from sums

  The mean and the variance of the mean from per-pixel sums of the samples and of their squares.

  m = Σx ÷ n; s² = max(0, (Σx² − n m²) ÷ (n − 1)); var(mean) = s² ÷ n

  Statistical comparison of rendered pictures compute/parity.js

  • BimCraft
  • ModelCraft
  64-bit

• Block means

  The mean of each block of pixels and the variance of that mean.

  block mean = Σ means ÷ count; block variance = Σ variances ÷ count²

  Statistical comparison of rendered pictures compute/parity.js

  • BimCraft
  • ModelCraft
  64-bit

• Two pictures compared

  Whether a test picture agrees with a reference picture.

  z = |μ_t − μ_r| ÷ √(σ²_r + σ²_t + (10⁻⁴ max(1, |μ_r|))²); ok if failed blocks ≤ 1 % and the whole-image z ≤ 4.5

  Statistical comparison of rendered pictures compute/parity.js

  • BimCraft
  • ModelCraft
  64-bit

• Furnace test

  Whether every block of a test picture has the expected mean within its noise.

  z = |block mean − expected| ÷ √(block variance + (10⁻⁴ max(1, |expected|))²); ok if no block has z > 4.5

  Statistical comparison of rendered pictures compute/parity.js

  • BimCraft
  • ModelCraft
  64-bit

• Samples read from a block

  Puts the samples of one unpacked block into each channel's picture.

  half: table[16-bit value]; float: the 32-bit value; uint: the 32-bit integer

  OpenEXR and Radiance HDR picture files exr/exr.js

  • BimCraft
  • ModelCraft
  64-bit

• Table for linear B44 channels

  The table that undoes the logarithmic step of B44 compression for channels marked linear.

  table[x] = half(8 × log(value of x))

  OpenEXR and Radiance HDR picture files exr/exr.js

  • BimCraft
  • ModelCraft
  64-bit

• Red, green, blue and alpha of a picture

  Gathers a layer's channels into one RGBA picture.

  subsampled channel: sample at (floor(x ÷ xSampling), floor(y ÷ ySampling))

  OpenEXR and Radiance HDR picture files exr/exr.js

  • BimCraft
  • ModelCraft
  64-bit

• Samples written to a block

  Writes each channel's samples in its type before the block is compressed.

  half: floatToHalf(v); float: nearest 32-bit float; uint: v as a 32-bit unsigned integer

  OpenEXR and Radiance HDR picture files exr/exr.js

  • BimCraft
  • ModelCraft
  64-bit

• RGBE pixel to a number

  Turns the four bytes of a Radiance pixel into red, green and blue values.

  value = (m + 0.5) × 2^(E − 136); E = 0 gives 0

  OpenEXR and Radiance HDR picture files exr/hdr.js

  • BimCraft
  • ModelCraft
  64-bit

• Number to an RGBE pixel

  Encodes red, green and blue values as three mantissas and a shared exponent.

  e: 2^(e−1) ≤ max(r, g, b) < 2^e; mantissa = floor(value × 256 ÷ 2^e); E = e + 128

  OpenEXR and Radiance HDR picture files exr/hdr.js

  • BimCraft
  • ModelCraft
  64-bit

• Distance of a point along the run

  How far along the run's drawn line a point lies, measured from the first point.

  s = Σ earlier leg lengths + clamp((p − a)·d, 0, L)

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Each part's place and center-line length

  The two ends of every pipe, bend and tee along the run, and the part's own center-line length.

  bend: R × θ; tee: 2 × m; straight: |b − a|

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Where the run is cut and where spools break

  The cut positions inside straight pipe and the breaks between spools, from the field welds and the longest spool length.

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Spool length

  The parts of each spool and the spool's total center-line length.

  length = Σ part lengths, in run order

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Cut length of a straight piece

  The length of each straight pipe piece on the spool sheet's cut list.

  length = |b − a|

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Bend angle on the fitting list

  The angle shown for each elbow and the count of like fittings.

  angle = round(θ × 180 ÷ π)

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Pipe length and weight on the bill of materials

  The total cut length of pipe in a spool and its steel weight.

  kg = (kg per m) × Σ cut lengths ÷ 1000

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Duct cut list

  How many stock-length duct pieces each straight duct needs, and the length left over.

  full = floor((L + 0.5) ÷ piece); rest = L − full × piece

  Pipe spools, cut lists and spool sheets mep/spools.js

  • BimCraft
  64-bit

• Lengths and weights printed on the spool sheet

  The spool length, cut lengths, pipe length and weight as they appear in the PDF.

  printed kg = round(kg × 10) ÷ 10

  Pipe spools, cut lists and spool sheets mep/spool-sheet.js

  • BimCraft
  64-bit

• Spool sheet CSV rows

  The cut lengths, quantities and pipe weight exported as CSV.

  exported kg = round(kg × 100) ÷ 100

  Pipe spools, cut lists and spool sheets mep/spool-sheet.js

  • BimCraft
  64-bit

• Distance from a point to a triangle

  The shortest distance from a point to a triangle, on a corner, an edge or the face.

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  64-bit

• Segment crossing a triangle

  Where along a straight piece of a run it passes through one triangle of a wall or slab.

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

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  64-bit

• Clearance between an MEP element and the structure

  The gap reported when a pipe or duct is closer to a structural element than its insulation plus the typed clearance.

  need = (insulation + gap) ÷ 1000; gap = min over vertices p and triangles T of dist(p, T), both ways

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  32-bit in part

• Run center line as straight pieces

  The center line of a run, part by part, that is tested against walls and slabs.

  bend point k = center + (−n cos(θk ÷ 8) + d₁ sin(θk ÷ 8)) × R, k = 0 … 8

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  64-bit

• Opening size

  The width and height of the opening a run needs in a wall or slab.

  opening = outside size + 2 × insulation + 2 × gap

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  64-bit

• Where a run passes through an element

  The position of each opening on the run, its place in plan, and the length of run inside the wall or slab.

  s = s₀ + u × piece length; through = s_leave − s_enter; at = midpoint of the two crossings

  MEP clearances and openings through walls and slabs mep/structure.js

  • BimCraft
  32-bit in part

• Rounding to three decimals

  The rounding applied to every stored length, spacing and symbol coordinate.

  r3(v) = round(v × 1000) ÷ 1000

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Reading a length

  A height, spread, spacing, volume or girth read from a file, a CSV cell or the editor.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Sizes, hardiness zone and spacing of one species

  The mature size, planting size, zone range and spacing kept for a species row.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Default planting size and spacing

  The planting size, container and spacing filled in when a species row leaves them empty.

  spacing = max(q, round(k × spread ÷ q) × q), by plant type

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Spacing from planting density

  The center-to-center spacing that gives a stated number of plants per square meter.

  square: s = 1 ÷ √d; triangular: s = √(2 ÷ (√3 × d))

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Spacing used for a species

  The spacing a bed or hedge is planted at: the species' own, else from its density, else the default.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Supplied sizes written to CSV

  Container volume, height, spread, girth and clear stem of each supplied size as one CSV cell.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Supplied sizes read from CSV

  The same cell read back into numbers.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• 2D planting symbol

  The circles, lines, arcs and outlines of a plant's plan symbol, sized to its spread.

  R = spread ÷ 2 × scale; point = (r cos a, r sin a)

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Library item size

  The width, depth and height of the library item placed for a plant.

  mm = round(m × 1000)

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Key times to one millisecond

  The rounding applied to every key time, clip start and the animation length.

  t = round(t × 1000) ÷ 1000

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Stretching an animation to a new length

  Every key time and clip start when the animation is made longer or shorter.

  k = new length ÷ old length; t′ = round₀.₀₀₁(t × k)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Spreading chosen keys about a time

  The new times of selected keys scaled away from or toward a pivot time.

  t′ = round₀.₀₀₁(pivot + (t − pivot) × f)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Frame count and frame times

  The number of frames a render of the animation has, and the time of each frame.

  frames = max(1, round(length × fps) + 1); time of frame i = i ÷ fps

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Position between two keys

  Which pair of keys a time falls between and how far along that stretch it is.

  s = (t − tᵢ) ÷ (tᵢ₊₁ − tᵢ)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Time on a repeating track

  The time inside the keys' span that a repeating, ping-pong or offset track plays at.

  x = (t − t₀) ÷ P; k = floor(x); f = x − k

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Speed through a key

  The rate of change of a value at a key, from its two neighbors.

  m = (pᵢ₊₁ − pᵢ₋₁) ÷ (tᵢ₊₁ − tᵢ₋₁)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Value between two keys

  A position or other value between two keys: held, straight, or on a smooth curve.

  p = h₀₀ p₀ + h₀₁ p₁ + h₁₀ Δt m₀ + h₁₁ Δt m₁

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Bézier handles of a property key

  The two handle points of a key on a property curve.

  automatic: handle = key ± (Δt ÷ 3, m × Δt ÷ 3)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Property value on a Bézier curve

  The value of an animated property at a time between two keys with Bézier handles.

  solve x(u) = t by bisection; v = B(v₀, p₁, p₂, v₃; u)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Property value at a time

  A property track's value at any time, with its cycles followed.

  offset cycle: v + n × (last key − first key)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Camera at a time

  The camera's eye, target, view angle and view height between camera keys.

  eye, target, fov on the curve; height = exp(curve of ln height)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Matrix to position, rotation and scale

  A placement matrix split into its position, a unit quaternion and three scale factors.

  s = column lengths; q from the rotation matrix (Shepperd)

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Position, rotation and scale to matrix

  The placement matrix built from a position, a quaternion and a scale.

  Animation keys, curves and poses render/anim.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit