Calculations

Calculations

• 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
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  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
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  • 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
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  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

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  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
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  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
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  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
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  64-bit

• Supplied sizes read from CSV

  The same cell read back into numbers.

  Plant sizes, spacing and symbols plants/plants.js

  • BimCraft
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  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
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  • 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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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• 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

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• 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

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  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

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  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

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  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

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  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

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  64-bit

• Blend of two rotations

  The rotation part of a pose between two keys.

  q = (sin((1 − k)θ) a + sin(kθ) b) ÷ sin θ, θ = acos(a · b)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Angles to quaternion

  XYZ rotation angles in degrees turned into a unit quaternion.

  q = q_z × q_y × q_x, each from sin and cos of half the angle

  Animation keys, curves and poses render/anim.js

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  64-bit

• Quaternion to angles

  XYZ rotation angles in degrees from a unit quaternion, chosen close to the key before.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Object placement at a time

  A moving object's placement matrix between two matrix keys.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Time inside a clip

  The time of a saved action that a clip plays at a given timeline time.

  x = (t − start) ÷ scale; local = from + (x mod span)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Mixing a clip into a pose

  An object's placement when a clip is blended in at less than full influence or added on top.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Driver expression

  The value of a property worked out from other properties by a typed expression.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Camera keys from saved scenes

  The key times of a walkthrough built from saved scenes with a move time and a hold time.

  t accumulates move and hold; key time = round₀.₀₀₁(t)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Default gravity for rigid bodies

  The gravity a physics bake starts from when the file sets none.

  g = (0, 0, −9.81) m/s²

  Animation keys, curves and poses render/anim.js

  • BimCraft
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  64-bit

• Rounding of manifest numbers

  The rounding every number in the manifest goes through.

  round(x, k) = round(x × 10ᵏ) ÷ 10ᵏ, k = 6 unless stated

  Render hand-off manifest render/handoff.js

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• Color to linear values

  A hex color turned into the linear Rec.709 triple the manifest carries.

  c ≤ 0.04045: c ÷ 12.92; else ((c + 0.055) ÷ 1.055)^2.4

  Render hand-off manifest render/handoff.js

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• Points and directions on the stage

  A point or a direction of the scene moved to the USD stage's coordinates.

  p′ = M p; d′ = (M d) ÷ |M d|

  Render hand-off manifest render/handoff.js

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• Camera entry

  The camera's matrices, position, view direction, aspect and view angle or height.

  fovY = 2 atan(1 ÷ |P₅|) × 180 ÷ π; ortho height = 2 ÷ |P₅|

  Render hand-off manifest render/handoff.js

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• Sun and lamp entries

  The sun's direction and strength, and each lamp's lumens, color, position, aim and size.

  Render hand-off manifest render/handoff.js

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• Scene, exposure and white balance entries

  The ground height, sample count, exposure in stops, shutter time and white-balance gains.

  Render hand-off manifest render/handoff.js

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• Animation frames entry

  The start and end time, frame rate, frame count and the camera matrix of each frame.

  Render hand-off manifest render/handoff.js

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• Material values

  The OpenPBR parameters written for each material, where they differ from the defaults.

  Render hand-off manifest render/handoff.js

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• Exposure value at ISO 100

  The EV figure the render dialog shows for the chosen f-stop, shutter time and ISO.

  EV = log₂(N² ÷ t) − log₂(S ÷ 100)

  Camera exposure and white balance render/pt-camera.js

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• Exposure in stops

  The stops added to the picture's exposure from the brightness slider and, in camera mode, the f-stop, shutter and ISO.

  stops = slider + log₂(t × S × 256 ÷ N²)

  Camera exposure and white balance render/pt-camera.js

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  64-bit

• Color temperature to chromaticity

  The CIE xy color of light of a given temperature in kelvin.

  x = a ÷ T³ + b ÷ T² + c ÷ T + d; y = polynomial in x

  Camera exposure and white balance render/pt-camera.js

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• White-balance gains

  The three color gains that make light of the chosen temperature show as white.

  gain = (ref ÷ w) × (Y(w) ÷ Y(ref))

  Camera exposure and white balance render/pt-camera.js

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• Picked focus distance

  The distance from the camera to the clicked surface, measured along the view direction.

  focus = round(−(d · view axis ÷ |view axis|) × t × 100) ÷ 100

  Focus distance picked in the view render/ipr.js

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  32-bit in part

• Linear picture to display values

  Each pixel's final red, green and blue from the linear picture and the post settings.

  AP1 × wb → Rec.709 × 2^exposure → vignette → highlights → contrast → view transform → sRGB → 8 bits

  Rendered picture: mixing, post and color render/pt-post.js

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  32-bit in part

• View transform

  The tone curve that brings scene light into the display's range.

  Rendered picture: mixing, post and color render/pt-post.js

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  64-bit

• Blur for bloom

  The wide blur of the picture that bloom blends back in.

  r = max(1, round(√(12σ² ÷ 3 + 1) ÷ 2)); three box blurs of radius r per axis

  Rendered picture: mixing, post and color render/pt-post.js

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  32-bit in part

• Bloom

  The picture blended with three blurs of itself.

  out = img × (1 − a) + (0.5 b₁ + 0.3 b₂ + 0.2 b₃) × a

  Rendered picture: mixing, post and color render/pt-post.js

  • BimCraft
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  32-bit in part