Calculations

Calculations

• 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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  • PartCraft
  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
  • ModelCraft
  • PartCraft
  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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• Material values

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

  Render hand-off manifest render/handoff.js

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  64-bit

• 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

  • BimCraft
  • ModelCraft
  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

  • BimCraft
  • ModelCraft
  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

  • BimCraft
  • ModelCraft
  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

  • BimCraft
  • ModelCraft
  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
  • ModelCraft
  32-bit in part

• Light group gains

  The color and brightness factor of each light group in a light mix.

  gain = color (linear AP1) × 2^stops

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

  • BimCraft
  • ModelCraft
  64-bit

• Mixed picture

  The linear picture made by adding the light groups with their gains.

  pixel = Σ over groups of (group sum × gain) ÷ samples

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

  • BimCraft
  • ModelCraft
  32-bit in part

• Picture or lighting element from the sums

  The linear picture, light mix or one lighting element as a mean per pixel, handed to the post step.

  mean = Σ samples ÷ sample count

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

  • BimCraft
  • ModelCraft
  32-bit in part

• ACES filmic curve

  The default tone curve.

  f(x) = (x (x + 0.0245786) − 0.000090537) ÷ (x (0.983729 x + 0.4329510) + 0.238081)

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

  • BimCraft
  • ModelCraft
  64-bit

• sRGB encoding

  The curve from display-linear values to stored sRGB values.

  c ≤ 0.0031308: 12.92 c; else 1.055 c^(1 ÷ 2.4) − 0.055

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

  • BimCraft
  • ModelCraft
  64-bit

• One pixel to 8 bits

  A pixel's mean color turned into display bytes with the default settings.

  byte = round(sRGB(ACES(Rec.709(AP1 × wb))) × 255)

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

  • BimCraft
  • ModelCraft
  64-bit

• Sample sums on the graphics card

  The running sum of every sample's color for a pixel when the render runs on the graphics card.

  ACC[pixel] += max(sample, 0)

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

  • BimCraft
  • ModelCraft
  32-bit in part

• Denoiser exposure scale

  The scale that brings the picture to the brightness the denoising network expects.

  scale = 0.18 ÷ 2^(mean of log₂ L over 16 × 16-pixel bins)

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

  • BimCraft
  • ModelCraft
  64-bit

• Denoiser input

  The color, albedo and normal values given to the denoising network.

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

  • BimCraft
  • ModelCraft
  32-bit in part

• Denoiser output

  The denoised picture taken back to linear values.

  linear = inverse curve(x ÷ norm) ÷ scale

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

  • BimCraft
  • ModelCraft
  32-bit in part

• Triangles in world space

  Every mesh triangle moved by its placement, cut by the section planes and stored for the kernels.

  Path tracer scene, light and sky buffers render/pt-pack.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Derived material values

  The relative refractive index, coat Fresnel average, extinction and dispersion term of a material record.

  μt = −ln(color) ÷ depth; B = (η − 1) ÷ ((abbe ÷ scale) × (1 ÷ 0.4861² − 1 ÷ 0.6563²))

  Path tracer scene, light and sky buffers render/pt-pack.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Physical sky color

  The sky's color in a direction for a sun position and haze.

  Path tracer scene, light and sky buffers render/pt-sky.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Light profile grid

  A lamp's IES light distribution as a grid of direction multipliers.

  multiplier = candela(θ, φ) × 4π ÷ flux

  Path tracer scene, light and sky buffers render/pt-ies.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Adding samples to a pixel

  The sum of sample colors and the sample count of every pixel in the reference tracer.

  Path tracer kernels render/pt-ref.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Fresnel reflectance of glass

  The share of light a clear surface reflects at an angle.

  F = (r∥² + r⊥²) ÷ 2

  Path tracer kernels render/pt-ggx.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Sample position

  The random-like number in [0, 1) used for one dimension of one sample of one pixel.

  Path tracer kernels render/pt-sampler.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Adaptive sampling test

  Decides when a pixel has had enough samples.

  v = max(0, (S₂ − S₁² ÷ J) ÷ (J − 1)); stop when √(v ÷ J) ≤ noise × (μ + 0.02)

  Path tracer kernels render/pt-adapt.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Lamp strength

  A lamp's lumens turned into the strength the renderers use.

  point or spot: Φ ÷ 4π × 0.02; panel: Φ ÷ π × 0.02

  Fast render lights and texture placement render/looks.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Sun direction for a pass

  A direction inside a small cone around the sun, one per render pass, for soft shadow edges.

  r = √h₅ × tan(degrees × π ÷ 180); a = 2π h₇

  Fast render lights and texture placement render/looks.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Texture coordinates of a point

  Where a point of a surface falls on its material's texture, in meters along and up the surface.

  u = p · (up × n) ÷ |up × n|; v = p · (n × u axis)

  Fast render lights and texture placement render/materials.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Axis placement

  The origin and three unit axes of an AXIS2_PLACEMENT_3D.

  z = axis ÷ |axis|; x = (ref − z (ref·z)) ÷ |…|; y = z × x

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Circle

  The point at an angle on a circle, and the angle of a point.

  P(t) = o + r cos t · x + r sin t · y; t = atan2(l_y, l_x)

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Polyline

  The point at a parameter along a polyline, and the parameter nearest a point.

  P(t) = (1 − f) Pᵢ + f Pᵢ₊₁, i = ⌊t⌋, f = t − i

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit