Calculations

Calculations

• 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

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• 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
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  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

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  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

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• 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

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• 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

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• 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

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  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

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• Denoiser input

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

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

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  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

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  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
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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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• 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
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  64-bit

• Knot span

  Which knot interval a B-spline parameter falls in.

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

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  Exact

• B-spline basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N_{i,0} = 1; N_{i,j} built from (t − u) ÷ (u_b − u_a) terms in the Cox–de Boor triangular table

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

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• B-spline curve point

  The point on a B-spline or rational B-spline curve at a parameter.

  C(t) = Σ Nᵢ wᵢ Pᵢ ÷ Σ Nᵢ wᵢ

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

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• Is a B-spline curve closed

  Decides whether a B-spline curve's two ends meet, so it can be treated as periodic.

  |C(t₀) − C(t₁)| < 1e-9 × max(1, max |Pᵢ|)

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

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• B-spline surface point

  The point on a B-spline or rational B-spline surface at (u, v).

  S(u, v) = Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ Pᵢⱼ ÷ Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ

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

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  64-bit

• Seams and poles of a B-spline surface

  Decides which direction of a B-spline surface closes on itself and which boundary rows collapse to a point.

  two points are the same when |a − b| < 1e-7 × max(1, |a|)

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

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  64-bit

• Cylinder

  The point at (angle, height) on a cylinder, and the angle and height of a point.

  S(u, v) = o + r cos u · x + r sin u · y + v · z

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

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  64-bit

• Cone

  The point at (angle, distance along the axis) on a cone, and the inverse.

  r(v) = R + v tan α; v = ((ρ − R) tan α + l_z) ÷ (tan² α + 1)

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

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  64-bit

• Sphere

  The point at (longitude, latitude) on a sphere, and the inverse.

  S(u, v) = o + r cos v cos u · x + r cos v sin u · y + r sin v · z

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

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  64-bit

• Torus

  The point at two angles on a torus, and the inverse.

  c = R + r cos v; S(u, v) = o + c cos u · x + c sin u · y + r sin v · z

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

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  64-bit

• Surface of revolution

  The point of a profile curve turned about an axis by an angle, and the inverse.

  S(u, v) = o + h z + rr cos u + (z × rr) sin u, h = (C(v) − o)·z, rr = (C(v) − o) − h z

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

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  64-bit

• Angle of a point on an ellipse

  The ellipse parameter of the point on the ellipse nearest a given point.

  f(t) = (E(t) − q)·E′(t) = 0, solved by t ← t − f ÷ f′

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

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• Parameter of a point on a B-spline curve

  The curve parameter nearest a given point.

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

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  64-bit

• Parameters of a point on a B-spline surface

  The (u, v) of the surface point nearest a given point.

  [S_u·S_u S_u·S_v; S_u·S_v S_v·S_v] d = −[r·S_u; r·S_v], r = S(u, v) − q

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

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  64-bit

• Surface of linear extrusion

  The point of a curve swept along a vector, and the (u, v) of a point.

  S(u, v) = C(u) + v · vec; v = (q − C(u))·d ÷ |vec|, u = project(q − v · vec)

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

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  64-bit

• Offset surface

  The point of a surface moved a distance along its normal.

  S_off(u, v) = S(u, v) + d · n, n = (ΔS_u × ΔS_v) ÷ |…|

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

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  64-bit

• Length and angle units

  The file's length unit in millimeters and its plane-angle unit in radians.

  mm = prefix × 1000 for a meter; a conversion-based unit = its value × its base unit

  STEP import: units, points and assembly placement step/read.js

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• Point coordinates

  A CARTESIAN_POINT in millimeters.

  (x, y, z) × unit in mm

  STEP import: units, points and assembly placement step/read.js

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  64-bit

• Radii and cone angles

  The radius of a cylinder, cone, sphere or torus in millimeters and a cone's half angle in radians.

  r × unit in mm; α × angle unit, accepted when 0 < |α| < π ÷ 2

  STEP import: units, points and assembly placement step/read.js

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  64-bit

• Knot vectors

  The full knot vector of a B-spline from the file's knots and multiplicities, or the one a uniform, quasi-uniform or Bézier form implies.

  STEP import: units, points and assembly placement step/read.js

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  Exact

• Model size and chord tolerance

  The size of the model and the distance a triangle may stray from a curved face.

  size = diagonal of the box around every point; tol = max(4e-4 × size, 1e-4 mm)

  STEP import: units, points and assembly placement step/read.js

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• Closing a loop of edges

  Drops the last point of an edge loop when it repeats the first.

  |first − last| ≤ 0.01 × tol

  STEP import: units, points and assembly placement step/read.js

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• Placement matrices

  A placement as a 4 × 4 matrix, the product of two and the inverse of one.

  M⁻¹ = [Rᵀ, −Rᵀ t]

  STEP import: units, points and assembly placement step/read.js

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• A component placed in its assembly

  The transform that carries a component's coordinates into the assembly that uses it.

  T = M(A₂) × M(A₁)⁻¹

  STEP import: units, points and assembly placement step/read.js

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• Placed vertex positions

  Every vertex of a solid moved to its place and stored for the app.

  p′ = R p + t

  STEP import: units, points and assembly placement step/read.js

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• Faceted faces

  Triangles for a flat face given as loops of points.

  n = Σ ((yᵢ − yⱼ)(zᵢ + zⱼ), (zᵢ − zⱼ)(xᵢ + xⱼ), (xᵢ − xⱼ)(yᵢ + yⱼ))

  STEP import: units, points and assembly placement step/read.js

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  64-bit