Calculations

Calculations

• Convert between two units

  A value in one unit as a value in another unit of the same kind.

  result = v × (n₁ × d₂) ÷ (d₁ × n₂)

  Unit conversion unit-table.js Every app
  Exact

• Typed text to a unit

  What was typed, with its unit, as a value in the unit the field wants.

  3 yd typed into a feet field = convert(3, yd, ft) = 9

  Unit conversion unit-table.js Every app
  Exact

• A value to the base unit

  A number in any unit as meters, square meters, cubic meters, radians or rise over run.

  base = v × n ÷ d (v × π × n ÷ d for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• A base value to a unit

  A stored base value as a number in the unit shown.

  value = base × d ÷ n (base ÷ π × d ÷ n for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• Add, subtract, multiply, divide

  The four operations on values that carry units.

  a + b and a − b need the same dimensions; a × b and a ÷ b combine them

  Calc sheets calc.js Every app
  64-bit

• Comparisons

  Pass or fail checks such as M <= 40 kN m.

  a ≤ b is true when a ≤ b + 1e-12 × max(1, |a|, |b|)

  Calc sheets calc.js Every app
  64-bit

• Number shown

  A result as text in the sheet.

  6 significant digits; exponent form from 1e9 up and below 1e-4

  Calc sheets calc.js Every app
  64-bit

• Knot span

  Which span of the knot vector a parameter falls in.

  i such that U[i] ≤ u < U[i + 1], by bisection

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Basis functions

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

  N(i, p)(u) by the Cox-de Boor recurrence (A2.2)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Basis function derivatives

  The basis functions and their derivatives up to a chosen order.

  N⁽ᵏ⁾(i, p)(u) for k = 0 … d (A2.3)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Inverse of a 4 × 4 matrix

  The inverse of the model's coordination matrix.

  M⁻¹ = adj(M) ÷ det(M)

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Viewer point to model point

  A camera position or a point on a section plane in the model's own coordinates, in meters with Z up.

  (x, y, z) = C⁻¹ v; IFC = (x, −z, y)

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Viewer direction to model direction

  A camera direction, up vector or section plane normal in the model's own axes.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Model point to viewer point

  A point read from an issue file, placed in the viewer.

  v = C (x, z, −y)

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Model direction to viewer direction

  A direction read from an issue file, in the viewer's axes.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Numbers written to the file

  How each coordinate, direction, field of view and scale is written in a viewpoint.

  round(v × 10⁶) ÷ 10⁶

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Numbers read from the file

  Reads the X, Y and Z of a point or direction.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Camera read from a viewpoint

  The camera of an issue as BimCraft keeps it.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
  • ModelCraft
  64-bit

• Corners of a piece

  The four plan corners of a piece, or of the zone kept clear round it.

  corner = (x + u·cos r − v·sin r, y + u·sin r + v·cos r), u ∈ ±(w ÷ 2 + side), v ∈ [−front, d]

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Do two pieces overlap

  Whether two rectangles overlap by more than a hair.

  separate when, on any edge normal of either, a_max ≤ b_min + 1 mm or b_max ≤ a_min + 1 mm

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Is a point in the room

  Whether a point lies inside the room's outline.

  even-odd count of edges crossed by a ray toward +x

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Does a piece fit in the room

  Whether a rectangle lies inside the room and outside its holes.

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Which way the outline runs

  The signed area of the room's outline, used to turn it counter-clockwise.

  area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Zone kept clear at a door

  The rectangle round a door where nothing is placed.

  width = door width + 300 mm; depth = one door width on each side of the wall

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• A piece against a wall

  The first free place for a piece along a wall, with its back on the wall.

  t = w ÷ 2 + side, then every 100 mm; (x, y) = wall start + direction × t + inward normal × depth; kept: round(x), round(y), rotation to 0.001°

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• A piece in the middle of the room

  The first free place for a free-standing piece, starting at the middle of the room.

  middle of the outline's box, then offsets of k × 150 mm (k = 1…20) along ±x and ±y, at the longest wall's angle and at 90° to it

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Footprint of a placed piece

  The four plan corners of a piece from its stored position.

  Furniture layout of a room furnish.js

  • BimCraft
  • ModelCraft
  64-bit

• Resampling the chosen part of the photo

  Averages the chosen part of the photo down, or interpolates it up, to the tile size.

  Material maps and statistics from a photo photo-material.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Normal map from the height estimate

  The surface direction at each pixel of the tile.

  v = (−Δh_x × S, −Δh_y × S, 1) ÷ |v|, S = 4 × bump; stored as (v ÷ 2 + ½) × 255

  Material maps and statistics from a photo photo-material.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Seam ratio

  How big the jump across the tile's join is compared with the usual jump between neighboring pixels.

  ratio = mean |jump across the join| ÷ max(mean |jump between neighbors|, 1e-6)

  Material maps and statistics from a photo photo-material.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Statistics returned with the maps

  The crop, seam ratios, mean color, height spread and mean roughness reported for the material.

  σ = √(Σ band² ÷ n); seam and σ rounded to 0.01, mean roughness to 0.001, crop to 0.001

  Material maps and statistics from a photo photo-material.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Longitude and latitude to local meters

  The position of a point in meters east and north of the address point.

  x = (lon − lon₀) × 111320 × cos(lat₀); y = (lat − lat₀) × 111320

  Site surroundings in local meters site-context.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Local meters back to longitude and latitude

  The longitude and latitude of a local point.

  lon = lon₀ + x ÷ (111320 × cos(lat₀)); lat = lat₀ + y ÷ 111320

  Site surroundings in local meters site-context.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Outlines and heights kept in the site file

  Converts each outline to local meters, thins and rounds it, and sets the item's height.

  ring = round(x × 100) ÷ 100, round(y × 100) ÷ 100; height = source height, else levels × 3

  Site surroundings in local meters site-context.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Thinning a long outline

  Reduces an outline of more than 400 points by keeping every n-th point.

  step = ⌈points ÷ 400⌉; keep indexes divisible by step

  Site surroundings in local meters site-context.js

  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Child boxes of a node

  Stores the boxes of a node's four children as 8-bit offsets from the node's corner.

  scale = 2^e, the smallest with 255 × 2^e ≥ extent; lo = floor((box − origin) ÷ scale), hi = ceil(…)

  Ray casting for rendering and the render bench compute/bvh-wide.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Ray through the hierarchy

  Walks the hierarchy along a ray and returns the nearest hit distance.

  per axis: t = (box − origin) × (1 ÷ direction); hit if max of the near values ≤ min of the far values × (1 + 4×10⁻⁷)

  Ray casting for rendering and the render bench compute/bvh-wide.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Preview shading of a tile

  Shades each pixel of a tile from the surface it sees, lit from the eye.

  value = 0.08 + 0.72 × |n · d| ÷ |n|

  Ray casting for rendering and the render bench compute/preview-kernel.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Camera rays for one tile

  Casts the camera rays of a tile and counts the rays, the hits and the sum of the hit distances, for timing.

  direction = normalize(camera matrix · ((2 (x + jitter) ÷ W − 1) ÷ f_x, (1 − 2 (y + jitter) ÷ H) ÷ f_y, −1))

  Ray casting for rendering and the render bench compute/raycast-job.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Mean and variance from batches

  The mean picture and the variance of that mean from two or more independent batch pictures.

  mean = Σ b ÷ k; var(mean) = Σ (b − mean)² ÷ (k − 1) ÷ k

  Statistical comparison of rendered pictures compute/parity.js

  • BimCraft
  • ModelCraft
  64-bit

• 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