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

• Which side of an edge

  Whether three points turn counter-clockwise, clockwise, or lie on one line.

  sign of (b − a) × (c − a)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Is an edge Delaunay

  Whether the point across an edge lies inside the circle through the triangle on this side.

  sign of the in-circle determinant of a, b, c, d

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Triangle under a point

  The triangle a point falls in and its three barycentric weights.

  w₁ = ((y_b − y_c)(x − x_c) + (x_c − x_b)(y − y_c)) ÷ d, w₃ = 1 − w₁ − w₂

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Elevation at a point

  The surface elevation at an easting and northing.

  z = w₁·z₁ + w₂·z₂ + w₃·z₃

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Slope of a triangle

  Rise over run and the downhill direction of one triangle.

  n = u × v; slope = √((n_x ÷ n_z)² + (n_y ÷ n_z)²)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Plan and surface area

  The surface's area in plan and along the slope, with its elevation range.

  area in plan = Σ |n_z| ÷ 2; surface area = Σ |u × v| ÷ 2

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  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

• Reproject points

  Every point of a file from one coordinate system to another.

  (x, y) in the source system → (x, y) in the target system, point by point

  Map coordinate reprojection reproject-batch.js

  • BimCraft
  • CivilCraft
  64-bit

• Map origin from latitude and longitude

  Eastings and northings of a typed latitude and longitude.

  (lon, lat) → (E, N), rounded to 0.000001 of the map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Model to map conversion

  Where a model sits on the map: origin, rotation and scale.

  X axis = (cos a, sin a) rounded to 1e-12; scale = model unit ÷ map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Normal of a tilted plane

  The unit normal of a plane from its tilt and compass bearing.

  n = (h_x·sin β, h_y·sin β, cos β), h = north·cos az + east·sin az

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Tilt and azimuth of a normal

  The tilt from horizontal and the compass bearing a plane faces.

  tilt = acos(clamp(n_z, −1, 1)); azimuth = (atan2(n·east, n·north) + 360) mod 360

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Axes of a plane from its normal

  The along-eave and up-slope directions of a roof face.

  u = (−n_y, nₓ, 0) ÷ √(nₓ² + n_y²); v = n × u

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Roof faces from triangles

  Groups upward triangles into planes and gives each plane's normal, tilt, azimuth, area and centroid.

  n = (B − A) × (C − A) ÷ |…|; area = |…| ÷ 2; face normal = unit(Σ area·n); centroid = Σ area·mid ÷ Σ area

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Point inside the face's triangles

  Tests whether a point in the face's own axes lies in any of its triangles.

  barycentric l₁, l₂, l₃ ≥ −eps ÷ √|d|

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Modules fitted in a roof face

  The positions and corners of the modules that fit inside a face with a setback.

  position = (s₀ + col × (across + gap), t₀ + row × (along + gap)); corner = centroid + s·u + t·v

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Row spacing without winter shade

  The distance between ground rows that leaves no shade at the winter solstice.

  h = slant × sin β; shadow = h ÷ tan α × max(0, cos Δaz); pitch = slant × cos β + shadow

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Ground rows of tables

  The tables and modules of a ground array with their corners, the pitch and the ground cover ratio.

  table width = n × across + (n − 1) × gap; row origin = origin − facing × pitch × r; GCR = slant ÷ pitch

  Solar array layout on roofs and ground pv.js

  • BimCraft
  • CivilCraft
  64-bit

• Compass frame of the site

  The north and east unit vectors in the model's plan.

  north = n ÷ |n|; east = (north_y, −northₓ)

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Clock hour for a solar hour

  The clock time at which the sun's time is a given hour.

  clock = solar − (equation of time + 4 × longitude − 60 × time zone) ÷ 60

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Sun direction in model axes

  The sun's altitude, azimuth and direction for a solar hour, turned to the model's north.

  dir = east × dₓ + north × d_y, with d_z unchanged; ω = 15° × (solar hour − 12)

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Daily extraterrestrial radiation

  The energy on a horizontal plane above the atmosphere in one day.

  δ = 23.45° × sin(2π(284 + n) ÷ 365); ω_s = acos(−tan φ · tan δ); H₀ = 24 ÷ π × G_on ÷ 1000 × (cos φ cos δ sin ω_s + ω_s sin φ sin δ)

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Monthly diffuse fraction

  The share of a month's daily light that is diffuse, from the clearness index.

  ω_s ≤ 81.4°: 1.391 − 3.56 k + 4.189 k² − 2.137 k³; else 1.311 − 3.022 k + 3.427 k² − 1.821 k³

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Hourly share of the day's light

  The share of a day's total and diffuse light that falls in one hour.

  r_d = π ÷ 24 × (cos ω − cos ω_s) ÷ (sin ω_s − ω_s cos ω_s); r_t = (a + b cos ω) × r_d

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Hourly light from a typed monthly total

  Spreads a typed daily total over the hours and splits it into diffuse and direct light.

  I = H × r_t ÷ Σ r_t; I_d = min(I, H_d × r_d ÷ Σ r_d); DNI = min((GHI − DHI) ÷ cos z, 0.8 × G_on)

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Light on a tilted plane

  The direct, sky and ground-reflected light on a module plane in one hour.

  beam = DNI × cos θ; sky = DHI × (1 + cos β) ÷ 2; ground = GHI × ρ × (1 − cos β) ÷ 2; IAM = max(0, 1 − b₀(1 ÷ cos θ − 1))

  Sun position and light on a tilted plane for solar pv-sun.js

  • BimCraft
  • CivilCraft
  64-bit

• Grid position from GeoTIFF tags

  The transform from cell column and row to map coordinates.

  x₀ = tie_X − tie_I × scale_X; y₀ = tie_Y + tie_J × scale_Y; pixel-is-point: x₀ −= ½(a + b), y₀ −= ½(d + e)

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• Cell size

  The single cell size of a square, north-up grid.

  cell = a when b = d = 0 and |a + e| < 1e-9 × |a|

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• ASCII grid header

  The size, cell size and lower-left corner of an ASCII grid.

  corner = center − cell ÷ 2 when the header gives cell centers

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• ASCII grid position

  The transform of an ASCII grid, with its origin at the top-left corner.

  top = yll + rows × dy

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• Elevation from a stored sample

  Turns one stored sample into an elevation, or into no value.

  z = v × scale + offset

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• Reading a window of cells

  Reads part of the grid, optionally every step-th cell, into an array of elevations.

  columns = ceil((c₁ − c₀) ÷ step); rows = ceil((r₁ − r₀) ÷ step)

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• Lowest and highest elevation

  The minimum and maximum elevation shown for a grid.

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• Position of a georeferenced picture

  The transform and bounding box of a picture read at a reduced size.

  sₓ = width ÷ w, s_y = height ÷ h; affine = [x₀, a·sₓ, b·s_y, y₀, d·sₓ, e·s_y]

  Elevation grids read from GeoTIFF and ASCII files raster.js

  • BimCraft
  • CivilCraft
  64-bit

• 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

• Local origin and merging of points

  Moves every point to a local origin and makes points that fall on the same spot one vertex.

  origin = ⌊min x⌋, ⌊min y⌋; key = round(x ÷ 1e-4), round(y ÷ 1e-4)

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Coordinates handed to the triangulator

  Packs the local coordinates for Delaunator and runs the Delaunay triangulation.

  coords = (x, −y) per vertex

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Boundary clipped to the data and given elevations

  Clips the outer boundary to the convex hull of the data and takes each corner's elevation from a first surface built without it.

  z_corner = elevationAt(x, y) with a 1e-6 edge tolerance

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Triangles kept inside the boundary

  Drops the triangles whose center lies outside the boundary.

  keep when the centroid (Σx ÷ 3, Σy ÷ 3) is inside the boundary polygon

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Surface from given triangles

  Builds a surface from vertices and faces read from a file, keeping every triangle counter-clockwise.

  o = (x_b − x_a)(y_c − y_a) − (y_b − y_a)(x_c − x_a); swap b, c when o < 0

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Crossing of two breakline segments

  Finds where two segments properly cross, as a fraction along each.

  t = cross(c − a, s) ÷ cross(r, s); u = cross(c − a, r) ÷ cross(r, s)

  Ground surface triangulation and contours tin.js

  • BimCraft
  • CivilCraft
  64-bit