Calculations

Calculations

• Page scale matrix

  The matrix that lets the rest of the page be written in millimeters.

  [K 0 0 K 0 0] with K written as num(K) = 2.8346

  Vector PDF output pdf.js

  • BimCraft
  • DetailCraft
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• Page size in points

  The page box from the sheet size in millimeters.

  points = num(mm × K)

  Vector PDF output pdf.js

  • BimCraft
  • DetailCraft
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• Circle as four curves

  A circle written as four cubic Bézier curves.

  control distance k = 0.5522847498 × r

  Vector PDF output pdf.js

  • BimCraft
  • DetailCraft
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• Text position and matrix

  Where a text starts and how it is turned and sized.

  x = x₀ + dx·cos θ, y = y₀ + dx·sin θ; matrix = [h·cos θ, h·sin θ, −h·sin θ, h·cos θ, x, y]

  Vector PDF output pdf.js

  • BimCraft
  • DetailCraft
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• Room grid

  Covers a room with square cells and marks the ones whose center is inside it.

  cell × 1.5 while (W ÷ cell + 1) × (H ÷ cell + 1) > 60000

  People movement and travel distance people-sim.js

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• Distance field over a room

  The walked distance from one cell to every other free cell of a room.

  d(n) = min over 16 neighbors of d(c) + √(Δi² + Δj²) × cell

  People movement and travel distance people-sim.js

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• Route between two places in a room

  The distance in meters between two points of a room and the path walked.

  straight: √(Δx² + Δy²) ÷ 1000; else Σ segment lengths of the string-pulled path ÷ 1000

  People movement and travel distance people-sim.js

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• Distance to the goal from every opening

  The remaining walking distance to an exit, or to a room, from each side of each door and stair.

  d(far side of j) = min(d(i) + walk(j → i) + length of opening j)

  People movement and travel distance people-sim.js

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• Travel distance of each room

  The farthest point of each room from an exit, in meters along the walls.

  max over cells and corners of min over openings (distance to opening + distance beyond it); rounded to 0.01 m

  People movement and travel distance people-sim.js

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• Speed factor from crowd density

  How much the crowd in a room slows a walker.

  D = people ÷ max(area, 0.5); factor = 1 for D ≤ 0.54, else max(min factor, (1 − 0.266·D) ÷ (1 − 0.266 × 0.54))

  People movement and travel distance people-sim.js

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• Flow rate of an opening

  How many people per second a door or stair lets through.

  rate = flow per meter × max(0.2, width ÷ 1000 − 2 × boundary)

  People movement and travel distance people-sim.js

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• Walking step

  How far a walker moves in one time step and the distance added to their total.

  Δs = v₀ × factor × Δt; walked += min(Δs, remaining length of the leg)

  People movement and travel distance people-sim.js

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• Passing an opening

  The moment each queued person passes a door or stair, and their wait.

  pass = max(next free, arrival, t) + counter-flow penalty; next free = pass + 1 ÷ rate; wait = pass − arrival

  People movement and travel distance people-sim.js

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• Reported times and distances

  Total time, median and 90th percentile times, walked distances and waits.

  percentile p = times[ceil(n·p) − 1]; times rounded to 0.1 s, distances to 0.01 m

  People movement and travel distance people-sim.js

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• Stair length

  The walked length of a stair from its plan walk line and its height.

  length = √(plan length² + height²)

  People movement and travel distance people-model.js

  • BimCraft
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• People per room

  How many people each room holds before a run.

  count = floor(area in m² ÷ m² per person)

  People movement and travel distance people-model.js

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• Rounding of stored picture values

  Rounds a position, scale or angle to 6 decimals.

  r6(v) = Math.round(v × 10⁶) ÷ 10⁶

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• New picture record

  The cleaned numbers of a picture, with defaults.

  default scale = r6(20000 ÷ width in pixels); angle = r6(((a mod 360) + 360) mod 360)

  Picture backgrounds: scale, placement and print picture.js

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• Scale by two points and a distance

  The new scale and center that make two picked points a given distance apart.

  k = mm ÷ |b − a|; scale′ = r6(scale × k); center′ = r6(a + (center − a) × k)

  Picture backgrounds: scale, placement and print picture.js

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• Corners of a placed picture

  The four corners of the picture in plan millimeters.

  corner = center + R(θ) · (± w·scale ÷ 2, ± h·scale ÷ 2), θ = angle × π ÷ 180

  Picture backgrounds: scale, placement and print picture.js

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• Pixel to plan matrix

  The transform from picture pixels to plan millimeters.

  [k·c, k·s, k·s, −k·c, x + k(−w/2·c − h/2·s), y + k(−w/2·s + h/2·c)]

  Picture backgrounds: scale, placement and print picture.js

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• Plan point to picture pixel

  The pixel of the picture under a plan point.

  (u, v) = (R(−θ)·(q − center)) ÷ scale; pixel = (u + w ÷ 2, h ÷ 2 − v)

  Picture backgrounds: scale, placement and print picture.js

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• Picture on a vertical plane

  Puts a picture on the plane standing over a line, as wide as the line.

  k = L ÷ w; center = (L ÷ 2, h·k ÷ 2)

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• Picture width typed

  The scale that makes the picture a typed width.

  scale = r6(mm ÷ w)

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• Turning a picture

  The new angle after a turn about the picture's center.

  angle′ = r6(((angle + deg) mod 360 + 360) mod 360)

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• Fit in a box

  Centers an unscaled picture in the view at 90 % of the box.

  k = min(box width ÷ w, box height ÷ h) × 0.9

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• Corners in 3D

  The picture's corners in level coordinates, for a level or a vertical plane.

  p = origin + u·û + v·v̂

  Picture backgrounds: scale, placement and print picture.js

  • BimCraft
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• Vertical picture in an elevation or section

  Decides whether a standing picture shows in a view and gives its matrix there.

  shown when |û·dir| ≤ sin(2°) and, in a section, |(a − o)·dir| ≤ 100 mm; x = a·right ± picture x, y = level difference + picture y

  Picture backgrounds: scale, placement and print picture-sheet.js

  • BimCraft
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• Number format for picture matrices

  Turns a matrix entry or opacity into the text written in the PDF.

  toFixed(6), trailing zeros removed; |v| < 5e-7 → 0

  Picture backgrounds: scale, placement and print picture-pdf.js

  • BimCraft
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• Pixels to print a picture at

  The pixel size a picture is drawn at on a sheet, limited by dots per inch and total pixels.

  paper mm per pixel = scale ÷ ratio; dpi = 25.4 ÷ that; k = min(1, max dpi ÷ dpi, √(max pixels ÷ (w·h)))

  Picture backgrounds: scale, placement and print picture-pdf.js

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• Matrix for the image's unit square

  Turns the pixel-to-plan matrix into the one PDF needs for an image drawn on a unit square.

  [a·w, b·w, −c·h, −d·h, e + c·h, f + d·h]

  Picture backgrounds: scale, placement and print picture-pdf.js

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• Scale from two points and a distance

  How many millimeters one page unit is, from two picked points and the true distance between them.

  scale = distance in mm ÷ √(Δx² + Δy²)

  Walls and openings from PDF pages and scans plan-pages.js

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• Circle through three points

  The center and radius of the circle through three points.

  d = 2(p_x(q_y − r_y) + q_x(r_y − p_y) + r_x(p_y − q_y)); center from the squared lengths ÷ d

  Walls and openings from PDF pages and scans plan-pages.js

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• Arc from a Bézier curve

  Recognizes a cubic curve that is a circular arc and gives its center, radius, start angle and sweep.

  circle through P(0), P(0.5), P(1); accepted when |dist(P(t), center) − r| ≤ 0.012 r at t = 0.25 and 0.75

  Walls and openings from PDF pages and scans plan-pages.js

  • BimCraft
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• Page lines and arcs in millimeters

  Scales a PDF page's line segments and arcs to millimeters and finds their bounds.

  x_mm = x_pt × mm per point

  Walls and openings from PDF pages and scans plan-pages.js

  • BimCraft
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• Scan lines and arcs in millimeters

  Scales a traced scan from pixels to millimeters and turns its y axis up.

  x = px × k; y = (height in px − py) × k

  Walls and openings from PDF pages and scans plan-pages.js

  • BimCraft
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• Joining lines cut at a tee

  Merges lines of one straight run that are split by a joining wall.

  merged when |cos angle| ≥ cos 1.2°, both ends within 35 mm of the first line, and the gap ≤ 350 mm

  Walls and openings from PDF pages and scans plan-pages.js

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• Hatching detection

  Finds bundles of short parallel lines so they are not taken for walls.

  Walls and openings from PDF pages and scans plan-pages.js

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• Merged wall line and thickness

  The center line and thickness of a wall made of collinear pieces with openings between them.

  t = Σ tᵢ·Lᵢ ÷ Σ Lᵢ; offset = Σ offsetᵢ·Lᵢ ÷ Σ Lᵢ; A, B = base + u·s + n·offset

  Walls and openings from PDF pages and scans plan-pages.js

  • BimCraft
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• Squaring a scanned page

  Decides whether a scan is turned back before it is traced.

  turned by the found angle when |angle| > 0.15° and confidence > 1.6

  Walls and openings from PDF pages and scans plan-pages.js

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• Stored position and angle

  The X, Y and Angle written for a symbol.

  X, Y = round(mm ÷ unit, 6); Angle = round(radians × 180 ÷ π, 4)

  Plan symbol positions plan-symbols.js

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• Point on a wall face

  The point on a wall's face at a distance along the wall, and the direction the face looks.

  at = a + s·u + side × (t ÷ 2)·n; face angle = atan2(side·n_y, side·nₓ)

  Plan symbol positions plan-symbols.js

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• Symbols following a wall

  Moves a symbol to where its stored side and distance are on the wall now.

  turn = atan2(sin Δ, cos Δ), Δ = new face angle − stored face angle

  Plan symbol positions plan-symbols.js

  • BimCraft
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• Program totals

  The net and gross floor area and the number of rooms in the program.

  net = Σ count × area; gross = net ÷ efficiency

  Massing options from a room program program-mass.js

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• Rectangle inside the envelope

  Tests whether a turned rectangle lies inside the envelope.

  Massing options from a room program program-mass.js

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• Rectangle of a wanted area

  The rectangle of a given proportion and area nearest the middle of the envelope.

  short = √(area ÷ ratio); long = short × ratio

  Massing options from a room program program-mass.js

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• L and U shapes of a wanted area

  An L-shaped or U-shaped footprint scaled to the wanted area and placed in the envelope.

  k = √(wanted area ÷ template area)

  Massing options from a room program program-mass.js

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• Rooms stacked on floors

  Assigns rooms to floors up to each floor's usable area.

  placed when floor net + room area ≤ usable area + 1e-6

  Massing options from a room program program-mass.js

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• Areas reported for an option

  Footprint, gross area, required area, surplus and coverage of one massing option.

  footprint = |shoelace|; gross = footprint × floors ÷ 10⁶; surplus = gross − required; coverage = footprint ÷ envelope area

  Massing options from a room program program-mass.js

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• Option handed to the site study

  The option's points and floor height in the project's units.

  x = mm × unit per mm

  Massing options from a room program program-mass.js

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