Calculations

Calculations

• Hardware and gasket quantities

  Handles, hinges, locking points, strikers, packers, connectors and the meters of each gasket for a set of windows.

  locks = max(2, ⌈perimeter ÷ lock pitch⌉); sash gasket = 2 (w + h) ÷ 1000 m; glass gasket = 4 (gw + gh) ÷ 1000 m

  Window and door pieces, glass and hardware addons/windows/lib/model.js

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• Millimeters per drawing unit

  The factor between the drawing's units and the builder's millimeters, and its reciprocal stored with each window.

  mm per drawing unit = per × 25.4 (inch rules) or per × 1; kmm = 1 ÷ that

  Window and door pieces, glass and hardware addons/windows/main.js

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• Typed opening size

  A size typed as width x height in drawing units, as millimeters.

  w = typed w × mm per drawing unit

  Window and door pieces, glass and hardware addons/windows/main.js

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• Pieces from quantities

  One entry per piece to cut, from rows with a quantity.

  Cutting profile pieces from stock bars addons/windows/lib/cutlist/bars.js

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• Length a bar takes

  The length of bar used by a set of pieces, with the saw kerf and the saving when mitred ends share a cut.

  used = Σ (length + kerf) − depth × min(⌊mitred ends ÷ 2⌋, mitred pieces − 1)

  Cutting profile pieces from stock bars addons/windows/lib/cutlist/bars.js

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• Fewest bars for one stock length

  Which pieces go on which bar, and whether the count is proven to be the fewest.

  bound = ⌈Σ (length + kerf) ÷ usable − 1e-9⌉; usable = stock − trim at both ends

  Cutting profile pieces from stock bars addons/windows/lib/cutlist/bars.js

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• Bars, cut positions, offcuts and waste

  The list of bars with each piece's start position, the offcut left on each bar and the totals.

  pos accumulates length + kerf; offcut = stock − trims − used; waste % = round((1 − piece length ÷ stock length) × 10⁴) ÷ 100

  Cutting profile pieces from stock bars addons/windows/lib/cutlist/bars.js

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• Totals over all profiles

  The number of bars, the piece length, the stock length and the waste over every profile and color.

  waste = Σ stock length − Σ piece length

  Cutting profile pieces from stock bars addons/windows/lib/cutlist/bars.js

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• Raw double, read

  A 64-bit value read from eight bytes of the bit stream.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Bit double, read

  A value stored as a two-bit code: a full 64-bit value, exactly 1.0 or exactly 0.0.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Double with default, read

  A value stored as the bytes in which it differs from a default value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• 64-bit whole number, read

  An unsigned 64-bit whole number from two 32-bit halves.

  value = high × 2³² + low

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Bit long long, read

  A whole number of up to seven bytes, the count given by three bits.

  value = Σ byteᵢ × 256ⁱ

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Handle reference, read

  An object handle, absolute or as an offset from another handle.

  value = Σ byte × 256ᵏ; codes 6, 8, A, C: ref + 1, ref − 1, ref + value, ref − value

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Modular char, read

  A signed whole number stored seven bits per byte.

  value = Σ (byte & 0x7F) × 2^(7i); the last byte carries 6 bits and the sign

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

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• Raw double, written

  A 64-bit value written as its eight bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Bit double, written

  A value written as a two-bit code for exactly 0.0 or 1.0, or as a full 64-bit value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Double with default, written

  A value written as a two-bit code when it equals its default, or in full.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Thickness, written

  A thickness written as one bit when it is zero, or as a bit double.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Bit long, written

  A 32-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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  Exact

• Bit short, written

  A 16-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Bit long long, written

  A 64-bit whole number as a three-bit byte count and its bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Point, written

  The three coordinates of a point as bit doubles.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Handle bytes, written

  A handle as its bytes, most significant first.

  byte = v mod 256; v = ⌊v ÷ 256⌋

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

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• Lineweight to stored index

  The index written for a lineweight, or the codes for by layer, by block and default.

  index of the list value with the smallest |value − lineweight|

  Lineweights in a DWG file dwg/model.js

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• Stored index to lineweight

  The lineweight for an index read from a file.

  Lineweights in a DWG file dwg/model.js

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• Variant changed to another type

  The value of a variant as a string, a double, a single, an integer, a long or a boolean.

  integer = round(n); double = n

  Number conversion in LISP variants lisp/vla.js

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• Longest clear part of a move

  The part of a straight move that stays clear of other solids.

  steps = min(160, max(4, ⌈|δ| ÷ (thin ÷ 2)⌉)); then 24 halvings between the last clear and the first blocked sample

  Where a moved solid stops against another m3d/collide.js

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• Thinnest dimension

  The smallest box side of the moving pieces and of everything in the way, which sets the sample spacing.

  thin = max(0.005, min over boxes of the smallest side above 1e-4)

  Where a moved solid stops against another m3d/collide.js

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• Edge through the inside of a face

  Whether a segment passes through a face's interior.

  sₚ = n·p − d, s_q = n·q − d; t = sₚ ÷ (sₚ − s_q)

  Where a moved solid stops against another m3d/collide.js

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• Point in a face

  Whether a point on a face's plane lies in one of its triangles.

  u, v from cross products ÷ |n|²; inside when u, v, 1 − u − v ≥ −1e-9

  Where a moved solid stops against another m3d/collide.js

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• Point inside a solid

  Whether a point lies inside the closed surface made of an object's faces.

  Where a moved solid stops against another m3d/collide.js

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• Ray against a triangle

  The distance along a ray to a triangle, or nothing when it misses.

  det = e₁·(d × e₂); u = (s·(d × e₂)) ÷ det; v = (d·(s × e₁)) ÷ det; t = (e₂·(s × e₁)) ÷ det

  Where a moved solid stops against another m3d/collide.js

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• Box of a placed component

  The bounding box in the model of a component's faces and edges, from its own coordinates and its placement matrix.

  X = (m₀x + m₄y + m₈z + m₁₂) ÷ w

  Where a moved solid stops against another m3d/collide.js

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• Triangles of a polygon with holes

  The triangles of a face as triples of vertex indexes.

  Triangulation of faces with holes m3d/earcut.js

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• Orientation of three points

  Twice the signed area of a triangle, whose sign says which way three points turn.

  (q_y − p_y)(r_x − q_x) − (q_x − p_x)(r_y − q_y)

  Triangulation of faces with holes m3d/earcut.js

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• Direction of a ring

  Whether a ring of points runs clockwise or counterclockwise.

  Σ (x_j − x_i)(y_i + y_j)

  Triangulation of faces with holes m3d/earcut.js

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• Is a corner an ear

  Whether the triangle at a corner is convex and holds no other point of the polygon.

  Triangulation of faces with holes m3d/earcut.js

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• Point in a triangle

  Whether a point lies in or on a triangle.

  three comparisons (c − p) × (a − p) ≥ 0, each written as product ≥ product

  Triangulation of faces with holes m3d/earcut.js

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• Duplicate and collinear points removed

  Drops points that repeat their neighbor or lie exactly on the line between their neighbors.

  Triangulation of faces with holes m3d/earcut.js

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• Bridge from a hole to the outer ring

  The outer-ring point a hole is connected to, found by a ray to the left from the hole's leftmost point.

  x = p_x + (h_y − p_y)(n_x − p_x) ÷ (n_y − p_y); tan = |h_y − p_y| ÷ (h_x − p_x)

  Triangulation of faces with holes m3d/earcut.js

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• Do two edges cross

  Whether two segments intersect, used to keep a diagonal from crossing the polygon.

  Triangulation of faces with holes m3d/earcut.js

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• Is a diagonal's midpoint inside

  Whether the middle of a candidate diagonal lies inside the polygon.

  crossing count of a ray from ((a_x + b_x) ÷ 2, (a_y + b_y) ÷ 2)

  Triangulation of faces with holes m3d/earcut.js

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• Start and finish of each step

  When each step of each batch starts and finishes, and on which vessel.

  release = start + b × interval; t₀ = max(ready, vessel free); t₁ = t₀ + duration; wait = t₀ − ready

  Batch schedule and batch inventory process/batch.js

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• Makespan, utilization, cycle time and batches per day

  The length of the whole schedule, how busy each vessel is, the steady cycle time and the batches per day it allows.

  makespan = max finish − start; utilization = Σ(end − start) ÷ makespan; cycle = (work + clean × uses) ÷ vessels; batches per day = 24 ÷ cycle

  Batch schedule and batch inventory process/batch.js

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• Hours to whole days for the Gantt view

  The start and finish day of each task for the schedule view, which works in whole days.

  start day = day0 + ⌊start ÷ 24 + 1e-9⌋; finish day = day0 + max(⌊(finish − 1e-9) ÷ 24⌋, ⌊start ÷ 24 + 1e-9⌋)

  Batch schedule and batch inventory process/batch.js

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• Batch mole inventory and atom balance

  The moles of each component in each vessel after every operation, what was discharged, and how well the atoms balance.

  transfer: q = n × fraction; react: ξ = conversion × n_key ÷ |ν_key|, n ← n + ν × ξ; atom error = max |charged − held| ÷ charged

  Batch schedule and batch inventory process/batch.js

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• Monod growth in a batch or fed-batch vessel

  Concentrations of biomass, substrate and product, the volume, and their masses at each output time.

  μ = μmax × S ÷ (Ks + S); dX/dt = (μ − kd) X − D X; dS/dt = −(μ ÷ Yxs + m) X + D (Sf − S); dP/dt = (α μ + β) X − D P; dV/dt = F; D = F ÷ V

  Fermentation (Monod kinetics) process/batch.js

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• Channel counts and counts with spare

  How many analog and digital inputs and outputs the I/O list needs, and how many once the spare allowance is added.

  with spare = ⌈n × (1 + spare)⌉

  I/O channel counts with spare allowance process/controls.js

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• Lower and higher heating value

  The heat released by burning one mole of a component, with the water formed as vapor or as liquid.

  LHV = h_f − n_C × h_f(CO2) − (n_H ÷ 2) × h_f(H2O gas); HHV uses h_f(H2O liquid)

  Process emissions (CO2 equivalent) process/emissions.js

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