Calculations

Calculations

• Newton iteration

  Finds the state that satisfies the equations at one time.

  J_ij = (F_i(x + h e_j) − F_i(x − h e_j)) ÷ 2h, h = 1e-7 × max(|x_j|, 1); stop when |Δxᵢ| ≤ rtol |xᵢ| + atol + 1e-11 × max(1, largest |x|)

  Lumped system models solve.js Every app
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• Operating point

  The steady state, or the state set by the given initial values.

  one Newton solve from zero

  Lumped system models solve.js Every app
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• One time step

  The state one step later.

  trapezoid: (2 ÷ h)(S − S₀) − Ṡ₀ + g = 0; backward Euler: (1 ÷ h)(S − S₀) + g = 0

  Lumped system models solve.js Every app
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• Step-size control

  Chooses the time step from an estimate of the error, when adaptive stepping is on.

  err = max |x(two half steps) − x(one step)| ÷ (rtol |x| + atol), ÷ 3 for the trapezoid; h ← h × clamp(0.9 err^(−1/3), 0.2, 5)

  Lumped system models solve.js Every app
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• Value between stored times

  A node value at a time between two computed points.

  y = y₀ + (y₁ − y₀)(t − t₀) ÷ (t₁ − t₀)

  Lumped system models solve.js Every app
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• Parameter expressions

  The value of a parameter written as an expression, such as 2*3.3e3.

  numbers, parameters, + − × ÷ ^, brackets, pi

  Lumped system models modelica.js Every app
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• Thermal conductance to resistance

  A thermal conductor from a model file as the engine's thermal resistance.

  R = 1 ÷ G

  Lumped system models modelica.js Every app
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• Center, radius and sweep of a bulge arc

  The circle an arc segment lies on, from its two end points and its bulge.

  θ = 4·atan(b); k = (c ÷ 2) ÷ tan(θ ÷ 2); center = midpoint + k × left normal; r = |p − center|

  Arcs from bulges and three points geom.js

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• Arc through three points

  The bulge of the arc that starts at one point, passes through a second and ends at a third.

  center = circumcenter(p, m, q); sweep = atan2(q − center) − atan2(p − center) ± 2π; b = tan(sweep ÷ 4)

  Arcs from bulges and three points geom.js

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• Segment lengths of a run and their sum

  The length of each segment of an open run, and the run's total.

  L = Σ segLength(pᵢ₋₁, pᵢ, bᵢ₋₁)

  Arcs from bulges and three points geom.js

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• Snap to the nearest 45° direction

  Moves a picked point onto the nearest of the eight 45° directions from the point before it, keeping its distance along that direction.

  u = round(cos, sin of k·45° to 12 decimals); p′ = base + u × ((p − base)·u)

  Arcs from bulges and three points geom.js

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• Circle as two half-circle arcs

  Stores a drawn circle as two points on opposite sides of the center, each starting a half-circle arc.

  r = |edge − c|; points (c.x + r, c.y) and (c.x − r, c.y), b = 1 each

  Arcs from bulges and three points takeoffcraft.js

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• Arc reshaped by dragging its middle

  Recomputes a segment's bulge when its arc handle is dragged.

  Arcs from bulges and three points takeoffcraft.js

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• Length and area shown while drawing

  The length of the last segment, the run so far, the area and perimeter so far, or a circle's radius and area, shown in the status line as the pointer moves.

  length = L × k; area = A × k²; circle preview = π·r²

  Arcs from bulges and three points takeoffcraft.js

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• Check distance between two points

  The real distance between two picked points, in feet-inches, decimal feet and meters.

  ft = |b − a| × k; m = ft × 0.3048

  Arcs from bulges and three points takeoffcraft.js

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• Dropping repeated points when a mark is finished

  Removes a point that sits on the one before it, as a double-click adds.

  drop pᵢ when |pᵢ − previous kept point| ≤ 2 ÷ zoom

  Arcs from bulges and three points takeoffcraft.js

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• Arcs broken into chords

  Replaces each arc of a run or outline by points on the arc, at most one step apart.

  k = max(2, ⌈|θ| ÷ step⌉); pⱼ = center + r·(cos tⱼ, sin tⱼ), tⱼ = a₀ + θ·j ÷ k

  Outline checks: inside, crossing, cut-outs geom.js

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• Point inside an outline

  Whether a point lies inside a closed outline.

  Outline checks: inside, crossing, cut-outs geom.js

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• Two segments cross or touch

  Whether two straight segments cross or touch.

  d₁ = (d − c) × (a − c), …; cross when d₁, d₂ lie beyond ±s on opposite sides and d₃, d₄ too

  Outline checks: inside, crossing, cut-outs geom.js

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• Outline crosses itself

  Whether a closed outline crosses itself.

  Outline checks: inside, crossing, cut-outs geom.js

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• Outline that cannot be measured

  Refuses an outline with too few points, one that crosses itself, or one with no area.

  Outline checks: inside, crossing, cut-outs geom.js

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• Cut-out inside its area

  Checks that a cut-out lies fully inside the outline and does not overlap another cut-out.

  Outline checks: inside, crossing, cut-outs geom.js

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• Convex outline test

  Whether a closed outline has no inward corner.

  Back-outs clipped out of areas geom.js

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• Polygon clipped to a convex polygon

  The part of a polygon that lies inside a convex polygon.

  t = d₁ ÷ (d₁ − d₂); corner = p + (q − p)·t

  Back-outs clipped out of areas geom.js

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• Area a back-out covers

  How much of an area mark (outline minus cut-outs) a convex back-out covers, in square PDF units.

  overlap = max(0, A(outline ∩ b) − Σ A(cut-out ∩ b))

  Back-outs clipped out of areas geom.js

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• Net area shown for a mark

  The area in a mark's properties after its back-outs are taken off.

  net = max(0, area − cut); plan area = max(0, plan − cut ÷ factor)

  Back-outs clipped out of areas takeoffcraft.js

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• Typed length to feet

  Normalizes what was typed and reads it as one length, or as arithmetic over lengths.

  Typed lengths and item units units.js

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• One length with a metric unit, or feet and inches

  Reads one normalized length into feet.

  ft = meters ÷ 0.3048

  Typed lengths and item units units.js

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• Numbers and fractions in a typed length

  Splits the text into numbers, fractions and foot or inch marks.

  fraction = a ÷ b (integers)

  Typed lengths and item units units.js

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• Feet and inches combined

  Combines a feet part and an inches part (whole number plus fraction) into feet.

  ft = feet + (inches + fraction) ÷ 12

  Typed lengths and item units units.js

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• Bare number in the default unit to feet

  Converts a number in feet, inches, meters, centimeters or millimeters to feet.

  ft = v ÷ 12; ft = v ÷ 0.3048; ft = v ÷ 1000 ÷ 0.3048

  Typed lengths and item units units.js

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• Item unit factors

  The factor from feet or square feet to each unit an item can be reported in.

  Typed lengths and item units units.js

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• Feet and square feet into an item's unit

  Converts a length in feet or an area in square feet to the unit that goes with an item.

  m = ft × 0.3048; area = sf × perSqFt

  Typed lengths and item units units.js

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• Round half away from zero

  Rounds a number to a given count of decimals.

  r = ⌊|v|·10ᵈ + 0.5 + 1e-9·max(1, |v|·10ᵈ)⌋ ÷ 10ᵈ

  Rounding for display and for money units.js

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• Grouped fixed-point text

  Writes a number with a fixed count of decimals and thousands separators.

  Rounding for display and for money units.js

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• Feet and inches to the nearest fraction

  Writes a length in feet as feet, inches and a fraction of an inch.

  n = round(|ft|·12·denom + 1e-9); feet = ⌊n ÷ (12·denom)⌋; inches = ⌊rem ÷ denom⌋

  Rounding for display and for money units.js

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• Length in a chosen display unit

  Writes a length in feet-inches, decimal feet, inches, meters or millimeters.

  m = ft × 0.3048; mm = ft × 0.3048 × 1000

  Rounding for display and for money units.js

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• Feet per sheet unit for a named scale

  The scale number for a preset such as 1/4 inch = 1 foot or 1:100.

  ftPerUnit = (userUnit ÷ 72) × ratio ÷ 12

  Sheet scale: presets, calibration and drawing units units.js

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• Scale from two points and a typed distance

  The scale that makes the distance between two picked points equal a typed real distance.

  ftPerUnit = realFt ÷ |b − a|

  Sheet scale: presets, calibration and drawing units units.js

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• Named scale a number matches

  Finds the preset scale closest to a scale number, if any is close enough.

  err = |ftPerUnit − s| ÷ s ≤ 0.005

  Sheet scale: presets, calibration and drawing units units.js

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• Feet per drawing unit by INSUNITS code

  The table that turns a drawing's unit code into feet per drawing unit.

  code 6: 1 ÷ 0.3048; code 21 as written: 1200 ÷ 3937

  Sheet scale: presets, calibration and drawing units cadcount.js

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• The drawing's length unit

  Reads the INSUNITS code and looks up feet per drawing unit.

  Sheet scale: presets, calibration and drawing units cadcount.js

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• Viewport transform, model to paper

  The matrix that places model space on the paper through a layout's viewport.

  k = h ÷ viewHeight; M = [k·c, k·s, −k·s, k·c, cx − k(c·vx − s·vy), cy − k(s·vx + c·vy)]

  Sheet scale: presets, calibration and drawing units cadset.js

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• Sheet size and the drawing-to-page transform

  Puts a layout on a page in points, or fits model space to a page about 2000 units on its longest side.

  layout: page = (paper − min) × ptU; model: k = 2000 ÷ extent, page = k·x + (24 − k·minX)

  Sheet scale: presets, calibration and drawing units cadset.js

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• Scale a drawing's sheet starts with

  Feet per page unit for a model sheet or a layout.

  model: f = ft ÷ k; layout: f = (1 ÷ ptU) ÷ s × ft, s = h ÷ viewHeight

  Sheet scale: presets, calibration and drawing units cadset.js

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• Drawing scale taken over by the takeoff

  Gives each sheet of an opened drawing the scale the drawing itself states, when the sheet has none yet.

  Sheet scale: presets, calibration and drawing units takeoffcraft.js

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• Calibration stored on the sheet

  Reads the typed distance, computes the scale and stores it with the two points and the text.

  Sheet scale: presets, calibration and drawing units takeoffcraft.js

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• Typed unit rate

  Reads a rate such as $1,234.5 into whole 1 ÷ 10000 dollars.

  rate4 = round(roundTo(v, 4) × 10000)

  Rates, percentages and extended amounts units.js

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• Typed percentage

  Reads a percentage such as 7.5 % into whole basis points.

  bp = round(roundTo(v, 2) × 100)

  Rates, percentages and extended amounts units.js

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• Quantity × rate to the cent

  The extended amount of a line in whole cents.

  cents = round(qty × rate4 ÷ 100)

  Rates, percentages and extended amounts units.js

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