Calculations

Calculations

• Scale and turn from two points

  The scale, the angle and the point mapping that carry two points on the picture onto the two points where they belong.

  scale = |b₂ − b₁| ÷ |a₂ − a₁|; angle = atan2(b₂ − b₁) − atan2(a₂ − a₁); map(p) = b₁ + scale · R(angle) · (p − a₁)

  Size of a copied area areaclip.js

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  64-bit

• Box round a turned picture

  The width and height of the upright box round a picture turned by an angle.

  w′ = w·|cos θ| + h·|sin θ|; h′ = w·|sin θ| + h·|cos θ|

  Size of a copied area areaclip.js

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  64-bit

• Outline of an elliptical area

  The points of the ellipse that fits a box, used as the outline of a round copied area.

  pᵢ = (cx + rx·cos(2π·i ÷ n), cy + ry·sin(2π·i ÷ n)), n = 96

  Size of a copied area areaclip.js

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  64-bit

• abs, round, ceil and floor

  The four functions an expression may call.

  round(x) = floor(x + 0.5) for x ≥ 0, −floor(−x + 0.5) for x < 0; ceil(x) = ⌈x − 1e-9·max(1, |x|)⌉; floor(x) = ⌊x + 1e-9·max(1, |x|)⌋

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Adding and subtracting

  Sums and differences of lengths, left to right.

  Arithmetic on typed lengths lencalc.js

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  64-bit

• Multiplying and dividing

  Products and quotients of a length and plain numbers, left to right.

  Arithmetic on typed lengths lencalc.js

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  64-bit

• The answer

  The final value and whether it is a length or a plain number.

  Arithmetic on typed lengths lencalc.js

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• Unit sizes

  Builds each unit's row: its fraction of the base unit and the factor used for quick conversion.

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

  Unit table: reading and writing values unit-table.js Every app
  Exact

• Numbers and fractions

  Reads 6, 6.5, 6 1/2, 6-1/2 or 1/2 as a number.

  whole + numerator ÷ denominator

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Feet and inches

  Reads 12'6", 12'-6 1/2" or 6" as inches.

  inches = feet × 12 + inches

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Degrees, minutes and seconds

  Reads 45°30'15" or 45d30m15s as degrees.

  degrees + minutes ÷ 60 + seconds ÷ 3600

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Slopes

  Reads 2%, 3H:1V, 1:3 or 6/12 as a grade (rise over run).

  percent ÷ 100; H:V → V ÷ H; rise:run → rise ÷ run

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing feet and inches

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

  n = round(|inches| × den); feet = ⌊n ÷ 12den⌋; inches = ⌊rest ÷ den⌋; fraction = rest ÷ den, reduced

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing degrees, minutes and seconds

  Writes an angle as degrees, minutes and seconds.

  t = round(|degrees| × 3600 × 10^dec); split by 3600 and 60

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Writing decimals

  Writes a number with a fixed count of decimals.

  text = v.toFixed(dec)

  Unit table: reading and writing values unit-table.js Every app
  64-bit

• Guard on every written measure

  Decides what is written when a value is missing or not a number.

  DWG object values written and read dwg/objects-write.js

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• Arc angles brought into one turn

  Brings an angle into the range from 0 up to 2π before it is written.

  r = a mod 2π; r + 2π when r < 0

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Arc

  Writes an arc's center, radius and two angles.

  DWG object values written and read dwg/objects-write.js

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• Line

  Writes a line's two end points.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Polyline vertices

  Writes the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Vertex widths

  Writes the start and end width at a polyline vertex.

  equal widths: one value, −|width|

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Linetype pattern length

  The total length of a linetype's dash pattern.

  length = Σ |element length|

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Text direction

  The x-axis direction written for a block of text.

  x axis = (cos θ, sin θ, 0)

  DWG object values written and read dwg/objects-write.js

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• Block scale

  Writes the three scale factors of a block reference in the format's compact form.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Single-line text

  Writes a text's insertion point, height, rotation, width factor and oblique angle.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Extension record values

  Writes the numbers kept in an extension record.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Block content matrix of a multileader

  The 4 × 4 matrix written for a multileader's block content when the model gives none.

  M = translate(l) · rotate_z(θ) · scale(s)

  DWG object values written and read dwg/mleader-write.js

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  64-bit

• Line read

  Reads a line's two end points.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Polyline vertices read

  Reads the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• North direction of older geolocation data

  Turns the north angle of a version 1 geolocation object into a direction vector.

  north = (sin a, cos a)

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Lineweight to the file's lineweight index

  Picks the stored lineweight that is nearest to the one the layer or object has.

  index = the i with the smallest |table[i] − lineweight|

  Values changed when a DWG is written dwg/write.js

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  Exact

• Ellipse with a ratio above 1

  Rewrites an ellipse whose minor axis is longer than its major axis so the file holds a ratio of 1 or less.

  major′ = (−m_y, m_x) × ratio; ratio′ = 1 ÷ ratio; start and end − π ÷ 2

  Values changed when a DWG is written dwg/write.js

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  64-bit

• Ray and construction line direction

  Writes the direction of a ray or construction line as a unit vector.

  d′ = d ÷ |d|

  Values changed when a DWG is written dwg/write.js

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• Drawing extents

  The lower left and upper right corner of model space written to the file header.

  min and max of x, y, z over the defining points

  Values changed when a DWG is written dwg/write.js

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• Default angles of a light

  The hotspot and falloff angles written for a light that has none of its own.

  hotspot = π ÷ 4; falloff = π ÷ 4 + 0.0872664626

  Values changed when a DWG is written dwg/solid-write.js

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• Standard gravity

  The acceleration that turns a member's or plate's mass into its own weight.

  g = 9.80665 m/s²

  Member axes and section properties frame/frame-model.js

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  Exact

• Member length and local axes

  The length of a member and the three unit vectors its forces and section are measured in.

  L = |P_j − P_i|; x = (P_j − P_i) ÷ L; z = unit(Z − (Z·x) x); y = z × x; then y and z turned by the roll

  Member axes and section properties frame/frame-model.js

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• Zero-length member test

  Removes a member whose two ends are at the same place.

  |P_j − P_i| < 1e-6 mm

  Member axes and section properties frame/frame-model.js

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  64-bit

• Area and second moments of a polygon section

  Area, centroid, second moments about the centroid, principal moments and their angle for a section outline with holes.

  A = ½ Σ cᵢ; Iyy = (1 ÷ 12) Σ (v₀² + v₀v₁ + v₁²) cᵢ − A c_v²; cᵢ = u₀v₁ − u₁v₀; I₁,₂ = (Iyy + Izz) ÷ 2 ± √(((Iyy − Izz) ÷ 2)² + Iyz²)

  Member axes and section properties frame/frame-model.js

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• Torsion constant

  The Saint-Venant torsion constant J of a catalog section or of an outline.

  I with fillets: J = ⅔ B T³ + ⅓ (D − 2T) t³ + 2 α D₁⁴ − 0.420 T⁴; open: Σ b t³ ÷ 3; tube: 4 Am² t ÷ p; round tube: π (D⁴ − (D − 2t)⁴) ÷ 32

  Member axes and section properties frame/frame-model.js

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• Section properties for the analysis

  Chooses the area, the two second moments, J and the roll a drawn section is analyzed with.

  A = the profile's exact area if given, else the polygon's; Iy, Iz = Iyy, Izz, or I₁, I₂ with roll = the principal angle

  Member axes and section properties frame/frame-model.js

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• A load split onto the member's axes

  Turns one point or distributed load in a global or local direction into pieces along the member's own three axes.

  piece = load × (direction · local axis); a, b clamped to [0, L]

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Inverse of the released block

  Inverts the small stiffness block of a member's released end actions.

  Gauss–Jordan with partial pivoting

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member stiffness with released ends

  Removes pinned or sliding end actions from a member's 12 × 12 stiffness.

  k_c = k_kk − k_kr · k_rr⁻¹ · k_rk

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Fixed-end loads with released ends

  Moves the load a released end cannot carry onto the member's other end actions.

  p_c = p_k − k_kr · k_rr⁻¹ · p_r

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member stiffness in global axes

  Turns a member's stiffness from its own axes to the model's.

  k_g = Tᵀ · k · T, by 3 × 3 blocks

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Directions nothing stiffens

  Finds translations and rotation directions at a node that no member resists, so they can be held instead of stopping the run.

  translation: diagonal + spring ≤ 1e-14 × largest diagonal; rotation: eigenvalue ≤ 1e-10 × largest rotational diagonal

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Eigenvalues of a symmetric 3 × 3 matrix

  The principal stiffnesses and directions of the rotation block at a node.

  cyclic Jacobi rotations; t = sign(θ) ÷ (|θ| + √(θ² + 1))

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Loads of a case or combination

  Adds up node forces and member loads of every case in a set, each times its factor, including the members' own weight.

  own weight w = −factor × f × ρ × 1e-9 × A × g (N/mm along −Z)

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member end forces and reactions

  The six forces at each end of every member and the forces at the supports, from the solved displacements.

  f = k_c · u_local − p_c; reaction = Σ Tᵀ f − loads applied at the node

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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  64-bit