Calculations

Calculations

• Numbers written to the PDF and SVG

  Turns a coordinate, width or size into the text that goes into the file.

  text = v.toFixed(4), trailing zeros dropped; |v| < 0.00005 → 0

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Arcs as cubic Béziers

  The Bézier pieces that draw an arc in the PDF.

  n = ⌈|sweep| ÷ 90°⌉; d = sweep ÷ n; k = 4⁄3 · tan(d ÷ 4); handles at r·k along the tangents

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Lines and arcs in the page

  Writes each line and arc of a stroke group as PDF path operators.

  arc start = c + r(cos a, sin a)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Text placement in the PDF

  The text matrix of each line of text: size, width factor, rotation, lean and position.

  size = h × FONT_EM; width = advance ÷ 1000 × size × wf; lean = size × tan(oblique)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Page size in points

  The PDF page's width and height from the sheet's millimeters.

  pt = mm × 72 ÷ 25.4, written with 4 decimals

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Lines and arcs in the SVG

  Writes each line and arc of a stroke group as SVG path data.

  arc pieces of at most 90°: A r r 0 0 flag x y

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Paper size from a printer's paper name

  The size in millimeters of a named paper such as A3, Letter or 24 x 36 in.

  A-series: start 841 × 1189, each step (w, h) → (⌊h ÷ 2⌋, w); inches × 25.4

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Matching a sheet to a printer paper

  Finds the printer paper that is the sheet's size.

  |short − a| ≤ 2 mm and |long − b| ≤ 2 mm

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• PNG size in pixels

  The pixel width and height of a sheet at a resolution, held under what a canvas can make.

  px = round(mm ÷ 25.4 × dpi), at least 1; while too big: dpi = ⌊dpi × 0.9⌋

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Plot extents of the model

  The box round everything the model would plot.

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Zero-size test

  Says whether a line has no length or a circle or arc no radius.

  line: |p₂ − p₁| ≤ 1e-9; circle or arc: r ≤ 1e-9

  Drawing audit: zero-size and duplicate objects problems-cadcraft.js

  • CadCraft
  64-bit

• Duplicate test

  A text key that two objects share only when they are the same shape on the same layer.

  Drawing audit: zero-size and duplicate objects problems-cadcraft.js

  • CadCraft
  Exact

• 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

• Writing a length

  A length as text in one of the five formats.

  engineering: r = round(|v| × 10^p) ÷ 10^p; feet = ⌊r ÷ 12 + 1e-12⌋; architectural: whole inches and a fraction of 1⁄2^p

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Whole part and fraction

  Splits a value into a whole number and a reduced fraction of a power of two.

  n = round(|v| × den); whole = ⌊n ÷ den⌋; numerator = n − whole × den

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Writing decimals

  A number with up to prec decimals, trailing zeros dropped.

  |v| < 0.5 × 10^−prec → 0; else v.toFixed(prec)

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Writing an area

  An area as text: square drawing units, or square inches and square feet in the feet-and-inches formats.

  sq ft = sq in ÷ 144

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Numbers and fractions typed

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

  whole + numerator ÷ denominator

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• One typed length

  Reads a single length such as 12'6" or 6-1/2 into drawing units.

  feet × 12 + inches, with the sign applied last

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Millimeters per drawing unit

  The size of one drawing unit in millimeters, from the drawing's units.

  inch 25.4; foot 304.8; yard 914.4; mile 1 609 344; micron 0.001

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  Exact

• A length typed with a unit

  Converts a number with a unit mark, such as 450mm or 6in, into drawing units.

  mm, cm, m, in, ft: value × mm per typed unit ÷ mm per drawing unit; other marks: parseTo into the drawing's unit

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Writing an angle

  An angle in degrees as the drawing's angle format writes it.

  decimal: deg.toFixed(4)

  Lengths as text: formats and typed lengths units.js

  • CadCraft
  64-bit

• Lengths with units inside an entry

  Replaces 900mm, 2ft or 3'6" in the typed text by the same length in drawing units.

  mm = feet × 304.8 + inches × 25.4, or value × mm per unit; text = (mm ÷ mm per drawing unit) to 12 significant digits

  Values typed in the Properties panel propexpr.js

  • CadCraft
  64-bit

• Degrees, minutes and seconds inside an expression

  Replaces 45°30'15" in an angle expression by decimal degrees.

  degrees + minutes ÷ 60 + seconds ÷ 3600, written with 12 significant digits

  Values typed in the Properties panel propexpr.js

  • CadCraft
  64-bit

• Full-precision text of a value

  The text a number field shows while it is edited.

  v.toPrecision(12)

  Values typed in the Properties panel propexpr.js

  • CadCraft
  64-bit

• Millimeters per drawing unit

  The size of one drawing unit, or of the sheet's paper unit, in millimeters.

  inch 25.4; foot 304.8; yard 914.4; mile 1 609 344

  Values typed in the Properties panel propexpr.js

  • CadCraft
  Exact

• Limits on a typed value

  Checks a result against the field's kind and limits.

  whole number: |v − round(v)| ≤ 1e-9; v ≥ min − 1e-12; v ≤ max + 1e-12

  Values typed in the Properties panel propexpr.js

  • CadCraft
  64-bit

• Angles typed in radians, grads or degrees-minutes-seconds

  Reads 0.5r, 100g or 45°30'15" as degrees.

  radians × (180 ÷ π); grads × 0.9; d + m ÷ 60 + s ÷ 3600

  Values typed in the Properties panel propexpr.js

  • CadCraft
  64-bit

• Length, area and perimeter of an object

  The measures the panel shows for one object.

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Centroid of a closed polyline

  The point a closed polyline is scaled about when its area is typed.

  a = Σ (xᵢyⱼ − xⱼyᵢ); c = Σ (pᵢ + pⱼ)(xᵢyⱼ − xⱼyᵢ) ÷ 3a

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Scaling a polyline to a typed area

  Resizes a closed polyline about its centroid so its area becomes the value typed.

  k = √(A ÷ A₀); p′ = c + (p − c)·k

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Line: length and angle

  Shows a line's length, angle and deltas, and moves its end point when length or angle is typed.

  L = |p₂ − p₁|; new end = p₁ + L·(cos a, sin a)

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Circle: diameter, circumference and area

  Shows a circle's diameter, circumference and area, and sets the radius when one of them is typed.

  d = 2r; C = 2πr; A = πr²; r = d ÷ 2, C ÷ 2π, √(A ÷ π)

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Arc: sweep, length, chord and segment area

  Shows an arc's angles, sweep, length, chord and segment area, and changes the arc when one is typed.

  θ = (a₁ − a₀) mod 2π; length = rθ; chord = 2r·sin(θ ÷ 2); area = r²(θ − sin θ) ÷ 2

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Ellipse: axes and angles

  Shows an ellipse's major and minor axis, ratio, axis angle and start and end angles, and changes them when typed.

  major = |m|; minor = |m| × ratio; ratio = minor ÷ major

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Ellipse parameter and true angle

  Converts between an ellipse's stored parameter and the angle seen from its center.

  angle = atan2(ratio·sin t, cos t); t = atan2(sin a ÷ ratio, cos a)

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Do several objects share a value

  Decides whether a field shows one value or 'varies' for a selection.

  |a − b| ≤ 1e-9 × max(1, |a|, |b|)

  Properties panel: measures and geometry edits propdefs.js

  • CadCraft
  64-bit

• Text held by a number field

  Chooses whether a field holds the formatted text (12'-6") or the plain number.

  formatted text kept when |read(text) − value| ≤ 1e-9 × max(1, |value|)

  Properties panel: measures and geometry edits props.js

  • CadCraft
  64-bit

• Rigid fit from point pairs

  The rotation and shift that best take the picked points of one scan onto their targets, with the error left at each pair.

  R from the top eigenvector (a quaternion) of Horn's 4 × 4 matrix; t = c_dst − R·c_src; rms = √(Σ rᵢ² ÷ n)

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Eigenvalues of a symmetric matrix

  Eigenvalues and eigenvectors of the 3 × 3 and 4 × 4 symmetric matrices the fit and the normals use.

  Jacobi rotations: θ = (a_qq − a_pp) ÷ 2a_pq; t = sign(θ) ÷ (|θ| + √(θ² + 1))

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Combining transforms

  The product of two 4 × 4 transforms, used to add each iteration's step to the running transform.

  o = a · b, each entry a sum of four products

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Inverse of a rigid transform

  The transform that undoes a rotation and shift.

  R⁻¹ = Rᵀ; t′ = −Rᵀ·t

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Transform in shifted coordinates

  Rewrites a transform for coordinates measured from the working origin, and back.

  t_local = t + R·o − o; back: t = t_local − R·o + o

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit