Calculations

Calculations

• 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
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• 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
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• 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

• Is a matrix a rigid transform

  Checks that a matrix is a pure rotation and shift before it is accepted.

  |rᵢ·rⱼ − δᵢⱼ| ≤ tol; |det − 1| ≤ tol; last row 0 0 0 1 within tol

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Rotation about an axis

  The 4 × 4 matrix for a turn by an angle about an axis through a center.

  R = c·I + s·[axis]ₓ + (1 − c)·axis·axisᵀ; t = center − R·center

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Angle and shift of a transform in words

  The rotation angle in degrees and the length of the shift that a result is reported with.

  angle = acos((trace − 1) ÷ 2); shift = |t|

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Working points of a scan

  The thinned copy of a scan's points, measured from the session origin, that registration works on.

  work = pos + cloud origin − session origin, every stride-th point

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Nearest point in the fixed scans

  For a moved point, the closest point of the fixed scans within the match distance.

  d² = Δx² + Δy² + Δz², smallest over the grid cells searched ring by ring

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Surface normals

  The direction square to the surface at each fixed point, from its 10 nearest neighbors.

  normal = eigenvector of the smallest eigenvalue of the neighbors' covariance

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Matching and the error of a step

  Pairs each moving point with its nearest fixed point and reports the root-mean-square distance, and the distance along the normals.

  rms = √(Σ d² ÷ matched); plane rms = √(Σ ((p − q)·n)² ÷ matched)

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Point-to-plane step

  The small turn and shift that reduce the distances along the normals in one iteration.

  minimize Σ ((ω × p + t + p − q)·n)²: 6 × 6 normal equations for (ω, t); R = rotation by |ω| about ω

  Scan registration register.js

  • CadCraft
  • BimCraft
  32-bit in part

• Solving the 6 × 6 system

  Solves the linear equations of the point-to-plane step.

  Gaussian elimination with partial pivoting, then back substitution

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• When the iteration stops

  Decides when a phase has settled and narrows the match distance for the next step.

  stop when |previous − error| ≤ tol × max(error, 1e-12) or error < 1e-12; cut = min(start, max(2.5 × rms, 1e-6 × diagonal))

  Scan registration register.js

  • CadCraft
  • BimCraft
  64-bit

• Pixels to drawing coordinates and back

  Maps a pixel of a placed picture to the drawing, a drawing point back to a pixel, and gives the size of one pixel.

  block = p + u·px + v·(h − py); drawing = M·block; pixel size = |M·u|

  Scanned drawings: size, scale, align, crop and trace scan.js

  • CadCraft
  64-bit

• Arc through three points

  The counterclockwise arc through a start, a middle and an end point.

  d = 2(aₓ(b_y − c_y) + bₓ(c_y − a_y) + cₓ(a_y − b_y)); center from |a|², |b|², |c|² ÷ d; r = |a − center|

  Scanned drawings: size, scale, align, crop and trace scan.js

  • CadCraft
  64-bit

• Traced lines and arcs as drawing objects

  Turns a trace result in pixels into lines, arcs and circles in drawing coordinates.

  circle r = r(px) × pixel size; arc through start, middle and end

  Scanned drawings: size, scale, align, crop and trace scan.js

  • CadCraft
  64-bit

• First size of an attached scan

  The size of one pixel when a scan is attached.

  per pixel = 25.4 ÷ dpi when the file states it and the drawing has units, else 1; typed width ÷ pixel width

  Scanned drawings: size, scale, align, crop and trace scan-cmd.js

  • CadCraft
  64-bit