Calculations

Calculations

• Text position, height and angle

  Where each run of text starts, how tall it is, its angle and its width, in displayed-page points.

  height = |M·(t₂, t₃)|; angle = atan2 of M·(t₀, t₁); width = |w| × |(m₀, m₁)|

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  64-bit

• Flattening curves to segments

  Turns every line and curve of the page into straight segments for snapping.

  B(t) = u³p₀ + 3u²t·c₁ + 3u·t²·c₂ + t³p₃, u = 1 − t, t = 1⁄8 … 8⁄8

  PDF page vectors and snapping to them pdf-vector.js

  • CadCraft
  • BimCraft
  32-bit in part

• Nearest point on a segment

  The point of a segment closest to the pointer.

  t = ((p − a)·(b − a)) ÷ |b − a|², held to [0, 1]; q = a + t(b − a)

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• Crossing of two segments

  The point where two segments of the page cross, or nothing when they are parallel, apart, or only meet end to end.

  den = r × s; u = ((c − a) × s) ÷ den; v = ((c − a) × r) ÷ den; point = a + u·r

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• Choosing the snap point

  Picks the end, crossing, middle or nearest point within the snap distance of the pointer.

  midpoint = (a + b) ÷ 2; candidate kept when |candidate − p| ≤ tol

  PDF page vectors and snapping to them pdf-snap.js

  • CadCraft
  • BimCraft
  32-bit in part

• 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

• 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

• A level written as text

  Reads a level such as 104.37, +104,37 or EL 104.37 from a text.

  value = Number(whole digits . decimal digits) × sign

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Levels into project units

  A level point's x, y and z in project units.

  x·k, y·k, z·k

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Extent and distance from the origin

  How far the data spans and how far it lies from zero, in meters.

  span = |(maxX − minX, maxY − minY)| × m per unit; far = max |coordinate| × m per unit

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Median

  The middle value of a list of levels.

  odd count: the middle value; even: mean of the two middle values

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Level outliers

  Counts levels that are far from the rest, and levels of exactly 0 that look like missing heights.

  MAD = median(|z − median|) × 1.4826; limit = max(6 × MAD, 5 m); outlier when |z − median| > limit

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Duplicate points

  Counts points that sit on top of another point, with the same or a different level.

  cell = 1 mm; key = (round(x ÷ cell), round(y ÷ cell)); same level when |Δz| ≤ cell

  Survey levels from text and the survey check survey.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Height unit factor

  The factor that turns a file's heights into the project's unit.

  factor = meters per file unit ÷ meters per project unit

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Cell center in the file's coordinates

  The x and y of a raster cell's center.

  x = g₀ + (col + 0.5)·g₁ + (row + 0.5)·g₂; y = g₃ + (col + 0.5)·g₄ + (row + 0.5)·g₅

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Cells inside a box

  The range of columns and rows whose centers can fall inside a picked box.

  (col, row) = G⁻¹·(x − g₀, y − g₃) − 0.5; c₀ = ⌈min − 1e-9⌉, c₁ = ⌊max + 1e-9⌋ + 1

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• A box through a coordinate conversion

  The box round a rectangle after its outline is converted to another coordinate system.

  17 points along each side, converted, then minimum and maximum

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Survey points from a grid

  One survey point per grid node that has a height.

  (x, y) = conversion(cell center); z = file height × unit factor

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Grid surface

  A triangulated surface from the grid: two triangles per cell.

  cell corners a (SW), b (SE), c (NE), d (NW): triangles a–b–c and a–c–d

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• 1-2-5 step

  Rounds a step up to 1, 2 or 5 times a power of ten.

  p = 10^⌊log₁₀ x⌋; m = x ÷ p; 1, 2, 5 or 10 × p

  Elevation grids into survey points and a grid surface terrain.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Turning a direction

  A direction turned by an angle in degrees.

  (x·cos a − y·sin a, x·sin a + y·cos a), each rounded to 6 decimals

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Turn between two walls

  The signed angle from one wall direction to the next, in degrees.

  atan2(u₀ × u₁, u₀ · u₁) × 180 ÷ π

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• End of a wall typed as length and angle

  The end point of a wall from its start, length and angle.

  (x + L·cos a, y + L·sin a), rounded to 6 decimals

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• End of an arc typed as radius and sweep

  The end point of an arc wall from its radius, sweep and the heading of its chord.

  chord = 2·R·sin(|sweep| ÷ 2); end = start + chord at the heading

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Sagitta

  How far the middle of a straight back of width w stands off a circle of radius r.

  s = r − √(r² − w² ÷ 4)

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Arc through three points

  The radius, center and signed sweep of the arc picked as start, middle and end.

  d = 2·(aₓ(bᵧ − cᵧ) + bₓ(cᵧ − aᵧ) + cₓ(aᵧ − bᵧ)); uₓ = Σ |p|²·(Δy) ÷ d; uᵧ = Σ |p|²·(Δx) ÷ d; r = |a − u|

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Arc from two ends and a sweep

  The radius, center, chord, arc length and sagitta of an arc wall.

  r = chord ÷ (2·sin(θ ÷ 2)); center = middle ± normal × r·cos(θ ÷ 2); arc length = r·θ; sagitta = r·(1 − cos(θ ÷ 2))

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Points along an arc wall

  An arc wall as n straight pieces.

  φᵢ = a₀ ± |θ|·i ÷ n; point = center + r·(cos φ, sin φ), rounded to 6 decimals

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Back line and half angle of each cabinet on an arc

  The radius of each cabinet's back line and the half angle it takes up.

  bay: H = √(R² − w² ÷ 4); bow or curved front: H = R; half angle = atan2(w ÷ 2, H), or w ÷ (2R) for a curved front; front radius = H − d

  Cabinets on angled and curved walls addons/cabinetcraft/lib/angled.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit