Calculations

Calculations

• Faceted faces

  Triangles for a flat face given as loops of points.

  n = Σ ((yᵢ − yⱼ)(zᵢ + zⱼ), (zᵢ − zⱼ)(xᵢ + xⱼ), (xᵢ − xⱼ)(yᵢ + yⱼ))

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Rounding of spoken numbers and angles

  Turns a number or an angle into the short text that is spoken.

  num(v) = round(v × 100) ÷ 100; deg(θ) = num(round(normAngle(θ) × (180 ÷ π) × 100) ÷ 100)

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Direction as one of eight words

  Which of the eight directions (right, up-right, up and so on) leads from one point to another.

  index = round(atan2(Δy, Δx) ÷ (π ÷ 4)) mod 8

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• A move in words

  The distance, angle and the two components of a move from one point to another.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx)

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Length along an object

  The length of a line, circle, arc or polyline that is read out with the object.

  line: √(Δx² + Δy²); circle: 2πr; arc: r × ((a₁ − a₀) mod 2π); polyline: Σ |pᵢ₊₁ − pᵢ|

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Extents of the visible objects

  The width, height and corners of the box round everything visible, said with the object counts.

  w = maxX − minX; h = maxY − minY

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Objects within a radius

  The objects within a given distance of a point, nearest first.

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Distance and side of each nearby object

  For each object near the cursor: whether the cursor is on it, or how far away it is and on which side.

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Measure between two points

  The length, angle, Δx and Δy said back after a command takes a point.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx)

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Snap points near the cursor

  The snap points within a radius of a point, nearest first, for stepping from one to the next with a key.

  d = √((sx − px)² + (sy − py)²)

  Spoken lengths, directions and extents access.js

  • CadCraft
  64-bit

• Transform from point pairs

  The move, turn, scale and mirror that carry one to three source points onto their destination points.

  turn = atan2(d₂ − d₁) − atan2(s₂ − s₁); k = |d₁d₂| ÷ |s₁s₂|; M = T(d₁) · R(turn) · S(k) · T(−s₁)

  Align and distribute align.js

  • CadCraft
  64-bit

• Bounding boxes in the coordinate system in force

  Each selected object's bounding box, measured along the axes of the user coordinate system when one is set.

  with a UCS: box of R(−angle) · T(−origin) applied to the object

  Align and distribute align.js

  • CadCraft
  64-bit

• Edge or middle of a box

  The left, center, right, top, middle or bottom coordinate of a bounding box.

  center = (minX + maxX) ÷ 2; middle = (minY + maxY) ÷ 2

  Align and distribute align.js

  • CadCraft
  64-bit

• Moves that line objects up

  For each object, the vector that brings its chosen edge or middle onto the reference.

  d = ref − edge(box)

  Align and distribute align.js

  • CadCraft
  64-bit

• Moves that space objects evenly

  For each object between the first and the last, the vector that makes the middles or the gaps equal.

  centers: step = (mid_last − mid_first) ÷ (n − 1); gaps: gap = (span − Σ sizes) ÷ (n − 1)

  Align and distribute align.js

  • CadCraft
  64-bit

• Degrees, minutes, seconds to decimal degrees

  The decimal angle for text such as 45°23'30", 45d23'30" or, inside a bearing, 45-23-30.

  deg = d + m ÷ 60 + s ÷ 3600

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Bearing or DMS text to an angle

  The angle in degrees, counter-clockwise from east, for a typed bearing such as N45°23'30"E or a plain DMS angle.

  N…E: 90 − off; N…W: 90 + off; S…E: 270 + off; S…W: 270 − off

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Decimal degrees to degrees, minutes and seconds

  The degrees, minutes and seconds text for an angle at the chosen precision.

  n = round(deg × unit + 10⁻⁹); d = ⌊n ÷ unit⌋; m = ⌊rest ÷ (unit ÷ 60)⌋; s = (rest − m × unit ÷ 60) ÷ 10^secDec

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Angle to a surveyor's bearing

  Bearing text such as N45°23'30"E for an angle in degrees.

  a = deg mod 360; offset = |90 − a| (north half) or |270 − a| (south half)

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Angle in the drawing's angle format

  An angle written as decimal degrees, degrees-minutes-seconds, grads, radians or a bearing.

  grads = a × 10 ÷ 9; radians = a × π ÷ 180

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Bearing of a vector

  The bearing text for a direction given by its x and y components.

  angle = atan2(dy, dx) × 180 ÷ π

  Bearings and degrees-minutes-seconds angles.js

  • CadCraft
  64-bit

• Size in millimeters

  The width and height of the copied area, on paper or at the drawing's real size.

  paper: mm = pixels × ptPerPx × (25.4 ÷ 72); real: mm = pixels × mmPerPx

  Size of a copied area areaclip.js

  • CadCraft
  • BimCraft
  • MarkupCraft
  • DetailCraft
  64-bit

• Render size in pixels

  The pixel width and height for an area given in points at a resolution, lowered so neither side passes the cap.

  d = dpi × min(1, maxPx ÷ (longer side × dpi ÷ 72)); w = round(wPt × d ÷ 72); h = round(hPt × d ÷ 72)

  Size of a copied area areaclip.js

  • CadCraft
  • BimCraft
  • MarkupCraft
  • DetailCraft
  64-bit

• 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

  • CadCraft
  • BimCraft
  • MarkupCraft
  • DetailCraft
  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

  • CadCraft
  • BimCraft
  • MarkupCraft
  • DetailCraft
  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

  • CadCraft
  • BimCraft
  • MarkupCraft
  • DetailCraft
  64-bit

• Unit factor and tolerance

  The factor between drawing units and the file's units, and the tolerance the comparison uses.

  factor = mm per unit ÷ 1000 (meters), ÷ 304.8 (feet), or 1; default tol = 10 ÷ mm per unit

  As-built points compared with the drawing asbuilt.js

  • CadCraft
  64-bit

• Comparison and conversion back to the drawing

  Compares the two point lists in the file's units and brings as-built points back into drawing coordinates.

  compare with tol × factor; drawn point = file point ÷ factor

  As-built points compared with the drawing asbuilt.js

  • CadCraft
  64-bit

• Marks and labels in the drawing

  A point at each as-built shot, a line from the design point when it is out of tolerance, and a label with the distance.

  label = (dh ÷ factor).toFixed(min(4, max(0, ⌈−log₁₀ tol⌉))); text height = max(1.5 × tol, 0.5 × current text height)

  As-built points compared with the drawing asbuilt.js

  • CadCraft
  64-bit

• Same object within a tolerance

  Whether two objects are the same shape on the same layer, apart from their ids.

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Next copy of a repeat

  The next copy when the last one to four objects were copied at one step twice or more.

  step = anchor(B₀) − anchor(A₀); tol = max(10⁻⁶, 0.004 × |step|); next = last group moved by step

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Next period of a stair

  The next riser and tread of an open polyline whose steps repeat.

  dⱼ = pⱼ − pⱼ₋₁; repeats when |dⱼ − dⱼ₋ₚ| ≤ max(10⁻⁶, 0.004 × longest step); new vertex = last vertex + dⱼ

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Chain of segments through the newest line

  The run of connected line and polyline segments that the newest line or polyline belongs to, as an ordered list of vertices.

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Do two segments cross

  Whether the closing side would cross another side of the figure.

  cross(o, a, b) = (a.x − o.x)(b.y − o.y) − (a.y − o.y)(b.x − o.x); cross when d₁·d₂ < 0 and d₃·d₄ < 0

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Closing sides of a nearly closed figure

  The two missing sides of a rectangle drawn as two sides, or the closing side of a chain of three or more.

  rectangle: p₄ = v₀ + (v₂ − v₁), offered when |u·w| ÷ (|u||w|) ≤ 0.01; chain: closing side v_last → v₀

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Next point of a running line

  The point offered after the points picked so far in LINE or PLINE: the next stair step, a rectangle's fourth corner, the first point again, or one more equal step.

  tol = max(10⁻⁶, 0.002 × longest step); square when |u·v| ≤ 0.002 × |u||v|; next = last point + a stored step

  Drawing autocomplete autocomplete.js

  • CadCraft
  64-bit

• Merging near-equal positions

  Sorts edge positions and merges those that are within a tolerance of each other.

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Level and plumb edges, circles and points

  Collects the positions to dimension from the chosen objects and the size of the box round them.

  diagonal = √(w² + h²); plumb when |Δx| ≤ tol; level when |Δy| ≤ tol

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Is the dimension already there

  A key that is equal for two dimensions of one kind with the same end points.

  key = kind : round(x₁ ÷ g), round(y₁ ÷ g), round(x₂ ÷ g), round(y₂ ÷ g), g = 20 × tol

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Chains and overall dimensions

  The dimensions strung between neighboring edge positions along each chosen side, and one overall dimension outside them.

  minLen = max(0.003 × diagonal, 10 × tol); chain line at box edge ∓ 6 × text height; overall line 5 × text height further out

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Opening widths

  The width of a gap between collinear lines on one layer, taken as a door or window opening.

  w = next.lo − this.hi; kept when minLen ≤ w ≤ 0.35 × diagonal and w ≤ 0.8 × the longer neighbor

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Hole diameters

  One diameter dimension for each size of circle.

  p₁ = center + r·(cos 45°, sin 45°); text at center + (r + 3 × text height)·(cos 45°, sin 45°)

  Automatic dimensions autodim.js

  • CadCraft
  64-bit

• Longitude and latitude to web Mercator

  The web Mercator x and y in meters for a longitude and latitude in degrees.

  x = R·λ·π ÷ 180; y = R·ln(tan(π ÷ 4 + φ·π ÷ 360)), R = 6378137

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Web Mercator to longitude and latitude

  The longitude and latitude in degrees for a web Mercator x and y in meters.

  λ = x ÷ R × 180 ÷ π; φ = (2·atan(e^(y ÷ R)) − π ÷ 2) × 180 ÷ π

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Standard tile grid

  The origin and the resolution of each zoom level of the usual 256-pixel web Mercator tiles.

  origin = (−πR, πR); res(z) = 2πR ÷ 256 ÷ 2^z, z = 0…22

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Tile that holds a place

  The tile column and row at a zoom level for a longitude and latitude.

  col = ⌊(x + πR) ÷ (res × 256)⌋; row = ⌊(πR − y) ÷ (res × 256)⌋

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Tiles covering a box

  The tiles of a grid that cover a box, each with its bounds in the grid's units.

  bounds = [ox + c·tw, oy − (r + 1)·th, ox + (c + 1)·tw, oy − r·th], tw = res × tile width

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Meters per unit of a coordinate system

  The length in meters of one degree, foot, US survey foot or meter, for turning a WMTS scale into a resolution.

  degree: 2π × 6378137 ÷ 360; foot: 0.3048; US survey foot: 1200 ÷ 3937; otherwise 1

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Tile grid of a WMTS tile matrix set

  The resolution and top-left corner of each level of a WMTS tile matrix set.

  res = scale denominator × 0.00028 ÷ meters per unit

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• World file to a pixel-corner transform

  The transform from pixel corners to coordinates for a picture, from the six numbers of its world file.

  x₀ = C − A ÷ 2 − B ÷ 2; y₀ = F − D ÷ 2 − E ÷ 2

  Map layers: web Mercator, tile grids and world files basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit