Calculations

Calculations

• 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

• Pixel to coordinates

  The coordinates of a pixel position in a picture's own system.

  x = g₀ + u·g₁ + v·g₂; y = g₃ + u·g₄ + v·g₅

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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Numbers in a map request

  How the corners of the requested box are written into a WMS or map-server export address.

  |v| ≥ 10,000: v.toFixed(3); otherwise v.toPrecision(12)

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

  • CadCraft
  • BimCraft
  • CivilCraft
  64-bit

• Affine through three points

  The affine transform that carries three source points onto three destination points.

  det = (u₁ − u₀)(v₂ − v₀) − (u₂ − u₀)(v₁ − v₀); a = ((x₁ − x₀)(v₂ − v₀) − (x₂ − x₀)(v₁ − v₀)) ÷ det

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Cells of a warped picture

  The cells a picture is cut into so that each can be drawn with one affine transform.

  split while max |affine(p) − exact(p)| > tol over the fourth corner, the middle and the four edge midpoints

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Tile level for the screen

  Which level of a tile grid is drawn at the current zoom.

  the coarsest level with res ≤ 1.5 × screen resolution

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Rounding of reported numbers

  Rounds a number before it is handed to an assistant or a script.

  num(v) = round(v × 10⁹) ÷ 10⁹; rnd6(v) = num(round(v × 10⁶) ÷ 10⁶)

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Extents, paper and viewport sizes

  The drawing's extents and each sheet's paper and viewport sizes in the drawing's description.

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• An object's fields, length and area

  One object as plain values, with its length and area where it has them.

  length = Σ primLength(primitive); area = entityArea(object).area

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• An object's state for change checks

  A flat set of an object's values that is compared later to see whether the object changed.

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Text height against the allowed list

  Whether a text's height is one of the heights a standards check allows.

  |h − e.h| ≤ 10⁻⁶ × max(1, h)

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Plot scale from text

  The scale number for text such as 1:100, an architectural scale label, or fit.

  a:b → n = b ÷ a

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Sheet a plot prints

  The sheet and its size for a plot asked for by an assistant: a layout, the extents, or a window of the model.

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Size and units of the SVG plot

  The SVG file's size in millimeters and the number of file units per millimeter.

  file units = mm × 10; width = sheet width in mm, rounded to 9 decimals

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Coordinates in the SVG plot

  How points, widths and matrix entries are transformed and written into the SVG file.

  (x′, y′) = M · (x, y, 1); text = round(v × 1000) ÷ 1000; width scale = √|det M|

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Arcs in the SVG plot

  An arc written as a run of straight chords.

  n = max(2, ⌈|sweep| ÷ 2°⌉); pᵢ = center + r·(cos, sin)(a₀ + sweep·i ÷ n)

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Text in the SVG plot

  The matrix, position and size written for a piece of text.

  transform = matrix(a b c d e f), each entry rounded to 3 decimals

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Insertion point of a block placed by an assistant

  The command-line input that places a block at a given point, scale and rotation.

  Values given to assistants and scripts agents.js

  • CadCraft
  64-bit

• Ortho: the point along the nearer axis

  The point a click stands for when ortho is on and there is a base point.

  u = c·Δx + s·Δy; v = −s·Δx + c·Δy; point = base + u·(c, s) if |u| ≥ |v|, else base + v·(−s, c)

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Coordinates, distance and angle in the status line

  The cursor's coordinates and, during a command, the distance and angle from the base point.

  Δ = √(Δx² + Δy²); angle = atan2(Δy, Δx) − UCS angle, normalized to 0…360°

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Length text in the model and on a sheet

  A length written for the status line, the tracking labels and the cursor box.

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Distance and angle beside the cursor

  The live distance or angle shown in the box by the cursor while a command waits for one.

  d = √(Δx² + Δy²); angle = atan2(Δy, Δx), normalized to 0…360°

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Size of the plot area on paper

  The width and height of the chosen area at the chosen scale, shown in the Plot dialog.

  w = (maxX − minX) × s, shown as round(w × 10) ÷ 10

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Number typed in Quick select

  The number a property is compared with in the Quick select dialog.

  Cursor point and status readouts cadcraft.js

  • CadCraft
  64-bit

• Turn sense of an angle constraint

  Whether the angle between two constrained lines is measured counter-clockwise or clockwise after a mirror or a trim.

  sign = +1 if atan2(u × v, u · v) ≥ 0, else −1

  Constraints kept through mirror and trim ckeep.js

  • CadCraft
  64-bit

• Which piece holds the original's ends

  The piece of a trimmed line or arc that still has the original's start, and the one that still has its end.

  start piece: t₀ ≤ 10⁻⁹; end piece: t₁ ≥ 1 − 10⁻⁹

  Constraints kept through mirror and trim ckeep.js

  • CadCraft
  64-bit

• Numbers in prompts and messages

  A number written for a prompt's default or a message in the command history.

  |n| < 10⁻¹⁰ → 0; otherwise n.toFixed(4) without trailing zeros

  Typed points, distances and angles commands.js

  • CadCraft
  64-bit

• Typed distance, number and angle

  The value a prompt takes when a distance, a number or an angle is typed.

  angle = degrees × π ÷ 180 (+ UCS angle for an absolute angle)

  Typed points, distances and angles commands.js

  • CadCraft
  64-bit

• Distance or angle shown by two clicks

  The distance or the angle a prompt takes when it is shown by picking two points.

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

  Typed points, distances and angles commands.js

  • CadCraft
  64-bit

• Typed point

  The point for typed coordinates: absolute, relative to the last point, polar, or a distance along the cursor direction.

  x,y → UCS to world; @x,y → last + R(ucs)·(x, y); d<a → from + d·(cos, sin)(a·π ÷ 180 + UCS angle); d → base + d·(cos, sin)(atan2(cursor − base))

  Typed points, distances and angles commands.js

  • CadCraft
  64-bit

• Calculation on the command line

  The result CAL prints for an expression such as 12'6"+3/4" or 2*450mm.

  raw = value.toPrecision(12)

  Typed points, distances and angles commands.js

  • CadCraft
  64-bit

• Circle through three points

  The center and radius of the circle through three picked points.

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

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Arc through three points

  The center, radius and end angles of the arc from a start point through a second point to an end point.

  center, r from circle3; counter-clockwise when (m − s) × (e − s) > 0; a₀ = atan2(s − center), a₁ = atan2(e − center)

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Arc segment of a polyline

  The bulge of a polyline's arc segment that leaves the last point along the previous segment's direction and ends at the picked point.

  φ = angle(p → q) − tangent at p, wrapped to −π…π; bulge = tan(φ ÷ 2)

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Circle from the two ends of a diameter

  The center and radius of a circle given by two opposite points on it.

  center = (a + b) ÷ 2; r = |b − a| ÷ 2

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Arc from center, start and end

  The radius and end angles of an arc given its center, its start point and a point toward its end.

  r = |start − center|; a₀ = atan2(start − center); a₁ = atan2(end − center)

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit

• Rectangle from two corners

  The closed four-corner polyline between two picked corners.

  Shapes made by the draw commands commands.js

  • CadCraft
  64-bit