Calculations

Calculations

• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Same object within a tolerance

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

  Drawing autocomplete autocomplete.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Merging near-equal positions

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

  Automatic dimensions autodim.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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
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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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