Calculations

Calculations

• Page box and rotation matrix

  The matrix from PDF user space to the page as displayed: origin at the lower-left corner, y up, the page's rotation applied.

  0°: [1 0 0 1 −x₀ −y₀]; 90°: [0 −1 1 0 −y₀ W + x₀]; 180°: [−1 0 0 −1 W + x₀ H + y₀]; 270°: [0 1 −1 0 H + y₀ −x₀]

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

  • CadCraft
  • BimCraft
  64-bit

• Matrix product

  Combines the current matrix with one the page's content applies.

  [a b c d e f] = m · k for 2 × 3 affine matrices

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

  • CadCraft
  • BimCraft
  64-bit

• Path points

  Each move, line and curve point of a path, taken through the current matrix into displayed-page points.

  x′ = a·x + c·y + e; y′ = b·x + d·y + f; quadratic to cubic: c₁ = p + ⅔(q − p), c₂ = x + ⅔(q − x)

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

  • CadCraft
  • BimCraft
  32-bit in part

• 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

• Drawing units per PDF point

  The scale that makes a PDF page its true size in the drawing's units.

  1 pt = 1⁄72 in; inches: 1 ÷ 72; feet: (1 ÷ 72) ÷ 12; millimeters: 25.4 ÷ 72; meters: 0.0254 ÷ 72

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Underlay scale and rotation for saving

  The insertion point, x and y scale and rotation of an underlay as a DWG stores them.

  sx = |(m₀, m₁)|; sy = |(m₂, m₃)| × sign(det); rotation = atan2(m₁, m₀)

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Underlay read from a file

  Builds the underlay block and its placing reference from a file's PDF underlay record.

  PDF underlay placement and PDF import pdfmodel.js

  • CadCraft
  64-bit

• Scale typed when attaching

  The scale an attached page gets: its true size, or the number typed.

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Inverse of a placement

  Takes clip points picked in the drawing back into the page's own points.

  det = a·d − b·c; inverse = [d −b −c a (c·f − d·e) (b·e − a·f)] ÷ det

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Is this curve a circular arc

  Decides whether one cubic curve of the page is an arc of a circle, and if so gives its center, radius and angles.

  circle through B(0), B(½), B(1); arc when | |B(t) − c| − r | ≤ 0.002·r at t = 0.25, 0.75, 0.1, 0.9

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Circle through three points

  Center and radius of the circle through three points.

  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

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Outline points of an imported shape

  The vertices of each imported path in drawing coordinates, with curves replaced by straight pieces.

  B(t) at t = 0.1 … 1.0, then (x, y) → placement matrix

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Stroke width kept on imported objects

  The line width stored on each imported line, arc and polyline.

  lw = round(width × scale × 100) ÷ 100

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Arcs and circles from curves

  Turns a path made only of circular curves into a circle or an arc object.

  same circle when |rᵢ − r₀| < 1e-3·r₀ and |cᵢ − c₀| < 1e-3·r₀; circle when Σ sweep = 2π within 1e-3

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  32-bit in part

• Imported text

  Position, height and angle of each imported text.

  h = PDF height × scale ÷ FONT_EM; angle = (PDF angle + placement rotation) mod 2π

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Scale from two points and a real distance

  Rescales an imported page so that two picked points are a stated distance apart.

  f = real ÷ |p₂ − p₁|

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Scaling an object about a point

  Moves and resizes one imported object by the factor f about the insertion point.

  x′ = aₓ + (x − aₓ)·f; r′ = r·f; h′ = h·f

  PDF underlay placement and PDF import pdfunder.js

  • CadCraft
  64-bit

• Picture corners

  The four corners of a picture in its block's coordinates.

  W = u × w; H = v × h; corners = p, p + W, p + W + H, p + H

  Pictures: size, placement and clip pictures.js

  • CadCraft
  64-bit

• Picture placement and clip for saving

  The insertion point, pixel vectors and clip boundary a DWG image record holds.

  det = u × v; a = (d × v) ÷ det; b = (u × d) ÷ det; pixel = (a − 0.5, h − 0.5 − b)

  Pictures: size, placement and clip pictures.js

  • CadCraft
  64-bit

• Pixel size of a picture in a PDF

  How many pixels a PNG keeps when a plot caps its resolution.

  w₂ = round(maxDpi × mm ÷ 25.4), at least 1

  Pictures: size, placement and clip pictures.js

  • CadCraft
  64-bit

• Pixel size when attaching

  The size of one pixel in drawing units from the width typed for the picture.

  px = W ÷ pixel width

  Pictures: size, placement and clip pictures-cmd.js

  • CadCraft
  64-bit

• Inverse of a placement

  Takes clip points picked in the drawing back into the picture block's coordinates.

  det = a·d − b·c; inverse = [d −b −c a (c·f − d·e) (b·e − a·f)] ÷ det

  Pictures: size, placement and clip pictures-cmd.js

  • CadCraft
  64-bit

• Rescaling a pasted area by two points

  Resizes a pasted picture so two picked points are a stated distance apart.

  f = wanted ÷ |b − a|; sx′ = sx·f; sy′ = sy·f

  Pictures: size, placement and clip pictures-cmd.js

  • CadCraft
  64-bit

• Finding the paper

  The four corners of the sheet of paper in the photo.

  corner = (pixel + 0.5) × k on a copy reduced by the whole factor k; fill = bright pixels inside ÷ outline area

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Perspective matrix from four corners

  The 3 × 3 matrix that takes the straight sheet's corners to the corners found in the photo.

  8 equations in h₁ … h₈ with h₉ = 1, solved by Gauss–Jordan elimination

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Applying the perspective matrix

  Where a point of the straight sheet lies in the photo.

  w = h₇x + h₈y + h₉; x′ = (h₁x + h₂y + h₃) ÷ w; y′ = (h₄x + h₅y + h₆) ÷ w

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Sheet proportions

  The long-to-short side ratio of the straightened sheet.

  w = mean of top and bottom; h = mean of left and right; ratio = max ÷ min

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Drawing units per pixel

  The size of one pixel of the straightened sheet in drawing units.

  units per px = long side (mm) ÷ long side (px) × scale ÷ mm per unit

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Traced arcs and circles

  Turns each traced arc into a circle or an arc object in drawing units.

  r = r(px) × units per px; arc through start, middle and end

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Layout sheet in millimeters

  The matrix that takes a layout and its viewports to millimeters on the sheet.

  toMm = [k 0 0 k −x₀k −y₀k], k = mm per paper unit; view matrix = toMm · viewport matrix

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Model plot scale and placement

  The scale and offset that put a window of the model on a paper size, fitted or at 1:n.

  fit: s = min((W − 10) ÷ w, (H − 10) ÷ h); offset = (W ÷ 2 − cₓs, H ÷ 2 − c_y·s)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Millimeters to points in the page

  The factor that lets the rest of the PDF page be written in millimeters.

  72 ÷ 25.4 = 2.834645669 (9 decimals)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Numbers written to the PDF and SVG

  Turns a coordinate, width or size into the text that goes into the file.

  text = v.toFixed(4), trailing zeros dropped; |v| < 0.00005 → 0

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Arcs as cubic Béziers

  The Bézier pieces that draw an arc in the PDF.

  n = ⌈|sweep| ÷ 90°⌉; d = sweep ÷ n; k = 4⁄3 · tan(d ÷ 4); handles at r·k along the tangents

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Lines and arcs in the page

  Writes each line and arc of a stroke group as PDF path operators.

  arc start = c + r(cos a, sin a)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Text placement in the PDF

  The text matrix of each line of text: size, width factor, rotation, lean and position.

  size = h × FONT_EM; width = advance ÷ 1000 × size × wf; lean = size × tan(oblique)

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Page size in points

  The PDF page's width and height from the sheet's millimeters.

  pt = mm × 72 ÷ 25.4, written with 4 decimals

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Lines and arcs in the SVG

  Writes each line and arc of a stroke group as SVG path data.

  arc pieces of at most 90°: A r r 0 0 flag x y

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Paper size from a printer's paper name

  The size in millimeters of a named paper such as A3, Letter or 24 x 36 in.

  A-series: start 841 × 1189, each step (w, h) → (⌊h ÷ 2⌋, w); inches × 25.4

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Matching a sheet to a printer paper

  Finds the printer paper that is the sheet's size.

  |short − a| ≤ 2 mm and |long − b| ≤ 2 mm

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• PNG size in pixels

  The pixel width and height of a sheet at a resolution, held under what a canvas can make.

  px = round(mm ÷ 25.4 × dpi), at least 1; while too big: dpi = ⌊dpi × 0.9⌋

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Plot extents of the model

  The box round everything the model would plot.

  Plotting: sheet scale, PDF, SVG and PNG output plot.js

  • CadCraft
  64-bit

• Zero-size test

  Says whether a line has no length or a circle or arc no radius.

  line: |p₂ − p₁| ≤ 1e-9; circle or arc: r ≤ 1e-9

  Drawing audit: zero-size and duplicate objects problems-cadcraft.js

  • CadCraft
  64-bit

• Duplicate test

  A text key that two objects share only when they are the same shape on the same layer.

  Drawing audit: zero-size and duplicate objects problems-cadcraft.js

  • CadCraft
  Exact

• 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