Calculations

Calculations

• Volume of each solid

  The volume listed for every closed solid.

  volume = |V of the definition| × |det of the placement's 3 × 3 part|

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Changing the drawing's units

  Scales the 3D model so it stays the same size on the drawing when the units change.

  ratio = m per new unit ÷ m per old unit; every vertex and placement offset × ratio

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• 3D faces written to a file

  The model's faces as 3D faces in drawing units for DWG and DXF.

  file point = placed vertex in meters ÷ k, k = meters per drawing unit

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• 3D faces inside blocks

  Places the 3D faces a block holds once for every reference of that block.

  (x′, y′) = M·(x, y); z′ = z

  3D solids, faces and units in a drawing model3d.js

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  64-bit

• 3D faces read from a file

  A file's loose 3D faces brought into the model in meters.

  corner in meters = file corner × k

  3D solids, faces and units in a drawing model3d.js

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  64-bit

• Axes of an object's own coordinate system

  The X, Y and Z axes of the plane an object was drawn on, from its extrusion direction.

  Z = N ÷ |N|; X = (Wy × Z or Wz × Z) ÷ length; Y = Z × X

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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• Is the plane the drawing's own

  Whether an extrusion direction is straight up, so that nothing needs mapping.

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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• Plane to plan matrix

  The transform from an object's own (x, y) at its elevation to the plan.

  M = [X.x, X.y, Y.x, Y.y, Z.x·elevation, Z.y·elevation]

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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  64-bit

• Mapped circle, arc or ellipse

  The circle, arc or ellipse that has exactly the points of the original after the map.

  t* = ½·atan2(2 u·v, |u|² − |v|²); U = u·cos t* + v·sin t*; V = −u·sin t* + v·cos t*; a = |U|, b = |V|

  Objects on a tilted or flipped plane mapped to the plan ocs.js

  • CadCraft
  64-bit

• Polyline arcs as points

  A polyline's arc segments as points, for a map that does not keep shapes.

  k = max(4, ⌈64 × |sweep| ÷ 2π⌉); point = c + r·(cos a, sin a)

  Objects on a tilted or flipped plane mapped to the plan ocs.js

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  64-bit

• Mapping each kind of object

  A drawing object in its own coordinate system put through the plane's matrix.

  text: rotation = atan2(M·d); height × |M·d × M·up| ÷ |M·d|; block: scales and rotation read from the mapped axes

  Objects on a tilted or flipped plane mapped to the plan ocs.js

  • CadCraft
  64-bit

• Automatic constraints

  Which ends are joined and which lines are made horizontal or vertical.

  coincident when |a − b| ≤ distance; horizontal when |sin α| ≤ sin(tolerance); vertical when |cos α| ≤ sin(tolerance)

  Automatic constraints and dimension conversion pcommands.js

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• Dimension to constraint

  The constraint a plain dimension stands for, when its points sit on constrainable geometry.

  tol = 1e-7 × max(1, largest |x| or |y|)

  Automatic constraints and dimension conversion pcommands.js

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  64-bit

• Underlay shapes and text

  The underlay's paths as plotted segments and fills, and its text.

  B(t) = (1 − t)³·P₀ + 3(1 − t)²t·P₁ + 3(1 − t)t²·P₂ + t³·P₃, t = s ÷ 8; line width = max(0.05, w × √|det|); text height = h × √|det| ÷ FONT_EM

  PDF underlay in a plot pdf-emit.js

  • CadCraft
  64-bit

• 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

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• 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

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  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

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• 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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  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

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  64-bit