Calculations

Calculations

• Hatches as solids or outlines (R12)

  Writes a filled triangle or quadrilateral as a SOLID and any other hatch as closed polylines of its rings.

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Text (R12)

  A text's position, height, rotation, width factor and oblique angle.

  rotation° = rot × 180 ÷ π; oblique° = obl × 180 ÷ π

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Multi-line text as single lines (R12)

  The position of each line of a multi-line text, written as separate single-line texts.

  p = T(x, y) × R(rot) × (anchor x of the text box, baseline of line k)

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Construction lines as segments (R12)

  Writes an infinite line or ray as a line across the drawing's extents.

  a = p + d × t₀; b = p + d × t₁

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Extents of what was written (R12)

  The smallest and largest x and y of the model-space objects written.

  min and max over the written points

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Linetype pattern (R12)

  A linetype's total pattern length and its dashes and gaps.

  length = Σ |elementᵢ|

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Center mark size (R12)

  The center mark size written with each dimension style.

  DIMCEN = arrow size ÷ 2 (2.5 when no arrow size is set)

  DXF writing: numbers as text dxf.js

  • CadCraft
  64-bit

• Number text (2000)

  The text every real number is written as in a DXF 2000 file.

  not finite → 0.0; |n| < 1e-12 → 0.0; otherwise n rounded to 15 significant digits

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Radians to degrees (2000)

  Converts an angle to the degrees DXF stores.

  degrees = (r × 180) ÷ π

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Point (2000)

  Writes a point as three groups.

  x → f(x); y → f(y); z → f(z); a missing coordinate → 0.0

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Arc (2000)

  An arc's center, radius and its angles in degrees.

  start° = (a₀ × 180) ÷ π; end° = (a₁ × 180) ÷ π

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Ellipse (2000)

  An ellipse's center, major axis, axis ratio and start and end parameters, with the ratio brought to 1 or less.

  ratio > 1: major′ = (−m_y × ratio, m_x × ratio); ratio′ = 1 ÷ ratio; parameters − π ÷ 2

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Hatch pattern lines (2000)

  A pattern hatch's angle, scale and each pattern line's angle, base, offset and dashes.

  angles° = (radians × 180) ÷ π

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Spline (2000)

  A spline's degree, knots, weights, control points and fit points.

  knot, control point and fit point tolerances written as 1E-10

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Extended data values (2000)

  Writes the extra values attached to an object: text, reals and whole numbers.

  code 1040: f(value); code 1070: value | 0

  DXF writing: numbers as text dxf2000.js

  • CadCraft
  64-bit

• Point in absolute coordinates

  One scan point, from Cartesian or spherical fields, moved by the scan's pose.

  spherical: x = r cos e cos a, y = r cos e sin a, z = r sin e; then p′ = R × p + T

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Rotation matrix from the pose quaternion

  The 3 × 3 rotation of a scan from its quaternion (w, x, y, z).

  q ÷ |q|; R = [1 − 2(y² + z²), 2(xy − zw), 2(xz + yw); 2(xy + zw), 1 − 2(x² + z²), 2(yz − xw); 2(xz − yw), 2(yz + xw), 1 − 2(x² + y²)]

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Field values of a packet

  Decodes one field of a data packet into numbers: 64-bit floats, 32-bit floats or scaled integers.

  float: as stored; integer: (raw + minimum) × scale + offset

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Bit-packed integer

  Reads an unsigned integer of up to 64 bits from a bit position in a packet.

  value = Σ chunkᵢ × 2^(32 i), chunks of up to 32 bits

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Bits from the stream

  Extracts up to 32 bits starting at any bit of a byte buffer, least significant bit first.

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Field description from the XML

  How many bits an integer field uses, and its minimum, scale and offset.

  bits = smallest b with 2ᵇ > maximum − minimum (at most 64)

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Thinning to the point budget

  How many points are kept and at what rate when a file has more points than the budget.

  cap = max(1, min(listed, ⌊budget⌋)); rate = min(1, cap ÷ listed); keep a point each time the running sum of rate reaches 1

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Points from the library's text

  Parses the x y z lines the web-e57 library returns and keeps the bounds and the thinned points.

  x, y, z = Number of each field of a line

  E57 point cloud reading e57.js

  • CadCraft
  • BimCraft
  64-bit

• Solve for a parameter

  The value of the named parameter for which the expression equals the target.

  secant: c = b − f(b) × (b − a) ÷ (f(b) − f(a)); accept when |f| ≤ 1e-9 × max(1, |target|, |x₀|)

  Solving an expression for one parameter expr.js

  • CadCraft
  • BimCraft
  • PartCraft
  64-bit

• Rounding of the answer

  Replaces the answer by its 13-digit rounding when the two are practically equal.

  r = x to 13 significant digits; use r when |r − x| ≤ 1e-12 × max(1, |x|)

  Solving an expression for one parameter expr.js

  • CadCraft
  • BimCraft
  • PartCraft
  64-bit

• View frames

  The right, up and toward-the-viewer directions of the seven standard views.

  iso: u = (1, 1, 0) ÷ √2; v = (−1, 1, 2) ÷ √6; w = (1, −1, 1) ÷ √3

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Edges and triangles in the view's frame

  Every vertex of the model as (right, up, depth) in the view, and which soft edges are outlines from this side.

  q = (p · u, p · v, p · w); soft edge kept when its two faces face opposite ways

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Working tolerance

  The tolerance every later step uses, from the size of the projected model.

  diag = √(Δu² + Δv²); tol = diag × 1e-6

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Is a point hidden

  Says whether a point of an edge lies behind some triangle of the model.

  l₁, l₂ from signed areas ÷ triangle area; l₃ = 1 − l₁ − l₂; hidden when all ≥ 1e-9 and l₁ z_a + l₂ z_b + l₃ z_c > depth + tol

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Visible and hidden runs of an edge

  Splits an edge into stretches that are visible and stretches that are hidden.

  n = clamp(⌈L ÷ (diag ÷ 300)⌉, 1, 400) samples; each change bisected until (hi − lo) × L ≤ tol

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Merging pieces on one line

  Groups segments that lie on the same line and joins their overlapping spans.

  key = round(atan2(dy, dx) × 1e6) and round(offset ÷ (20 × tol)); spans joined when the gap ≤ tol

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Merged lines back to segments

  Writes each merged span as a segment, and removes from hidden lines what a visible line covers.

  point = offset × (−dy, dx) + t × (dx, dy)

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Scale from text

  Reads a scale typed as 1:50, 2:1, 1/20 or 0.5.

  a:b → a ÷ b; accepted when 0 < scale < 1e6

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• Paper units per meter and automatic scale

  How many drawing or paper units one meter of the model is, and the largest standard scale that fits the sheet.

  inch sheet: 1 ÷ 0.0254; mm sheet: 1000; model: 1 ÷ meters per unit; fit: size × units per meter × scale ≤ 0.45 × paper size

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• View lines in drawing units

  The 2D lines of the view, centered on the block's base point and scaled.

  x = (u − u_center) × f; y = (v − v_center) × f; f = units per meter × scale

  Flat views of 3D solids flatview.js

  • CadCraft
  64-bit

• GEODATA fields from the drawing's location

  The design point, reference point, units and north vector written to the file.

  rot = degrees × π ÷ 180; north = (round₉(sin rot), round₉(cos rot))

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Location from another program's GEODATA

  Turns a file's GEODATA into the drawing's coordinate system, placement point and rotation.

  rot = round₉(atan2(north.x, north.y) × 180 ÷ π)

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Fingerprint of the written location

  Decides whether the GEODATA in a file is still the one CadCraft wrote or another program moved the location since.

  Drawing location written to DWG and DXF geodwg.js

  • CadCraft
  64-bit

• Cut parameters

  Where along an object each cutting edge crosses it.

  t = segment index + parameter of the intersection; t = round(t × 10⁹) ÷ 10⁹

  Trim geom.js

  • CadCraft
  64-bit

• Piece between two parameters

  The part of a line, arc, open polyline or construction line between two parameters, as a new object.

  line: p₁ + (p₂ − p₁)·t; arc: a0 + sweep·t; polyline arc piece: bulge = tan(atan(b)·(t_b − t_a))

  Trim geom.js

  • CadCraft
  64-bit

• Trim at a pick

  The pieces that stay when the part of an object round the picked point is cut away.

  circle: remaining arc from angle t_hi × 2π to t_lo × 2π

  Trim geom.js

  • CadCraft
  64-bit

• Splitting into pieces

  An object divided at all its cutting edges into pieces that can be removed one by one.

  Trim geom.js

  • CadCraft
  64-bit

• Points along an ellipse

  The points that stand in for an ellipse or elliptical arc.

  n = max(8, ⌈steps × sweep ÷ 2π⌉); t = t0 + (t1 − t0)·i ÷ n; p = c + R(rot)·(a·cos t, a·ratio·sin t)

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Ellipse area and perimeter

  The area and perimeter of an ellipse, or the area and length of an elliptical arc.

  full: area = π·a·b; perimeter ≈ π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))), h = ((a − b) ÷ (a + b))²; arc: 720 chords per turn

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• A piece through a transform

  A line or arc piece of a block put through the block reference's transform.

  uniform: center M·c, radius r·s, start angle = atan2(M·start − M·c); otherwise n = max(8, ⌈48 × sweep ÷ 2π⌉) chords

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Outline of an object as points

  The outline a revision cloud follows, with arcs and curves in short pieces.

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Even spacing along a path

  Points spaced evenly along a run of points, about a given step apart.

  n = max(1 or 3, round(L ÷ step)); target k = L·k ÷ n; point = linear interpolation on the segment that holds the target

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Revision cloud

  A polyline of equal arcs along a path, bulging outward.

  bulge = ± tan(2π ÷ 3 ÷ 4) = ± tan 30°; outward side from the sign of Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Exploding a spline

  Replaces a spline with a polyline along the curve.

  Ellipses and curves as short segments geom.js

  • CadCraft
  64-bit

• Solver tolerance

  The size below which an equation counts as met.

  tol = 1e-9 × max(1, width, height, |min x|, |max x|, |min y|, |max y|)

  2D constraint solver gsolve.js

  • CadCraft
  64-bit