Calculations

Calculations

• 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
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• 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
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• 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

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• Linetype pattern (R12)

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

  length = Σ |elementᵢ|

  DXF writing: numbers as text dxf.js

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• 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

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• 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

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• Radians to degrees (2000)

  Converts an angle to the degrees DXF stores.

  degrees = (r × 180) ÷ π

  DXF writing: numbers as text dxf2000.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Pointer with the locked fields applied

  The point shown and used while fields are locked: a typed distance or angle from the base point, or a typed X or Y.

  polar: p = base + d × (cos a, sin a), a = typed degrees × π ÷ 180 or atan2 of the pointer; xy: (typed X or pointer x, typed Y or pointer y)

  Dynamic input: points from typed distance, angle, X and Y dynin.js

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• Field values while the pointer moves

  The distance and angle from the base point, or X and Y, that the fields show.

  d = √(Δx² + Δy²); a = atan2(Δy, Δx) × 180 ÷ π, brought into 0 to 360

  Dynamic input: points from typed distance, angle, X and Y dynin.js

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• Text sent on Enter

  The coordinate text the command line receives: a relative distance and angle, or an absolute X and Y.

  @d<a from the last point; otherwise #x,y with x = base_x + d × cos a, y = base_y + d × sin a

  Dynamic input: points from typed distance, angle, X and Y dynin.js

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• Number text for the command line

  Writes a value taken from the pointer as text.

  text = value rounded to 10 decimals, trailing zeros dropped

  Dynamic input: points from typed distance, angle, X and Y dynin.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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
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• Unit factors

  What a number with a unit is multiplied by to give millimeters or degrees.

  mm 1; cm 10; m 1000; in 25.4; ft 304.8; deg 1; rad 180 ÷ π

  Parameter expressions expr.js

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• A number with a unit

  Reads a number and converts it by the unit written right after it.

  v = Number(text) × factor

  Parameter expressions expr.js

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• Sine, cosine and tangent in degrees

  sin, cos and tan of an angle given in degrees.

  sin(x) = clean(Math.sin(x × π ÷ 180)); tan refused when |cos| < 1e-12

  Parameter expressions expr.js

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• Trig result rounded to 15 digits

  Rounds a sine, cosine or tangent to 15 significant digits and turns a value below 1e-15 into 0.

  |x| < 1e-15 → 0; otherwise Number(x.toPrecision(15))

  Parameter expressions expr.js

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• Inverse trigonometry and square root

  asin, acos, atan, atan2 in degrees, and sqrt.

  asin(x) = Math.asin(clamp(x, −1, 1)) ÷ (π ÷ 180); sqrt(x) = √max(0, x)

  Parameter expressions expr.js

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• Power, exponential and logarithms

  pow, exp, ln and log.

  pow(a, b) = aᵇ; ln and log need x > 0

  Parameter expressions expr.js

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• ceil, floor and int

  Rounds up, down or toward zero to a whole number.

  ceil(x) = ⌈x − ε⌉; floor(x) = ⌊x + ε⌋; int(x) = trunc(x + sign(x) × ε); ε = 1e-9 × max(1, |x|)

  Parameter expressions expr.js

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• round(x, digits)

  Rounds to a number of decimal places from 0 to 10, halves away from zero.

  y = |x| × 10ᵈ; r = ⌊y + 0.5 + 1e-9 × max(1, y)⌋; result = ± r ÷ 10ᵈ

  Parameter expressions expr.js

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• Comparisons

  The six comparison operators, giving 1 or 0.

  tol = 1e-9 × max(1, |a|, |b|); a == b when |a − b| ≤ tol; a < b when a < b − tol

  Parameter expressions expr.js

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• Arithmetic operators

  Addition, subtraction, multiplication, division, remainder and power.

  a % b = a − b × ⌊a ÷ b⌋; division and remainder refuse b = 0

  Parameter expressions expr.js

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• Number as text

  The text a number is written as when it is put back into an expression or a message.

  Number(v.toPrecision(12)) as text

  Parameter expressions expr.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• What a dimensional constraint measures

  The current horizontal, vertical or direct distance, radius, diameter or angle of the constrained geometry.

  hdist = s × (x₁ − x₀); dist = |p₁ − p₀| or s × signed distance from the line; diameter = 2r; angle = s × atan2(u × v, u · v) × 180 ÷ π; arc angle = (a₁ − a₀) mod 2π in degrees

  Constraint values and signs gcons.js

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• Signed distance from a line

  How far a point or a circle's center lies from an infinite line, positive on the left of its direction.

  d = (u_x × (p_y − l_y) − u_y × (p_x − l_x)) ÷ |u|

  Constraint values and signs gcons.js

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