Calculations

Calculations

• 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

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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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• Rounding of stored values

  The two rounding helpers every stored number of an item goes through.

  r4(v) = round(v × 10000) ÷ 10000; r2(v) = round(v × 100) ÷ 100

  Library item geometry from blocks, meshes and tables libitem.js

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• Block to millimeters

  The transform from a block's own coordinates and units to the item's millimeters.

  M = [mm, 0, 0, mm, −base.x·mm, −base.y·mm]; scale of a transform = √|det|; angle = r4(atan2(y, x) in degrees)

  Library item geometry from blocks, meshes and tables libitem.js

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• Lines, circles and arcs of a block

  A block's lines, circles and arcs as plan entities in millimeters.

  Library item geometry from blocks, meshes and tables libitem.js

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• Nested block references

  The transform of a block reference inside the block, with its rows and columns.

  local = T(insert) · R(rotation) · S(sx, sy) · T(−base + column·spacing ÷ sx, row·spacing ÷ sy)

  Library item geometry from blocks, meshes and tables libitem.js

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• Mesh body from a block's 3D faces

  The item's 3D body and height when the block holds 3D faces.

  Library item geometry from blocks, meshes and tables libitem.js

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• Arcs as points, 5° steps

  The points that replace a circle or arc that cannot stay an arc.

  n = max(8, ⌈sweep ÷ 5°⌉); point = c + r·(cos a, sin a)

  Library item geometry from blocks, meshes and tables libitem.js

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• Splines as points

  The points that replace a block's spline in the item's plan.

  de Boor recurrence at u = lo + (hi − lo)·s ÷ n, n = max(16, 8 × control points)

  Library item geometry from blocks, meshes and tables libitem.js

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• Arc points for bounds

  The points of an arc used to find a plan's bounds.

  n = max(4, ⌈sweep ÷ 10°⌉); point = c + r·(cos a, sin a)

  Library item geometry from blocks, meshes and tables libitem.js

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• Bounds of a plan

  The box round an item's plan, from which its width and depth are taken.

  Library item geometry from blocks, meshes and tables libitem.js

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• Box body on a plan

  A box body standing on the plan's bounds with a given height.

  w = max x − min x; d = max y − min y; each of x, y, z, w, d, h rounded to 4 decimals

  Library item geometry from blocks, meshes and tables libitem.js

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• Item from a mesh file

  A mesh file's triangles as an item body in millimeters, with its footprint and sizes.

  p = file point × unit in mm; Y-up: (x, y, z) → (x, −z, y); stored = r4(p − (center x, max y, min z))

  Library item geometry from blocks, meshes and tables libitem.js

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• STL corners

  The corner points of an STL file's triangles.

  Library item geometry from blocks, meshes and tables libitem.js

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• glTF accessor values

  The numbers of a glTF position or index accessor.

  Library item geometry from blocks, meshes and tables libitem.js

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• glTF node transforms

  The matrix that places each glTF node's mesh.

  M = T · R(q) · S, R from the quaternion (qx, qy, qz, qw)

  Library item geometry from blocks, meshes and tables libitem.js

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• Convex hull footprint

  The outline round a mesh item's points, used as its plan.

  monotone chain; turn = (a − o) × (b − o), popped when ≤ 0

  Library item geometry from blocks, meshes and tables libitem.js

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• Box of a body

  The box a body fills: a box body itself, or the extent of a mesh body.

  Library item geometry from blocks, meshes and tables libitem.js

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• Box item from width, depth and height

  The rectangle plan and box body of an item given only by its sizes.

  x from −w ÷ 2 to +w ÷ 2; y from −d to 0; label height = min(0.18 d, 150), reduced to fit 90% of w

  Library item geometry from blocks, meshes and tables libitem.js

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• Sizes from attributes

  Width, depth and height read from a block's attributes.

  Library item geometry from blocks, meshes and tables libitem.js

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• Millimeters per unit

  The table of unit factors.

  in 25.4; ft 304.8; yd 914.4; mi 1 609 344; mil 0.0254; µm 0.001

  Lengths and units for Library items libitem.js

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• Length from text

  A typed or stored length in millimeters.

  with a unit: n × factor; feet and inches: (feet × 12 + inches + a ÷ b) × 25.4; bare number: n × mm per unit

  Lengths and units for Library items libitem.js

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• Drawing units

  The millimeters per drawing unit of a DWG or DXF file.

  Lengths and units for Library items libitem.js

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• Pipe size in eighths of an inch

  A decimal inch size written as a fraction, such as 1 1/2".

  eighths = round((n − ⌊n⌋) × 8)

  Lengths and units for Library items libitem.js

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• Size label on the preview

  The overall width and depth printed on an item's preview picture.

  millimeters: round(mm); inches: round(mm ÷ 25.4 × 8) eighths

  Lengths and units for Library items libitem.js

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• Warp plan for a picture or tile

  How finely a map image is cut into patches so that it lands in the right place on screen.

  tolerance = 0.25 ÷ zoom project units, rounded to a half power of 2

  Map layers under the drawing maplayers.js

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• Drawing the patches

  Each patch of a map image drawn on the canvas.

  Map layers under the drawing maplayers.js

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• Local placement record

  The scale, rotation and point pair that tie drawing coordinates to grid coordinates.

  scale = m per drawing unit ÷ m per grid unit; from = (x, y); to = (e, n)

  Drawing placed on a map grid maplayers.js

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• 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

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• 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

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• 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

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  32-bit in part