Calculations

Calculations

• Ellipse as line pieces

  The line pieces an ellipse is drawn and plotted with.

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

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• Linetype scale of an object

  The factor that sizes the dashes of an object's linetype.

  scale = LTSCALE × object scale × plot style table scale (non-ISO linetypes only)

  Wide polylines, text masks and ellipses as drawn and plotted walk.js

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• Meters to drawing units

  A 3D point in meters as a point in drawing units.

  x = pₓ ÷ k; y = pᵧ ÷ k; z = p_z ÷ k

  Points and heights picked in the 3D view ws3d-core.js

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• Point on a level plane

  Where the ray under the pointer meets the plane at height z.

  plane: n = (0, 0, 1), d = z × k; point = ray hit ÷ k

  Points and heights picked in the 3D view ws3d-core.js

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• Height following the cursor

  A height above a base outline, taken from where the pointer's ray passes nearest the vertical line through the outline's middle.

  c = Σ ring points ÷ n; height = s ÷ k, with s the distance along the vertical line from (cₓ·k, cᵧ·k, z₀·k)

  Points and heights picked in the 3D view ws3d-core.js

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• 2D drawing as snap edges

  The visible lines and arcs of the 2D drawing as edges on the ground, in meters, for the 3D snaps.

  edge = (x·k, y·k, 0); arc: m = max(2, ⌈sweep ÷ 7.5°⌉) chords

  Points and heights picked in the 3D view ws3d-core.js

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• Height shown beside the cursor

  The text of the height tip while a height is being picked.

  |n| < 1e-9 → 0; else n to 4 decimals, trailing zeros removed

  Points and heights picked in the 3D view ws3d-core.js

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• Unit vector

  A direction scaled to length 1, or nothing when it has no length.

  l = √(x² + y² + z²); l > 1e-300 → (x, y, z) ÷ l

  ACIS solids turned into triangles acis/geom.js

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• Plane

  A point of a plane from its (u, v), and back.

  P = O + U·u + V·v; u = (p − O)·U; v = (p − O)·V

  ACIS solids turned into triangles acis/geom.js

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• Cone and cylinder

  A point of a cone or cylinder (circular or elliptical base) from its angle and its height along the axis, and back.

  t = ± sin ÷ cos; q = 1 + h·t ÷ R; P = C + X·R·q·cos u + Y·R_m·q·sin u + N·h; u = atan2((d·Y) ÷ R_m, (d·X) ÷ R); apex at h = −R ÷ t

  ACIS solids turned into triangles acis/geom.js

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• Sphere

  A point of a sphere from longitude and latitude, and back.

  P = C + r·cos v·(X·cos u + Y·sin u) + Z·r·sin v; u = atan2(y, x); v = atan2(z, √(x² + y²))

  ACIS solids turned into triangles acis/geom.js

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• Torus

  A point of a torus from its two angles, and back.

  w = A + B·cos v; P = C + X·w·cos u + Y·w·sin u + N·B·sin v; u = atan2(y, x); v = atan2(z, √(x² + y²) − A)

  ACIS solids turned into triangles acis/geom.js

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• Spline surface and its inverse

  A point of a B-spline surface from (u, v), its normal, and the (u, v) nearest to a given point.

  first guess: nearest of 25 × 25 samples; then Gauss–Newton on |p − S(u, v)|² with partial derivatives by central differences of 1e-6 of the range

  ACIS solids turned into triangles acis/geom.js

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• Knot span

  Which knot interval a parameter falls in.

  ACIS solids turned into triangles acis/geom.js

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• De Boor evaluation

  A point of a B-spline at a parameter, in homogeneous coordinates.

  a = (t − Uᵢ) ÷ (Uᵢ₊ₚ₋ᵣ₊₁ − Uᵢ); d_j = (1 − a)·d_j₋₁ + a·d_j

  ACIS solids turned into triangles acis/geom.js

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• Spline curve point

  A point of a rational B-spline curve.

  P = (X, Y, Z) ÷ W from de Boor

  ACIS solids turned into triangles acis/geom.js

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• Spline surface point

  A point of a rational B-spline surface.

  de Boor along v on each of the p + 1 rows, then de Boor along u on the results

  ACIS solids turned into triangles acis/geom.js

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• Model size and tolerances

  The size of the whole solid and the tolerances every later test uses.

  size = max(|hi − lo|, 2·rad, 1e-9); tol = max(10 × resabs, 1e-5 × size); weld = 1e-9 × size

  ACIS solids turned into triangles acis/tess.js

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• Circular and elliptical edges

  The points along an edge that is part of a circle or an ellipse.

  step = min(7.5°, 2·acos(1 − min(1, 0.0002·size ÷ R))); n = min(4096, ⌈|t₁ − t₀| ÷ max(step, 0.001)⌉)

  ACIS solids turned into triangles acis/tess.js

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• Spline edges

  The points along an edge that follows a B-spline curve.

  N = min(4000, max(64, 24 × spans)) samples; end parameters by nearest sample, then 60 steps of a three-way search; n = ⌈|t₁ − t₀| ÷ (b − a) × N ÷ 4⌉ pieces, 2 to 4096

  ACIS solids turned into triangles acis/tess.js

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• Merging vertices

  Gives points that coincide one shared vertex, so a closed solid stays closed.

  key = round(x ÷ weld), round(y ÷ weld), round(z ÷ weld)

  ACIS solids turned into triangles acis/tess.js

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• Does the surface fit its edges

  Checks that every edge point of a face lies on the face's surface before the face is used.

  |S(inv(p)) − p| ≤ tol for every loop point

  ACIS solids turned into triangles acis/tess.js

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• Which way the triangles face

  Turns a face's triangles so they point the way the face's outward normal does.

  n = (t₁ − t₀) × (t₂ − t₀); vote by the sign of n · normal at the triangle's center

  ACIS solids turned into triangles acis/tess.js

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• Flat faces

  Triangles of a flat face with its holes.

  ACIS solids turned into triangles acis/tess.js

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• Outer loops and holes

  Sorts the loops of a face into outlines and the holes inside them.

  A = |Σ (uᵢ·vᵢ₊₁ − uᵢ₊₁·vᵢ)| ÷ 2; depth = number of larger loops that contain the middle of the loop's first edge

  ACIS solids turned into triangles acis/tess.js

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• Loop unwound on a curved surface

  The (u, v) of a loop's points with the angles unwound along the loop, and how many times it goes round the surface.

  du wrapped to ±half a turn and added up; windings = round(total ÷ period); pole when cos v · r ≤ tol (sphere) or the cone radius ≤ tol

  ACIS solids turned into triangles acis/tess.js

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• Step sizes on a curved face

  How finely a curved face is divided.

  angles: 7.5°; spline: range ÷ 16 each way; along a cone's axis: max(3 × 7.5° in radians × R, 1e-12)

  ACIS solids turned into triangles acis/tess.js

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• Extra points inside a curved face

  The points added inside a curved face so that its triangles follow the surface.

  one point per grid cell at (i + 0.5 + 0.3·jitter)·cell; kept when inside the face and at least 0.35 × the smaller cell side from every boundary edge

  ACIS solids turned into triangles acis/tess.js

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• Fixed offset of a grid point

  A repeatable offset between −0.5 and 0.5 for each grid cell.

  s = sin(12.9898·i + 78.233·j) × 43758.5453; s − ⌊s⌋ − 0.5

  ACIS solids turned into triangles acis/tess.js

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• Distance from a point to a segment

  How far an inside point is from a boundary edge, in scaled (u, v).

  t = clamp(((q − s₀)·d) ÷ |d|², 0, 1); distance = |q − s₀ − t·d|

  ACIS solids turned into triangles acis/tess.js

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• Body transform

  Moves the mesh by the transform stored with the body.

  p′ = s·(p · M) + t; mirrored when det(M)·s³ < 0

  ACIS solids turned into triangles acis/tess.js

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• Is the mesh closed

  Decides whether the mesh is a closed solid: every edge used once in each direction.

  key(a, b) = a × 2³² + b

  ACIS solids turned into triangles acis/tess.js

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• Volume and surface area of the mesh

  The volume enclosed by the triangles and their total area.

  V = Σ p · (q × r) ÷ 6; A = Σ |(q − p) × (r − p)| ÷ 2

  ACIS solids turned into triangles acis/tess.js

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• One span of the curve

  The 16 points of the curve between two neighboring outline points.

  tᵢ₊₁ = tᵢ + max(√|pᵢ₊₁ − pᵢ|, 1e-9); point = three levels of linear interpolation between p₀ … p₃ at t

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Sampled path of one edge

  The points of one seam edge from corner to corner.

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Length of a path

  The length of a sampled edge or outline.

  L = Σ |pᵢ − pᵢ₋₁|

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Running length

  The distance from the start of a path to each of its points.

  cᵢ = cᵢ₋₁ + |pᵢ − pᵢ₋₁|

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Point at a distance along a path

  The point and direction at a given distance along a path (used to place notches).

  k = (d − cᵢ₋₁) ÷ (cᵢ − cᵢ₋₁); point = a + (b − a)·k; tangent = (b − a) ÷ |b − a|

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Nearest point of a path

  The point of a path nearest to a given point, and its distance along the path.

  k = clamp(((p − a)·v) ÷ |v|², 0, 1); q = a + v·k

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Area of an outline

  The signed area inside a closed sampled outline.

  A = Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) ÷ 2

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Centroid of an outline

  The middle of area of a closed outline (where the label goes by default).

  kᵢ = xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ; a = Σ kᵢ; c = Σ (pᵢ + pᵢ₊₁)·kᵢ ÷ (3·a)

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Where two lines meet

  The meeting point of two lines and where it lies along each.

  den = r × s; u = ((c − a) × s) ÷ den; point = a + r·u

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Outline direction

  Makes an outline run counter-clockwise.

  Pattern outlines: curves, lengths and areas addons/apparel/lib/curve.js

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• Offset of one edge

  One edge of the sew line moved outward by its allowance.

  n = (tᵧ, −tₓ); moved point = p + n·w; join = meeting point of the two moved lines when |t₁ × t₂| > 1e-9 and it is within 4·w of the vertex

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• Offset of an outline with no corners

  A closed smooth outline moved outward by one allowance all round.

  vertex = meeting point of the two moved lines; fallback = b + right((t₁ + t₂) ÷ 2)·w

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• Dropping repeated points

  Removes points that repeat the one before.

  kept when |p − previous| > 1e-9

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• Dropping points on a straight line

  Removes cut line points that lie on the straight line between their neighbors.

  kept when |u × v| > 1e-9 × |u| × |v|

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• Inside corner

  Trims two cut lines where they cross at an inside corner.

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• Corner between two allowances

  The points that join the cut lines of two edges at a corner, by the corner's rule.

  turn = tₐ × t_b; turn < −1e-9 → inside corner; meeting points used only within 20 × the wider allowance of the corner

  Seam allowance: the cut line addons/apparel/lib/seam.js

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• The whole cut line

  The closed cut line of a piece from its sew line and allowances.

  Seam allowance: the cut line addons/apparel/lib/seam.js

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