Calculations

Calculations

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  • ModelCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• Outline direction

  Makes an outline run counter-clockwise.

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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• 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

  • CadCraft
  64-bit

• Inside corner

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

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

  • CadCraft
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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

  • CadCraft
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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

  • CadCraft
  64-bit

• Rounding to 6 decimals

  Rounds a value to 6 decimals.

  r6(v) = number(v.toFixed(6))

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Even step per size

  A rule that moves a point by the same step for each size away from the base.

  move of size i = (i − base) × step

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Rules in proportion between two points

  Rules for the outline points between two ruled points, by distance along the outline.

  s = distance from A ÷ distance A to B; rule = r_A + (r_B − r_A)·s

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Position of each point along the outline

  How far along the sampled outline each outline point lies, and the outline's length.

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• The sew line in a size

  The outline points of a piece in one size.

  point = base + move; unruled curve point: move = m_lo + (m_hi − m_lo)·s, s = d ÷ d_hi

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• The whole piece in a size

  The piece with its grain line, fold line, internal lines, drill holes, buttons and label moved to one size.

  mark = base + move of its rule, rounded to 6 decimals

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Rules after a change of sizes or base

  The rules re-measured from a new base size.

  new move = old move − old move of the new base

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Rules read from a file

  Checks the grade rules of a piece that came from a file or the clipboard.

  Grading a pattern across sizes addons/apparel/lib/grade.js

  • CadCraft
  64-bit

• Polyline arc to curve points

  The curve points that stand for one arc segment of a polyline.

  θ = 4·atan(bulge); r = L ÷ (2·sin(|θ| ÷ 2)); n = max(2, ⌈|θ| ÷ 15°⌉) points at equal angles

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Outline from a closed polyline, circle or ellipse

  The sew line of a new piece from one closed object.

  circle: 12 curve points at 30°; ellipse: 16 curve points at 22.5° of its parameter

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Outline from lines and arcs that join up

  The sew line of a new piece from separate lines, arcs and open polylines.

  arc: n = max(2, ⌈sweep ÷ 15°⌉) steps; ends join when |p − q| ≤ 1e-4

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Smooth vertices

  Decides which vertices of the outline are curve points.

  turn = |atan2(next − p) − atan2(p − previous)|; next to an arc and turn < 2° → curve point

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Stored outline points

  Cleans the new outline and rounds it for storage.

  x, y → toFixed(6)

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Where a notch sits

  The position of a notch on the cut line, opposite its place on the sew line.

  q = point at distance d along the edge; notch = q + outward normal × allowance

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• A new notch from a picked point

  The edge and distance along it for a notch placed at a picked point.

  nearest point over all edges; d → toFixed(4)

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Notch geometry

  The lines of a slit, V or T notch.

  a = ang × π ÷ 180; direction = −R(a)·normal; tip = notch + direction × depth

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Moving a point by its grip

  The piece after one of its points or marks was dragged.

  new position → toFixed(6)

  Pattern pieces from drawing geometry addons/apparel/lib/piece.js

  • CadCraft
  64-bit

• Thinning a cut line

  A closed cut line with the points removed that lie within a tolerance of the simplified line.

  d = |(P − A) × (B − A)| ÷ |B − A|; kept when d > tol

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
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• A piece's copy for the marker

  The cut line, grain angle and area stored with a piece placed on a marker.

  x, y → toFixed(2) in mm, toFixed(4) in inches; area = |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| ÷ 2 → toFixed(4); grain = atan2 of the grain line

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
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• Piece to drawing coordinates

  The position in the drawing of a point of a placed piece.

  x′ = x + cos(rot)·sx·pₓ − sin(rot)·sy·pᵧ; y′ = y + sin(rot)·sx·pₓ + cos(rot)·sy·pᵧ

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
  64-bit

• Grain direction check

  Whether a placed piece's grain line runs along the fabric within the marker's tolerance.

  tol = degrees × π ÷ 180 + 1e-6; ok when the angle between grain and fabric length ≤ tol (or to the opposite direction, on fabric that may be turned)

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
  64-bit

• Is the piece on the fabric

  Whether a piece lies on the fabric, off it, or across an edge.

  bounding box in the marker's frame against 0 … width and 0, with eps = 0.01 mm or 0.0005 in

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
  64-bit

• Do two edges cross

  Whether two cut line segments cross strictly inside both.

  den = r × q; u = ((c − a) × q) ÷ den; v = ((c − a) × r) ÷ den; crossing when 1e-9 < u, v < 1 − 1e-9

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
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• Do two pieces overlap

  Whether two placed pieces overlap.

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
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• Marker length and efficiency

  The fabric length used, the area of the pieces and the share of the fabric they cover.

  len = max x of the pieces + end allowance; used = Σ area; eff = used ÷ (len × width)

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
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• Marker label

  The width, length and efficiency written under the marker.

  inches: 2 decimals; mm: 1 decimal, or meters with 3 decimals from 1000 mm; efficiency: percent with 1 decimal

  Markers: pieces laid on fabric addons/apparel/lib/marker.js

  • CadCraft
  64-bit

• Paper sizes

  The two home-printer papers, in millimeters.

  A4 = 210 × 297; Letter = 215.9 × 279.4 (8.5 × 11 in × 25.4)

  Printing patterns at full size addons/apparel/lib/print.js

  • CadCraft
  Exact

• Test square

  The side of the square printed on the cover page to check the scale.

  10 cm = 100 mm; 4 in = 101.6 mm

  Printing patterns at full size addons/apparel/lib/print.js

  • CadCraft
  Exact

• Full-size page

  The page size and the placement of the pieces on a single full-size page.

  w = (maxX − minX)·k + 2·margin; h = (maxY − minY)·k + 2·margin + 8; matrix = [k, 0, 0, k, margin − minX·k, margin − minY·k]

  Printing patterns at full size addons/apparel/lib/print.js

  • CadCraft
  64-bit