Calculations

Calculations

• Resolution in dots per inch

  The horizontal and vertical resolution of a page, in dots per inch.

  dpi = resolution × 2.54 (centimeter files); dpi = resolution (inch files)

  TIFF scan resolution raster/tiff.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  • PartCraft
  64-bit

• Triangulate a polygon with holes

  The triangles of a polygon, as indexes into its point list.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Winding of a ring

  Whether a ring runs clockwise or counterclockwise.

  sum = Σ (xⱼ − xᵢ) × (yᵢ + yⱼ)

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Orientation of three points

  On which side of the line through two points a third point lies.

  area = (q.y − p.y) × (r.x − q.x) − (q.x − p.x) × (r.y − q.y)

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Removing duplicate and collinear points

  Drops points that repeat their neighbor or lie on the line between their neighbors.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Ear test

  Whether a corner can be cut off as a triangle: it is convex and no other point of the ring lies inside it.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Point in triangle

  Whether a point lies inside or on a triangle.

  three cross products of (corner − point) pairs, each ≥ 0

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Bridge from a hole to the outer ring

  The point of the outer ring a hole is connected to.

  x = p.x + (h.y − p.y) × (p′.x − p.x) ÷ (p′.y − p.y); tan = |h.y − p.y| ÷ (h.x − p.x)

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Do two segments cross

  Whether two edges intersect, including touching and collinear overlap.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
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  64-bit

• Is the middle of a diagonal inside the polygon

  Whether a diagonal between two corners runs through the inside of the polygon.

  midpoint = ((a.x + b.x) ÷ 2, (a.y + b.y) ÷ 2); even-odd count of edges crossed by a ray

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
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  • ModelCraft
  • TakeoffCraft
  • CivilCraft
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  64-bit

• Unit factor

  Chooses the factor that turns the file's unit into meters.

  k = meters per file unit, or 0.001 when that is missing, zero or not finite

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Mesh points in meters

  Scales a mesh's points and builds the model's mesh from them.

  p′ = p × k

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Block placement matrix

  Turns a block reference's matrix into the model's placement.

  m[c × 4 + r] = x[r × 4 + c]; translation × k

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Curve points in meters

  Scales each curve's points and drops segments of no length.

  a′ = a × k; kept when a′ ≠ b′

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Mesh record: 32-bit and 64-bit points

  Reads a mesh's points from the file and chooses between its 32-bit and 64-bit copies.

  use the 64-bit points when |aᵢ − vᵢ| ≤ |vᵢ| × 10⁻⁶ + 10⁻⁶ × max(1, |aᵢ|) for every coordinate

  3DM import: units, placements and mesh points io/3dm/objects.js

  • ModelCraft
  • PartCraft
  64-bit

• Orientation of three points

  Twice the signed area of a triangle: which way it turns.

  (b − a) × (c − a) = (bₓ − aₓ)(c_y − a_y) − (b_y − a_y)(cₓ − aₓ)

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Point inside a circumcircle

  Whether a fourth point lies inside the circle through three others.

  determinant of the rows (pₓ − dₓ, p_y − d_y, |p − d|²) for p = a, b, c; inside when it is positive

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Triangle that holds a point

  Walks from the last triangle toward the point, then searches all triangles if the walk fails.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Point insertion

  Splits the triangle a new point falls in, or the side it falls on.

  ε = 10⁻⁶ × orient(a, b, c); on a side when a sub-area ≤ ε

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Edge flip

  Flips an edge while the opposite corner lies inside the circumcircle.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Flips with no new points

  Makes an existing triangulation Delaunay by flipping edges.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Multiplicity of a knot

  How many times a value occurs in a knot vector.

  count of |Uᵢ − u| ≤ tol × (U_last − U_first)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Distinct knots

  The different knot values with their multiplicities.

  a knot joins the previous one when |x − u_prev| ≤ tol × range

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Clamped uniform knot vector

  A knot vector with evenly spaced interior knots.

  uᵢ = a + (b − a) × i ÷ (inner + 1)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Knot vector check

  Checks a knot vector read from a file: count, order, clamping and multiplicities.

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Greville abscissae

  One parameter value per control point.

  gᵢ = (Uᵢ₊₁ + … + Uᵢ₊ₚ) ÷ p

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Binomial coefficients

  The number of ways to choose k of n.

  C(n, k) = C(n − 1, k − 1) + C(n − 1, k)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  Exact

• A number as IGES text

  Writes one real number into the file.

  whole numbers below 10¹⁵: digits and a point; otherwise String(x) with E → D

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Global section

  States the file's number sizes, unit, resolution and largest coordinate.

  unit flag 2 (MM); scale 1; resolution 10⁻⁶; largest coordinate = max |x| over all control points

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• A number from IGES text

  Reads one real number from the file.

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Unit factors

  Millimeters per file unit, by the file's unit flag.

  1: 25.4; 2: 1; 3: 25.4 × 12; 4: 1609344; 5: 10⁶; 6: 1000; 7: 10⁶; 8: 10⁻³; 9: 10⁻³; 10: 10; 11: 25.4 × 10⁻⁶

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Scales a curve's control points by the file's unit as it is read.

  scale = UNITS[flag] (1 when unknown); p′ = p × scale

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit

• End derivative coefficients

  The factors that link the first control points to the first and second derivative at the start.

  a₁ = p ÷ (U[p+1] − U[1]); b₁ = p ÷ (U[p+2] − U[2]); c₂ = (p − 1) ÷ (U[p+1] − U[2])

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Point, direction and curvature at an end

  What a curve's end looks like to the curve that must meet it.

  t = C′ ÷ |C′|; out = ± t; speed = |C′|

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Curve end matched to a target

  Moves the first one to three control points so the curve meets the target with G0, G1 or G2.

  P₀ = T; P₁ = P₀ + out × speed ÷ a₁; D₂ = out (C″ · out) + κ speed²; P₂ = P₁ + (a₁ (P₁ − P₀) + D₂ ÷ c₂) ÷ b₁

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend curve between two ends

  A Bézier curve joining two curve ends with the asked continuity.

  G1: P₁ = A + out × α ÷ 3; G2: P₁ = A + out × α ÷ 5, P₂ = 2 P₁ − A + κ α² ÷ 20; α = chord × bulge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Frame on the target's edge

  For a point near the target's edge: the nearest edge point, the outward direction in the tangent plane, the normal and the normal curvature.

  d = (n × tₑ) ÷ |n × tₑ|, turned to point away from the surface

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Surface edge matched to a target

  Refits the first one to three rows of control points so the surface meets the target along an edge.

  stations vᵢ = i ÷ M, M = max(samples, 4 × columns); rows fitted by least squares; accepted when gap ≤ g0, angle ≤ g1 and Δκ ≤ g2 (or relative ≤ g2rel)

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• Blend surface between two edges

  A surface joining two surface edges, built from blend sections along them.

  sections at t = i ÷ n, n = 8, 16, … ≤ 128; each a Bézier of degree 1, 3 or 5; interpolated along the edge

  Matching and blending NURBS curves and surfaces nurbs/match.js

  • ModelCraft
  • PartCraft
  64-bit

• A number as STEP text

  Writes one real number into the file.

  String(x), with a decimal point added when it has none

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Knots and multiplicities

  Turns a full knot vector into STEP's list of distinct knots with their counts.

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Edges that have collapsed to a point

  Leaves a zero-length boundary edge, such as a pole, out of the face loop.

  skipped when arc length ≤ 10⁻⁹ × surface size

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
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  64-bit

• Trim loop written as a curve

  Writes a trim loop as one closed curve on the face.

  3D point = S(u, v) for each loop point; curve = straight segments through them

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
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  64-bit

• Length unit of a file

  Millimeters per file unit.

  SI: prefix × 1000; conversion-based: value × its base unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Control points in millimeters

  Reads each control point and scales it by the file's unit.

  p′ = p × unit

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Trim loops read back

  Turns a face's boundary edges into trim loops in the surface's own parameters.

  33 samples per curved edge; each projected to (u, v); outer when |area| < 0.999 × (u range × v range)

  STEP files of NURBS curves and surfaces nurbs/step.js

  • ModelCraft
  • PartCraft
  64-bit

• Volume and box of each body

  The volume and bounding box the body list shows.

  volume = Σ volume of the body's solids; box = min and max over them

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Move matrix

  The matrix a Move body feature applies: by distances, point to point, about an axis, or plane to plane.

  points: T(to − from); align: rotate n_a onto ∓n_b about c, then t = n_b ((o_b − c) · n_b)

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Shape under a move or a uniform scale

  Applies the matrix or the scale to a solid in the kernel.

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit

• Scale center and affine map

  Chooses the center of a scale and builds the map for unequal factors.

  c = picked point, or (box min + box max) × 0.5; A = diag(sₓ, s_y, s_z); t = c (1 − s)

  Bodies: volume, move and scale brep-bodies.js

  • PartCraft
  64-bit