Calculations

Calculations

• Making a curve family compatible

  Gives every profile (or every guide) the same degree and knots, exactly when possible and by fitting otherwise.

  exact: raise degree, merge knots; otherwise: cubic interpolation of C(r(s)), points doubled until error ≤ fitTol

  Surface through a network of curves gordon.js

  • PartCraft
  64-bit

• Boolean sum of the three surfaces

  Adds the surface through the profiles and the surface through the guides and subtracts the surface through the grid points.

  P = Lu + Lv − T, pole by pole

  Surface through a network of curves gordon.js

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  64-bit

• Tangency to a plane along a boundary

  Makes the surface leave a boundary curve tangent to a plane and reports the angle it made with the plane before.

  second pole row: q ← a + (d − n × (d·n)), d = q − a; angle = asin(|d·n| ÷ |d|)

  Surface through a network of curves gordon.js

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  64-bit

• Deviation from the input curves

  The largest distance between the finished surface and any profile or guide, shown in the panel.

  dev = max over 401 samples per curve of |S(s) − C(r(s))|

  Surface through a network of curves gordon.js

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  64-bit

• Fold check

  Refuses a surface whose normal flips between neighboring sample points.

  n = (S(u + h, v) − S(u − h, v)) × (S(u, v + h) − S(u, v − h)), h = 1e-7; fold if n₁·n₂ < 0

  Surface through a network of curves gordon.js

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  64-bit

• Point on the handle axis nearest the pointer ray

  How far along the handle's axis the pointer is, read from the pointer's ray.

  t = (b·e − c·d) ÷ (a·c − b²)

  Depth and angle set by dragging a handle handle-math.js

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  64-bit

• Dragged extrude depth

  The depth stored on the Extrude after a drag.

  depth = max(step, round((v₀ + scale × (t − t₀)) ÷ step) × step), step = max(0.01, 10^⌊log₁₀|v₀|⌋ ÷ 10)

  Depth and angle set by dragging a handle handle-math.js

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  64-bit

• Dragged revolve angle

  The angle stored on the Revolve after a drag.

  angle = atan2(q·w, q·u) in degrees, unwrapped toward the previous value, rounded to 1°, kept in 1…360

  Depth and angle set by dragging a handle handle-math.js

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  64-bit

• Hull mesh taken from the built part

  Copies the part's triangle mesh for the hull calculation.

  Hull hydrostatics and righting arms hydrostatics.js

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  32-bit in part

• Volume, moments and waterplane under a waterline

  Clips the mesh at the waterline and sums the wet volume, its first moments, the wetted area and the waterplane's boundary integrals.

  V = Σ a·(b × c) ÷ 6; ∫x dV = ½ ∮ x² nₓ dS; A = ½ Σ (pₓ q_z − qₓ p_z); ∫z² dA = Σ (p_z² + p_z q_z + q_z²)(pₓ q_z − qₓ p_z) ÷ 12

  Hull hydrostatics and righting arms hydrostatics.js

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  32-bit in part

• Hydrostatic properties at a draft

  Volume, displacement, KB, LCB, waterplane area and centroid, BM, KM, block and waterplane coefficients, and mass per unit immersion.

  V = V_skin + h × A ÷ 3; I_T = ∫z² dA − A × c_z²; BM = I ÷ V; KM = KB + BM; Δ = V × 1e-9 × ρ

  Hull hydrostatics and righting arms hydrostatics.js

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  32-bit in part

• Righting arms at fixed displacement

  The righting arm GZ at each heel angle, the metacentric height GM, the largest GZ and the angle where stability vanishes.

  GZ = z(B) − (keel + KG) × sin θ; GM = KM_T − KG; vanishing angle by linear interpolation between samples

  Hull hydrostatics and righting arms hydrostatics.js

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  32-bit in part

• Units and rounding of the answer

  Converts the hull results to meters, square meters, cubic meters and tonnes for the tool's answer.

  value_m = round(value_mm ÷ 1000 × 10⁴) ÷ 10⁴; volume_m³ = round(V ÷ 1e9 × 10⁴) ÷ 10⁴

  Hull hydrostatics and righting arms hydrostatics.js

  • PartCraft
  32-bit in part

• Coordinate and direction text

  Formats a coordinate or a direction component as a STEP real.

  text = v.toFixed(digits), trailing zeros removed; digits = 6 for points, 12 for directions

  IFC export of parts, assemblies and library items ifc-export.js

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  64-bit

• Quantity and property value text

  Formats a volume, area, mass, length or parameter value as a STEP real.

  text = Number(v.toPrecision(12))

  IFC export of parts, assemblies and library items ifc-export.js

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  64-bit

• Triangulated body

  Writes the vertices and triangles of the built mesh as an IfcTriangulatedFaceSet.

  (x, y, z) → (x, −z, y), each written with 6 decimals

  IFC export of parts, assemblies and library items ifc-export.js

  • PartCraft
  32-bit in part

• Volume, area, weight and overall size

  The quantity sets of an element: net volume in m³, net surface area in m², net weight in kg and the box dimensions in mm.

  V_m³ = V_mm³ × 1e-9; A_m² = A_mm² × 1e-6; weight_kg = V_mm³ × density_g/cm³ × 1e-6; size = |max − min|

  IFC export of parts, assemblies and library items ifc-export.js

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  64-bit

• Volume and area of a library item's body

  Volume, surface area and bounding box of a library item's mesh for its quantities.

  V = |Σ a·(b × c) ÷ 6|; A = Σ |(b − a) × (c − a)| ÷ 2

  IFC export of parts, assemblies and library items ifc-export.js

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  64-bit

• Placement of an assembly instance

  Turns an instance's 4 × 4 matrix into an IFC axis placement: location, local z axis and local x axis.

  location = C(column 3); Axis = C(column 1); RefDirection = C(column 0); C(x, y, z) = (x, −z, y)

  IFC export of parts, assemblies and library items ifc-export.js

  • PartCraft
  64-bit

• Extrude height and start

  The height of the extruded tool and where it starts along the sketch normal, for blind, two-sided, mid-plane and through-all ends.

  two-sided: h = depth + depth₂; mid-plane: z₀ = −h ÷ 2; through all: h = extent + 1 + 0.01 × (max − min)

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Revolve profile in axis coordinates

  Turns each profile point into a radius from the axis and a height along it before the profile is revolved.

  r = max(0, d·n); z = d·u; d = p − a, u = unit axis, n = unit normal to the axis on the profile's side

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Plane, cylinder or cone swept by a profile line

  Classifies the surface a straight profile line sweeps and, for a cone, finds its apex and slope.

  slope = (r₂ − r₁) ÷ (z₂ − z₁); apex height = z₁ − r₁ ÷ slope

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Hole depth

  The depth of a drilled hole, including a through hole's length past the body.

  through: depth = extent on the drilling side + 1 + 0.01 × (max − min)

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Countersink cone height

  The height of a countersink from its diameter, the hole diameter and the included angle.

  h = (R − r) ÷ tan(angle × π ÷ 360)

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Thread zone of a tapped hole

  The inner and outer radius of a tapped hole's thread, used when parts are checked against each other.

  r₀ = min(drill, d − 1.082532 × p) ÷ 2; r₁ = d ÷ 2

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Circular pattern step

  The angle between copies of a circular pattern and the rotation applied to each copy.

  full turn: step = 360 ÷ count; otherwise step = angle ÷ (count − 1); copy i turned by step × i

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Extent of the body along a normal

  How far the body reaches in front of and behind a sketch plane, for through-all cuts and through holes.

  range = min and max of (corner − o)·n over the 8 corners of the bounding box

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Reference stored for a picked flat face

  Chooses the stable face reference for a picked flat triangle, or stores a fixed plane when none matches.

  match: c.n·n > 1 − 1e-6 and |(p − c.o)·c.n| < 1e-4 × max(1, |p|); else o rounded to 9 decimals, n to 12

  Feature tool dimensions on the mesh kernel kernel.js

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  64-bit

• Body mesh read back for named faces and edges

  Copies the current body's vertices and triangles so its faces and edges can be named and measured by later features.

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Fillet tangent points and arc

  Where a round of radius r touches the two faces of an edge, its center, and the points of its arc.

  s_A = ±r × (1 − n_A·n_B) ÷ (u_A·n_B); center = O + u_A × s_A ± n_A × r; angle = acos(v₀·v₁ ÷ r²)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  32-bit in part

• Chamfer distances

  The two setback distances of a chamfer, for equal, two-distance and distance-and-angle modes.

  θ = acos(u_a·u_b); angle mode: d_b = d × sin α ÷ sin(π − α − θ)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Section and prism of a fillet or chamfer

  The cross-section that is extruded along the edge and cut from (or added to) the body.

  section pushed past both faces by 0.01 × size; prism length = edge length

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  32-bit in part

• Shell by offsetting the body

  Hollows the body to a wall thickness, or thickens it outward, and opens the picked flat faces.

  inward: body − (body ⊖ ball(t)); outward: (body ⊕ ball(t)) − body

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  64-bit

• Draft: new normal and hinge line

  The plane a drafted face turns to, and the line where the face meets the neutral plane.

  n₂ = unit(n_perp × cos a + pull × sin a); q = n × (d₁ − d₂ n·m) ÷ (1 − (n·m)²) + m × (d₂ − d₁ n·m) ÷ (1 − (n·m)²)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  64-bit

• Draft: outline of the face

  The outline of the drafted face, extruded into the prism that is trimmed by the old and new planes.

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Thread: extent of the round face

  Where a picked cylinder face starts and ends along its axis and whether it is a shaft or a bore, which set the thread's start, length and callout.

  t₀, t₁ = min and max of (p − o)·d over the face's vertices; full length = t₁ − t₀

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Mesh body from an imported file

  Scales an imported mesh to millimeters, turns it Y up and makes it a solid body.

  (x, y, z) → (x × s, z × s, −y × s) for a Z-up file; s = unit factor

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Matching tolerance and chord sag

  The distance within which a triangle's corners count as lying on a surface.

  sag = 1.5 × (1 − cos(π ÷ max(8, segments))); tol = 2e-6 × diagonal + 1e-7

  Named faces and edges from the body mesh meshfaces.js

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  64-bit

• Triangle on a surface

  Decides whether a triangle lies on a plane, cylinder, cone, sphere, torus or thread zone.

  plane: |(p − o)·n| ≤ tol; cylinder: −(tol + sag × r) ≤ ρ − r ≤ tol + 1e-6 × r

  Named faces and edges from the body mesh meshfaces.js

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  64-bit

• Face area and center

  The area of each named face and its area-weighted center.

  area = Σ |(b − a) × (c − a)| ÷ 2; center = Σ (triangle center × area) ÷ area

  Named faces and edges from the body mesh meshfaces.js

  • PartCraft
  32-bit in part

• Edge length, ends and middle

  The length of an edge between two faces, its two ends and its middle point.

  len = Σ |pᵢ − pᵢ₋₁|; middle = the point at len ÷ 2 along the polyline

  Named faces and edges from the body mesh meshfaces.js

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  32-bit in part

• Straight edge or circle

  Classifies an edge as a line, a circle or a curve and gives a circle its center, axis and radius.

  line: every point within 10 × tol of the chord; circle: every point at one radius and one height about a round face's axis

  Named faces and edges from the body mesh meshfaces.js

  • PartCraft
  32-bit in part

• Circle edge points put on the circle

  Moves a circular edge's ends and middle exactly onto its circle.

  p′ = c + r × w ÷ |w|, w = (p − c) − n × ((p − c)·n); closed circle: middle = 2c − a

  Named faces and edges from the body mesh meshfaces.js

  • PartCraft
  32-bit in part

• Surfaces of pattern and mirror copies

  Moves a tool's named planes, axes and centers with the matrix of a pattern or mirror copy.

  point′ = M × point; direction′ = unit(linear part of M × direction)

  Named faces and edges from the body mesh meshfaces.js

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  64-bit

• Circle of an edge

  Center, radius and axis of a circular edge from its two ends and its middle point.

  c = A + ((n × ab) |ac|² + (ac × n) |ab|²) ÷ (2 |n|²), n = ab × ac; full circle: c = (a + middle) ÷ 2, r = |a − middle| ÷ 2

  Measuring and mass properties in 3D measure.js

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• Area of a picked group of facets

  The area and normal of a picked face when no face table is available.

  area = Σ |(b − a) × (c − a)| ÷ 2; plane if |Σ cross| > 0.999 × 2 × area

  Measuring and mass properties in 3D measure.js

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  32-bit in part

• Closest point on a triangle

  The point of a facet nearest to a given point, for the fallback distance.

  Measuring and mass properties in 3D measure.js

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• Closest points of two segments

  The nearest pair of points on two mesh edges, for the fallback distance.

  s = (b·f − c·e) ÷ (a·e − b²), t = (b·s + f) ÷ e, clamped to 0…1

  Measuring and mass properties in 3D measure.js

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• Minimum distance over the facets

  The smallest distance between two picked entities when the exact kernel's distance is not available.

  min over point–point, point–triangle, point–segment and segment–segment pairs

  Measuring and mass properties in 3D measure.js

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  32-bit in part

• Distance, angle and gaps between two picks

  Point-to-point distance, the angle between planes, lines or axes, the gap between parallel planes and the distance between centers or parallel axes.

  angle = acos(clamp(d₁·d₂)) in degrees; plane gap = |(o₂ − o₁)·n|; axis gap = |w − d (w·d)|

  Measuring and mass properties in 3D measure.js

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  64-bit