Calculations

Calculations

• Faces in the cutting plane

  Finds the faces created by the cut, whose outlines bound the hatched area.

  ||n · d| − 1| ≤ 1e-9 and |(p − o) · d| ≤ 1e-6 × diagonal

  Drawing views of the exact solid brep-draw.js

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• Cutting-plane line in the neighboring view

  The point and direction of the line that shows where the section was cut.

  along = unit(d_section × d_view); p = (o · right, o · up); dir = (along · right, along · up)

  Drawing views of the exact solid brep-draw.js

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• Number of steps round a circle

  How many interpolation points a round direction gets, from its effective radius.

  h = (384 × DEV ÷ R)^¼; n = ceil(2π ÷ h) rounded up to a multiple of 4, between 32 and 720

  Non-uniform scale of an exact body brep-gtrsf.js

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• Cubic spline through the unit circle

  The poles of a periodic cubic B-spline that passes through the circle at every knot.

  pole i = k × (cos((i − 1)h), sin((i − 1)h)), k = 6 ÷ (4 + 2 cos h), err = h⁴ ÷ 384

  Non-uniform scale of an exact body brep-gtrsf.js

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• Largest stretch of the map

  The largest factor by which the scale lengthens any direction, used to size the deviation.

  σ = |M x| after 60 power iterations of x ← unit(Mᵀ M x)

  Non-uniform scale of an exact body brep-gtrsf.js

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• Image of a curve

  A line, circle, ellipse, Bézier or B-spline curve after the scale, as a B-spline in the same parameter.

  line, Bézier, B-spline: poles ↦ T × pole (exact); circle, ellipse: T × (c + r₁ x cos + r₂ y sin) at the spline's poles

  Non-uniform scale of an exact body brep-gtrsf.js

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• Image of a surface

  A plane, cylinder, cone, sphere, torus, extrusion, revolution or spline surface after the scale.

  plane ↦ plane with frame from T x, T x × T y; cylinder, cone: ring(u) × (R + v sin α) + v cos α Z; sphere, torus: ring(u) × (R₁ + R₂ cos v) + R₂ sin v Z

  Non-uniform scale of an exact body brep-gtrsf.js

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• Inverse of a placement

  Inverts a 3 × 4 placement matrix found in the shape.

  M⁻¹ = adj(M) ÷ det(M); t⁻¹ = −M⁻¹ t

  Non-uniform scale of an exact body brep-gtrsf.js

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• Parameter range of straight directions

  How far a line or the straight direction of a surface must extend to cover the body.

  t₀ = ((center − o) · d) ÷ |d|²; range = t₀ ± (1.02 × D ÷ |d| + 1)

  Non-uniform scale of an exact body brep-gtrsf.js

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• Raised shape tolerances

  The tolerance written on every vertex, edge and face of the scaled body.

  tol = max(old, max(1e-7, 3 × largest deviation + 1e-9))

  Non-uniform scale of an exact body brep-gtrsf.js

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• Numbers written to the shape text

  Turns each coordinate, weight and knot into the text OpenCascade reads back.

  text = String(v); −0 → 0

  Non-uniform scale of an exact body brep-gtrsf.js

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• Matching equal placements

  Decides when two uses of the same curve or surface share one mapped copy.

  key = entries rounded to 1e-12, joined

  Non-uniform scale of an exact body brep-gtrsf.js

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• Rib slab outline

  The flat region the rib is cut from: the sketch chain carried past both ends and offset to one side.

  E = 4 × max(1, diagonal + |box center − sketch origin|); ends ± E along the end directions; far side = + E × D

  Rib feature brep-rib.js

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• Choosing the rib piece

  Picks the piece of the trimmed slab that lies beside the sketch line and checks it is closed off by the body.

  distance(probe, piece) < 1e-6 × max(1, t); open when reach > 0.5 × E

  Rib feature brep-rib.js

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• Edge flange frame

  The axes a flange is built in: along the edge, out of the sheet's face, and out of its side.

  e = unit(B − A); square when |n_big · n_side| ≤ 1e-6; P = (A + B) ÷ 2 − v × t ÷ 2 must lie on the side face

  Sheet metal base and edge flanges brep-sheet.js

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• Flange cross-section

  The corner points of the bend and the leg in the plane square to the edge.

  C = (0, r); point(ρ, φ) = C + ρ (cos φ, sin φ); φ from −π ÷ 2 to −π ÷ 2 + θ; leg end = arc end + length × (−sin φ₁, cos φ₁)

  Sheet metal base and edge flanges brep-sheet.js

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• Bend relief slot

  The box cut beside an offset flange end so the bend can form.

  depth into the sheet = r + t; box length along u = (r + t) + 1e-3 × max(1, r + t); through 2 × t

  Sheet metal base and edge flanges brep-sheet.js

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• Side of the first bend

  Which side of an open profile the sheet's thickness goes on.

  turn = sign((p₁ − p₀) × (p₂ − p₁))

  Sheet metal base and edge flanges brep-sheet.js

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• Cone from its derivatives

  A cone's axis point, axis direction, radius and slope from the surface's derivatives at one point.

  ρ = |P_uu|; e = −P_uu ÷ ρ; z = P_v − (P_v · e) e; axis point = P − ρ e; slope k = (P_v · e) ÷ |z|

  Exact surface records for meshing brep-surf.js

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• Torus from its derivatives

  A torus's center, axis and two radii from the surface's derivatives at one point.

  ρ = |P_uu|; A = P − ρ e; d = unit(e × P_u); tube center M = P + P_vv; c = A + ((M − A) · d) d; R = |M − c|; r = |P_vv|

  Exact surface records for meshing brep-surf.js

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• Check against the kernel's face

  Confirms the record describes the same surface as the kernel's face before it is used.

  25 points on a 5 × 5 grid (2 % to 98 % of the parameter box); each within 1e-9 × size of the record's surface

  Exact surface records for meshing brep-surf.js

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• Closed loops with their curves

  Collects a sketch's curves into closed loops, keeping each curve's exact definition.

  tol = 1e-6 × max(1, extent); ends within 10 × tol share a node; arc mid = c + r (cos, sin)(a₀ + sweep ÷ 2)

  Sketch profiles for exact solids brep.js

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• Arc edge through three points

  Builds an arc of the profile in the kernel.

  arc through a (start node), mid, b (end node)

  Sketch profiles for exact solids brep.js

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• Loop orientation

  Whether a loop runs counterclockwise, from the sign of its area.

  A = ½ Σ (x_j y_i − x_i y_j)

  Sketch profiles for exact solids brep.js

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• Point just inside a loop

  A test point a small step inside the loop's longest edge, used to decide nesting.

  p = midpoint of longest edge + normal × 1e-6 × max(1, edge length)

  Sketch profiles for exact solids brep.js

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• Which loops are holes

  Counts how many other loops contain each loop; odd depth is a hole.

  depth_i = number of loops whose polygon contains innerPoint(loop_i); face when depth is even

  Sketch profiles for exact solids brep.js

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• Stored edge and face reference points

  The point saved with a picked edge or face so it can be found again after a rebuild.

  at = coordinates rounded to 6 decimals

  Sketch profiles for exact solids brep.js

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• Extrude depth and start

  Where an extrude starts and how far it runs for blind, mid-plane and through-all ends.

  blind: z₀ = 0 or −h; mid: z₀ = −h ÷ 2; through: h = far side + 1 + 0.01 × body range

  Exact-kernel feature dimensions brep.js

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• Revolve axis checks

  Checks that the axis lies in the sketch plane and that the profile stays on one side of it.

  side s = −u_y (p_x − a_x) + u_x (p_y − a_y); crosses when some s > 1e-7 × scale and some s < −1e-7 × scale

  Exact-kernel feature dimensions brep.js

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• Revolve angle

  The sweep angle given to the kernel.

  full turn when |angle − 360| < 1e-9; else angle × π ÷ 180

  Exact-kernel feature dimensions brep.js

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• Through-hole depth

  How deep a through hole is drilled.

  depth = distance to the body's far side + 1 + 0.01 × body range

  Exact-kernel feature dimensions brep.js

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• Countersink depth

  How deep the cone of a countersunk hole goes.

  h = (D_csink − d) ÷ 2 ÷ tan(angle × π ÷ 360)

  Exact-kernel feature dimensions brep.js

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• Cosmetic thread radii

  The major and minor radius recorded for a tapped hole's thread.

  r₁ = D ÷ 2; r₀ = min(tap drill, D − 1.082532 × P) ÷ 2

  Exact-kernel feature dimensions brep.js

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• Linear pattern positions

  Where each copy of a linear pattern sits.

  copy i moved by direction × spacing × i

  Exact-kernel feature dimensions brep.js

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• Circular pattern angles

  The angle of each copy of a circular pattern.

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

  Exact-kernel feature dimensions brep.js

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• Shell wall offset

  Hollows a body to a wall thickness.

  offset = +t (outward) or −t; kernel tolerance 1e-3

  Exact-kernel feature dimensions brep.js

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• Fillet and chamfer sizes

  The radius, distances or distance and angle given to the kernel for each picked edge.

  two distances: (d, d₂); distance and angle: (d, angle × π ÷ 180); fillet: r, or (r, r₂) along the edge

  Exact-kernel feature dimensions brep.js

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• Sewing an imported surface model

  Joins the faces of a STEP file that has no solid into a closed shell.

  sewing tolerance = 1e-6

  Exact-kernel feature dimensions brep.js

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• Mesh tolerances

  How closely the triangle mesh follows the exact faces.

  linear = 2e-4 × bounding-box diagonal; angular = 2π ÷ max(16, resolution)

  Exact solid: mesh, volume, area and tables brep.js

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• Mesh read from the kernel

  Copies the mesh's vertices, normals and triangles out of the kernel's memory.

  vertices and normals: Float32Array view of the WebAssembly buffer

  Exact solid: mesh, volume, area and tables brep.js

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• Welding the mesh

  Merges vertices that different faces share so the mesh is watertight.

  same vertex when x, y and z are identical

  Exact solid: mesh, volume, area and tables brep.js

  • PartCraft
  32-bit in part

• Volume and surface area

  The solid's volume and total surface area, from the exact faces.

  V = ∭ dV (Gauss–Kronrod, tolerance 1e-12, when a non-line, non-circle edge exists); A = ∬ dA

  Exact solid: mesh, volume, area and tables brep.js

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• Genus

  The number of through holes (handles) of the solid, from the welded mesh.

  χ = V − E + F; genus = round(bodies − χ ÷ 2)

  Exact solid: mesh, volume, area and tables brep.js

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• Face area and center

  Each face's area and center point, listed with the face.

  area = ∬ dA over the face; center = its centroid

  Exact solid: mesh, volume, area and tables brep.js

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• Edge length and points

  Each edge's length, end points and middle point.

  length = ∫ |C′(t)| dt; middle = C((t₀ + t₁) ÷ 2)

  Exact solid: mesh, volume, area and tables brep.js

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• Bounding box

  The smallest axis-aligned box around a shape.

  min, max per axis from the kernel's optimal box

  Exact solid: mesh, volume, area and tables brep.js

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• Facts about an imported STEP file

  The solid count, face count, volume and bounding box shown when a STEP file is imported.

  volume = ∭ dV (default integration)

  Exact solid: mesh, volume, area and tables brep.js

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• Stress at the integration points

  The six stress components at every integration point of every element, from the displacements.

  σ = D (ε − α ΔT I) per point, by the element's own routine

  Buckling load factors and thermal stress setup buckling.js

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• Geometric stiffness times a vector

  Applies the stress-dependent stiffness Kσ to a vector, element by element.

  y = s × Σ_elements G_e x_e on each of the three directions

  Buckling load factors and thermal stress setup buckling.js

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• Buckling factors by subspace iteration

  The lowest positive buckling load factors and their mode shapes.

  Z = K⁻¹ G X; reduce (Zᵀ G Z, Zᵀ K Z); Kr = L Lᵀ; eig(L⁻¹ Gr L⁻ᵀ) = μ; λ = 1 ÷ μ

  Buckling load factors and thermal stress setup buckling.js

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