Calculations

Calculations

• Developed length and bend lines

  The total flat length of a bent profile and where each bend line lies on the flat.

  length = Σ straights + Σ BA; bend line at x + BA ÷ 2; angle° = θ × 180 ÷ π

  Sheet metal bends and flat pattern sheetmetal.js

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• Joining straight segments

  Joins two straight segments of the flat outline that continue each other through a point no third segment touches.

  collinear when |d₁ × d₂| ≤ 1e-9 × |d₁| × |d₂| and d₁ · d₂ > 0

  Sheet metal bends and flat pattern sheet-flat.js

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• Points along an arc for the extent

  Sampled points on an arc or circle of the flat, used to find its box.

  point = c + r × (cos a, sin a), a = (a₀ + sweep × k ÷ n) × π ÷ 180

  Sheet metal bends and flat pattern sheet-flat.js

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• Flat of a bent base

  The developed rectangle of an open bent profile: flat length by depth, with its bend lines and area.

  box = L × D; area = L × D

  Sheet metal bends and flat pattern sheet-flat.js

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• Faces that lie on the plate's plane

  Decides which flat faces of the body belong to the plate, by distance from the sketch plane and by direction.

  size = max(1, largest distance of a face centroid from the origin); tol = 1e-6 × size; on the plane when |n · (c − o)| ≤ tol and | |n_face · n| − 1 | ≤ 1e-9

  Sheet metal bends and flat pattern sheet-flat.js

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• Unfolding a flange

  Turns a flange's leg back into the plate's plane and adds the bend strip between the plate and the leg.

  leg: rotate by −θ about the bend axis through C, then move by BA along u; strip: S₀, S₀ + e·run, + u·BA; bend line at S₀ + u × BA ÷ 2

  Sheet metal bends and flat pattern sheet-flat.js

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• Outline edges as lines, arcs and circles

  Reads each edge of the fused flat as a 2-D line, arc, circle or sampled curve in the sketch plane.

  2-D point = ((p − o) · x, (p − o) · y); arc angles = atan2(Δy, Δx) × 180 ÷ π, brought to [0, 360)

  Sheet metal bends and flat pattern sheet-flat.js

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• Box and area of the flat pattern

  The overall size of the flat pattern and its area.

  box = min and max over line ends, sampled curve points and 65 points per arc; area = surface area of the fused flat

  Sheet metal bends and flat pattern sheet-flat.js

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• Roots of a function along a span

  Finds the parameters where a signed distance along the spline is zero.

  sample n steps; bisect each sign change until b − a ≤ 1e-16 × max(1, |a|)

  Sketch intersections with splines sketch-intersect.js

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• Spline against a line or circle

  Collects the spline parameters where it crosses a line or a circle, span by span.

  24 samples per knot span of F(S(u)); line: F = signed distance; circle: F = |S − c| − r

  Sketch intersections with splines sketch-intersect.js

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• Spline against spline

  The parameter pairs at which two splines cross.

  Newton: [S₁′(u) −S₂′(v)] · (Δu, Δv) = −(S₁(u) − S₂(v))

  Sketch intersections with splines sketch-intersect.js

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• Polyline within a chord tolerance

  Points along a spline such that every chord stays within a tolerance of the curve.

  split a piece while the curve at ¼, ½ or ¾ of it is farther than tol from its chord

  Sketch splines: length, nearest point, polyline and offset sketch-spline.js

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• Length of a spline

  The arc length of a spline.

  L = Σ over 8 pieces per span of Σ wᵢ × h × |C′(uᵢ)| (5-point Gauss–Legendre)

  Sketch splines: length, nearest point, polyline and offset sketch-spline.js

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• Nearest point on a spline

  The parameter of the point on a spline nearest a given point.

  start at the nearest of 13 samples per span; Newton: u ← u − ((C − p) · C′) ÷ (C′ · C′ + (C − p) · C″)

  Sketch splines: length, nearest point, polyline and offset sketch-spline.js

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• Offset of a spline

  A new fit spline that follows a spline at a fixed distance, with the largest gap found.

  true offset point = C(u) + d × n(u); err = max over 40 points per span of the distance to the fitted curve; accept when err ≤ tol

  Sketch splines: length, nearest point, polyline and offset sketch-spline.js

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• Points along an arc

  An arc as a run of points from its start to its end.

  n = max(1, ⌈sweep ÷ 2π × segFull − 1e-9⌉); pointᵢ = c + r × (cos(a₀ + sweep·i ÷ n), sin(…))

  Sketch profiles as closed polygons sketch.js

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• Points around a circle

  A circle as a closed polygon with its corners on the circle.

  pointᵢ = c + r × (cos(2π·i ÷ n), sin(2π·i ÷ n))

  Sketch profiles as closed polygons sketch.js

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• Closed loops of a sketch

  Chains the sketch's curves into closed loops, or reports an open end or a branch.

  ends are one node when closer than 10 × tol, tol = 1e-6 × max(1, extent)

  Sketch profiles as closed polygons sketch.js

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• Point inside loops

  Tests whether a point lies inside a set of loops by the even-odd rule.

  count edges with (a_y > p_y) ≠ (b_y > p_y) and p_x < (b_x − a_x)·(p_y − a_y) ÷ (b_y − a_y) + a_x

  Sketch profiles as closed polygons sketch.js

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• Area of a loop

  The signed area of a polygon loop.

  area = ½ × Σ (xⱼ·yᵢ − xᵢ·yⱼ)

  Sketch profiles as closed polygons sketch.js

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• Jacobian by central differences

  The rate of change of every relation's residual with each variable it depends on.

  ∂f/∂x ≈ (f(x + h) − f(x − h)) ÷ 2h, h = 1e-6 × max(1, |x|)

  Sketch solver solver.js

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• Cholesky solve

  Solves the symmetric system of each Newton step.

  A = L·Lᵀ; L·y = b; Lᵀ·z = y

  Sketch solver solver.js

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• Damped Newton iteration

  Repeats least-norm steps until every residual is within the tolerance.

  dx = Jᵀ · (J·Jᵀ + μ·max(scale, 1e-12)·I)⁻¹ · (−r); x ← x + α·dx, α halved until Σr² falls

  Sketch solver solver.js

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• Solving a sketch

  Solves the sketch in place and classifies it as under, fully or over defined.

  tol = 1e-9 × max(1, extent)

  Sketch solver solver.js

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• Writing the solution back

  Copies the solved variables to the sketch's points, radii and spline end angles.

  angle ← atan2(sin a, cos a)

  Sketch solver solver.js

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• Rank, redundancy and determined geometry

  Counts the degrees of freedom left, names relations that repeat or conflict, and marks which entities are fully defined.

  dof = variables − rank; row redundant when its remainder after projection is under 1e-7; variable determined when Σ q_k² > 1 − 1e-6

  Sketch solver solver.js

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• Residual: an arc's two ends on one radius

  The equation that keeps an arc's end point at the same distance from the center as its start point.

  f = |e − c| − |s − c|

  Sketch solver sketch.js

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• Residuals: parallel and perpendicular

  The equations for two lines being parallel or square to each other.

  parallel: (u × v) ÷ (|u|·|v|); perpendicular: (u · v) ÷ (|u|·|v|)

  Sketch solver sketch.js

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• Residuals: tangent

  The equations for a line touching a circle or arc, and for two circles or arcs touching.

  shared end, line and arc: (u · (p − c)) ÷ (|u|·r); shared end, two arcs: ((p − a) × (p − b)) ÷ (rₐ·r_b); else: ±dist(c, line) − r, or d − (rₐ + r_b), or d − ±(rₐ − r_b)

  Sketch solver sketch.js

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• Residuals: distance dimensions

  The equations that hold a distance between two points, the length of a line, or the distance from a point or a line to a line.

  f = |b − a| − value; f = sign × signed distance − value

  Sketch solver sketch.js

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• Residual: angle dimension

  The equation that holds the angle between two lines at a value in degrees.

  f = atan2(u × v, u · v) − sign × value × π ÷ 180, wrapped to (−π, π]

  Sketch solver sketch.js

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• Normal distribution tail

  The probability that a standard normal value exceeds z, used for the share of assemblies outside a limit.

  P(Z > z) = ½ × erfc(z ÷ √2); erfc(x) ≈ k × exp(−x² − 1.26551223 + k·(1.00002368 + …)), k = 1 ÷ (1 + ½|x|)

  Tolerance stack-ups stackup.js

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• Stack-up analysis

  The gap's nominal and mean, its worst-case and statistical limits, each link's share and rank, and the check against the limits.

  a = sign × sens; m = nominal + (up + lo) ÷ 2; t = (up − lo) ÷ 2; WC = Σ|a|·t; RSS = √Σ(a·t)²; σ = RSS ÷ 3; ppm = (P_below + P_above) × 10⁶

  Tolerance stack-ups stackup.js

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• Links from picked points

  Makes the links of a chain from points picked in order, measured along one direction.

  Δs = d · b − d · a; nominal = round(|Δs|, 9 decimals); sign = sign of Δs

  Tolerance stack-ups stackup.js

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• Rounding of measured nominals

  Rounds a measured length or direction component before it is stored on a link.

  round(v) = Math.round(v × 10⁹) ÷ 10⁹

  Tolerance stack-ups stackup.js

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• Vector stack projected on a direction

  Turns a chain whose links run in different directions into a one-direction chain along an output direction, with each link's angular swing as an extra link.

  sens = (u · d) × sens; swing: c = nominal × ((axis × u) · d); up, lo = c × angle° × π ÷ 180

  Tolerance stack-ups stackup.js

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• Vector stack analysis

  The nominal gap as a vector and its length, and the analysis along each output direction and along the gap itself.

  gap = Σ sign × nominal × u; |gap| = √(g_x² + g_y² + g_z²)

  Tolerance stack-ups stackup.js

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• Links from picked points in 3-D

  Makes links that run straight from each picked point to the next.

  nominal = round(|b − a|, 9 decimals); dir = round((b − a) ÷ |b − a|, 9 decimals)

  Tolerance stack-ups stackup.js

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• Tolerance from a feature control frame

  How a geometric tolerance enters a stack: plus and minus half the tolerance zone.

  up = tol ÷ 2; lo = −tol ÷ 2

  Tolerance stack-ups stackup.js

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• General tolerance for a nominal size

  The deviation a title block's general tolerance gives one nominal size: a class of ISO 2768-1, or a typed ± value.

  class f, m, c or v: the table row for the first range with nominal ≤ upper end (0.5 to 4000 mm); typed: parseFloat of the number after ±

  Tolerance stack-ups stackup.js

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• Stack-up as CSV

  Writes the links and the results of a stack-up to a CSV file.

  every number written as +v.toFixed(4)

  Tolerance stack-ups stackup.js

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• Sizes of a template

  One row of parameter values (mm) per size of a standard part family.

  value = Number(value.toFixed(4))

  Standard part templates stdparts.js

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• Hexagon across flats

  Draws a hexagon from its across-flats size and holds that size with a dimension.

  e = s × 2 ÷ √3; first corner at (e ÷ 2, 0)

  Standard part templates stdparts.js

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• Circle of a given diameter

  Draws a circle about the origin and holds its diameter with a dimension.

  r = d ÷ 2

  Standard part templates stdparts.js

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• Parallel key outline

  The round-ended outline of a parallel key from its width and length.

  half = (l − b) ÷ 2; center distance = l − b; end radius = b ÷ 2

  Standard part templates stdparts.js

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• Patch through boundary edges

  A surface that fills a closed loop of edges, optionally tangent to the faces next to it.

  MakeFilling(3, 15, 2, false, 1e-5, 1e-4, 0.01, 0.1, 8, 9)

  Surface features built with the kernel surface.js

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• Loft through wires

  A surface or solid through a series of curves in order.

  ThruSections(solid, ruled, 1e-6)

  Surface features built with the kernel surface.js

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• Thickening a surface

  Turns a surface body into a solid of a given thickness on one side or half on each side.

  both sides: offset the sheet by −t ÷ 2, then thicken by t; in: thicken by −t

  Surface features built with the kernel surface.js

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• Offsetting a surface

  A new surface at a distance along the normals of an existing one.

  MakeOffsetShape.PerformBySimple(sheet, d)

  Surface features built with the kernel surface.js

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• Stitching surfaces

  Sews surface bodies together along their edges, into a solid when they close.

  edges closer than tol are joined; tol = 0.01 by default

  Surface features built with the kernel surface.js

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