Calculations

Calculations

• 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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• Automatic element size

  A tidy element size that gives about the target number of elements for a solid of a given volume.

  raw = ∛(volume ÷ target); size = next step ≥ raw among 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 × 10ⁿ

  Study set-up: element size and the mesh handed to the solver sim.js

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• Estimated element count

  The element size a study will use and the number of elements that size is expected to give.

  estimate = round(volume ÷ h³) (× 12 for tetrahedra)

  Study set-up: element size and the mesh handed to the solver sim.js

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• Mesh copy for the solver

  Copies the body's triangle mesh for the worker that meshes and solves the study.

  Study set-up: element size and the mesh handed to the solver sim.js

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• Design box and size of a generative study

  The design space of a generative study (the body's box grown by a margin), its volume, element size and estimated element count.

  box = bbox ± margin; volume = Δx × Δy × Δz (box) or the body's volume; estimate = round(volume ÷ h³)

  Study set-up: element size and the mesh handed to the solver sim.js

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• Point on a curve at a parameter

  The point of a line, circle, arc or spline at a given parameter.

  line: a + (b − a)·t; circle: c + r·(cos t, sin t); arc: c + r·(cos(a₀ + t), sin(a₀ + t))

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Parameter of the nearest point

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

  line: t = ((p − a) · d) ÷ (d · d); circle and arc: angle = atan2(p_y − c_y, p_x − c_x)

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Line and line

  The parameters at which two lines cross.

  den = d₁ × d₂; tₐ = (w × d₂) ÷ den; t_b = (w × d₁) ÷ den, w = B.a − A.a

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Line and circle

  The parameters along a line at which it crosses a circle.

  A·t² + B·t + C = 0; q = −½·(B + sign(B)·√(B² − 4AC)); t₁ = q ÷ A, t₂ = C ÷ q

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Circle and circle

  The points where two circles cross.

  a = (r₁² − r₂² + d²) ÷ 2d; h = √(r₁² − a²); points = c₁ + a·û ± h·û⊥

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Is a crossing on the bounded curve

  Decides whether a crossing lies on the drawn part of a line or arc, and snaps it to the end when it is within the tolerance.

  line: −eps ÷ L ≤ t ≤ 1 + eps ÷ L; arc: angle ≤ sweep + eps ÷ r

  Sketch intersections of lines, arcs and circles sketch-intersect.js

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• Crossing points of two curves

  All crossing points of two sketch curves, sorted along the first.

  eps = 1e-9 × max(1, size of A, size of B); duplicates within 10 × eps removed

  Sketch intersections of lines, arcs and circles sketch-intersect.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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• Knot span of a parameter

  Finds which knot span a parameter falls in.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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  Exact

• Basis functions

  The values of the B-spline basis functions that are not zero at a parameter.

  t = N[r] ÷ (right[r+1] + left[j−r]); N[r] = saved + right[r+1] × t; saved = left[j−r] × t (Cox–de Boor)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Basis function derivatives

  The basis functions and their derivatives up to a given order at a parameter.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Point on a spline

  The point of a spline at a parameter.

  C(u) = Σⱼ Nⱼ(u) × Pⱼ (degree + 1 terms)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Derivatives of a spline

  The point, tangent and higher derivatives of a spline at a parameter.

  C⁽ⁱ⁾(u) = Σⱼ Nⱼ⁽ⁱ⁾(u) × Pⱼ

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Chord-length parameters

  Gives each fit point a parameter equal to the summed straight distances up to it.

  tᵢ = tᵢ₋₁ + |Qᵢ − Qᵢ₋₁|

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Dense linear solve

  Solves the linear system that gives a fit spline its poles.

  A · X = B by Gauss–Jordan elimination with partial pivoting

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Spline through fit points

  The cubic spline that passes through the given points, open with natural or given end directions, or closed and periodic.

  rows: C(tᵢ) = Qᵢ; ends: C″ = 0 or C′ = (cos a, sin a); solve for the poles

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Area term of a spline

  The area term of a spline by Green's formula: the enclosed area when the spline is closed.

  ½ ∫ (x·y′ − y·x′) du, 3-point Gauss–Legendre per knot span (nodes ±√(3 ÷ 5), 0; weights 5/9, 8/9, 5/9)

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Curvature of a spline

  The signed curvature of a spline at a parameter.

  κ = (x′·y″ − y′·x″) ÷ |C′|³

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Knot insertion

  Adds one knot to a spline without changing its shape.

  a = (u − Uᵢ) ÷ (Uᵢ₊ₚ − Uᵢ); Qᵢ = (1 − a)·Pᵢ₋₁ + a·Pᵢ

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Exact piece of a spline

  The part of a spline between two parameters as a curve of its own, used by trim.

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.js

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• Knots rescaled to start at zero

  Shifts and scales a spline's knots so that they start at 0 and span the number of its pieces.

  u′ = (u − a) × (poles − degree) ÷ L

  Sketch splines: evaluation, fitting and exact pieces sketch-spline.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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• Sketch tolerance

  The base length tolerance of the sketch tools.

  tolOf = 1e-7 × max(1, extent of the sketch)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Crossings of a curve with the others

  Where a curve is cut by every other curve of the sketch, for trim.

  crossings within tol of the curve's own ends, or of one already found, are left out; tol = 1e-4 × max(1, extent)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Extending a line or arc

  Moves the picked end of a line or arc to the nearest curve it meets when continued.

  line: gap = (t − 1) × L or −t × L; arc: gap = (t − sweep) × r or (2π − t) × r; nearest gap > tol wins

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Offset of each curve

  The offset copy of each line, circle, arc or spline of a chain, before the corners are joined.

  line: S = a + d·n, E = b + d·n, n = (−Δy, Δx) ÷ L; circle: r − d; arc: r ∓ d, ends c + r·(cos, sin)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Corners of an offset chain

  Joins neighboring offset curves at the point where they meet when extended.

  corner = the intersection of the two full lines or circles nearest the midpoint of the gap

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Mirror about a line

  Reflects points about a sketch line, and reflects a spline's end angles.

  t = (p − a) · u; q = a + t·u; p′ = 2q − p; angle′ = 2φ − angle

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Linear pattern

  Copies geometry a number of times along a step.

  copy i: (x + Δx × i, y + Δy × i)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Circular pattern

  Copies geometry a number of times around a center.

  step = (360 ÷ count, or angle ÷ (count − 1)) × π ÷ 180; copy i: rotate about c by step × i

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Regular polygon

  The corners of a regular polygon from its center, one corner and the number of sides.

  cornerᵢ = c + r × (cos(a₀ + 2π·i ÷ n), sin(a₀ + 2π·i ÷ n))

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Straight slot

  The four corner points of a slot from its two centers and its width.

  r = w ÷ 2; corners = center ± n × r, n = unit normal of the center line

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Projecting a model edge

  Puts a straight or circular model edge onto the sketch plane as fixed geometry.

  (u, v) = ((p − o) · x, (p − o) · y)

  Sketch tools: trim, extend, offset, mirror, pattern, polygon, slot, project sketch-tools.js

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• Radius of a circle or arc

  The radius of a circle (stored) or of an arc (the distance from its center to its start point).

  r = |s − c|

  Sketch geometry and measured values sketch.js

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• Arc angles

  The center, radius, start angle and sweep of an arc from its three points.

  a₀ = atan2(s − c); sweep = atan2(e − c) − a₀, brought to (1e-12, 2π]

  Sketch geometry and measured values sketch.js

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• New arc: end point on the radius

  Places a new arc's end point on the circle through its start point.

  r = |s − c|; e = c + r × (cos a, sin a), a = atan2(e₀ − c)

  Sketch geometry and measured values sketch.js

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• Current value of a relation or dimension

  The value a new dimension starts with: a distance, a horizontal or vertical distance, a radius, a diameter, an offset or an angle, and which side it is on.

  distance = |b − a|; point to line = |signed distance|; diameter = 2r; angle° = |atan2(u × v, u · v)| × 180 ÷ π

  Sketch geometry and measured values sketch.js

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