Calculations

Calculations

• Scale matrix about a fixed point

  The 4 × 4 matrix that scales a body about the center, minimum corner, base center or origin of its box.

  a = (min + max) ÷ 2 (center); translation = a × (1 − f)

  Scale feature for printing print-features.js

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  64-bit

• Scaled triangles

  Applies the scale to every corner of a body's triangles, with the box found from the triangles themselves.

  x′ = f_x·x + t_x; y′ = f_y·y + t_y; z′ = f_z·z + t_z

  Scale feature for printing print-features.js

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  64-bit

• Rotation about an axis

  Turns a vector about a unit axis by an angle; used for a plane at an angle to another.

  v′ = v·cos a + (u × v)·sin a + u·(u · v)·(1 − cos a)

  Reference planes, axes and points refgeom.js

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  64-bit

• Point where three planes meet

  The single point that lies on three planes.

  solve [nₐ; n_b; n_c] · p = (nₐ·oₐ, n_b·o_b, n_c·o_c)

  Reference planes, axes and points refgeom.js

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  64-bit

• Axis where two planes meet

  The line of intersection of two planes: its direction and a point on it.

  d = nₐ × n_b; solve nₐ·x = cₐ, n_b·x = c_b, d·x = d·(oₐ + o_b) ÷ 2

  Reference planes, axes and points refgeom.js

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  64-bit

• Offset plane and plane at an angle

  A plane parallel to another at a distance, and a plane turned about an axis by an angle in degrees.

  o′ = o + n·d; n′ = rotate(n ⊥ axis, axis, angle × π ÷ 180)

  Reference planes, axes and points refgeom.js

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  64-bit

• Plane through three points

  The plane through three points, with its center at their mean.

  n = (b − a) × (c − a); center = (a + b + c) ÷ 3

  Reference planes, axes and points refgeom.js

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  64-bit

• Mid plane between two planes

  The plane half way between two planes, or the one that bisects them when they are not parallel.

  n = (nₐ − n_b) ÷ |nₐ − n_b|; c = (oₐ·nₐ − o_b·n_b) ÷ |nₐ − n_b|; parallel and same facing: o = oₐ + n·((o_b − oₐ)·n) ÷ 2

  Reference planes, axes and points refgeom.js

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  64-bit

• Plane tangent to a cylinder

  The plane that touches a cylindrical face on the side of a chosen point.

  radial = w − d·(w · d); n = radial ÷ |radial|; o = foot on the axis + n·r

  Reference planes, axes and points refgeom.js

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  64-bit

• Plane through a line and a point

  The plane that contains a straight edge or axis and one more point.

  n = d × (p − o)

  Reference planes, axes and points refgeom.js

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  64-bit

• Where a line meets a plane or a cylinder

  The point where a straight edge or axis crosses a plane, a flat face or a cylindrical face.

  plane: t = ((o_s − o) · n) ÷ (d · n); cylinder: A·t² + B·t + C = 0, t = (−B ± √(B² − 4AC)) ÷ 2A

  Reference planes, axes and points refgeom.js

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  64-bit

• Slider range of the section

  The lowest and highest value of n·p over the model's box, the two ends of the section slider.

  lo, hi = min, max of n · p over the 8 box corners

  Section view plane section.js

  • PartCraft
  32-bit, display only

• Section through a picked face

  The cutting plane through a flat face, placed by the instance matrix of an assembly.

  n = unit(M · n_face); c = n · (M · o_face)

  Section view plane section.js

  • PartCraft
  32-bit, display only

• Bend allowance

  The length of the neutral layer through a bend, which the flat pattern gives to that bend.

  BA = θ × (r + k × t)

  Sheet metal bends and flat pattern sheetmetal.js

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  64-bit

• Chain of profile lines

  Orders the lines of an open profile end to end, or reports that they are not one chain.

  two ends are the same point when their distance ≤ 1e-6 × max(1, largest coordinate size)

  Sheet metal bends and flat pattern sheetmetal.js

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  64-bit

• Bend angle, center and tangent length

  For every corner of the profile: the signed bend angle, the radius on the path side and the offset side, the distance the bend takes from each leg, and the bend's center.

  θ = atan2(u × w, u · w); d = r_path × tan(|θ| ÷ 2); c = P + left(u) × sign(θ) × r_path − u × d

  Sheet metal bends and flat pattern sheetmetal.js

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  64-bit

• Arc of a bend in the profile

  The arc of a bend between its two tangent points, with its middle point and 17 points along it.

  sweep = a₁ − a₀ brought to (0, 2π] or [−2π, 0); point = c + r × (cos t, sin t)

  Sheet metal bends and flat pattern sheetmetal.js

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  64-bit

• Straight runs between bends

  The flat length of each leg: its drawn length less what the bends at its ends take.

  straightⱼ = |Pⱼ₊₁ − Pⱼ| − dⱼ − dⱼ₊₁

  Sheet metal bends and flat pattern sheetmetal.js

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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  32-bit in part

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit