Calculations

Calculations

• Smoothed surface mesh and its volume

  Smooths the surface, restores its volume, and returns the mesh with its volume.

  Generative design by topology optimization topopt.js

  • PartCraft
  32-bit in part

• Surface smoothing

  Smooths the surface mesh without shrinking it as much as plain averaging would.

  p ← p + f (mean of neighbors − p), f alternating 0.5 and −0.53

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Volume restored after smoothing

  Moves every vertex along its normal by one distance so the mesh volume is the voxel volume again.

  d = (V₀ − V) ÷ area, three steps

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Volume of a closed mesh

  The volume enclosed by a triangle mesh.

  V = Σ a · (b × c) ÷ 6

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Check of the solid design

  The mass, volume, largest stress and displacement and the safety factor of the kept elements, solved again with the real material.

  volume = elements × h³; mass in kg = volume × ρ (t/mm³) × 1000; safety = yield ÷ max von Mises

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Compliance for another material

  Rescales the compliance of a shape when an outcome uses a different material.

  c = c₀ × E₀ ÷ E

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Unit vector

  Scales a vector to length 1.

  a ÷ |a|

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Frame of a plane

  The origin and the x, y and normal axes given to a sketch plane.

  o = (p·n) n; x = Y × n (or world X made square to n when |n_y| > 0.9); y = n × x

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Sketch point to model space

  The 3D position of a sketch point (u, v) on its plane.

  p = o + u x + v y

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Model point to sketch coordinates

  The sketch coordinates of a 3D point on a plane.

  u = (p − o)·x; v = (p − o)·y

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Rotation about an axis

  The matrix that turns geometry by an angle about an axis through a point.

  R = c I + s [d]× + (1 − c) d dᵀ, c = cos a, s = sin a, a = deg × π ÷ 180; translation = p − R p

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Reflection in a plane

  The matrix that mirrors geometry in a plane.

  M = I − 2 n nᵀ; translation = 2 (o·n) n

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Matrix applied to a point

  Transforms a point by a 4 × 4 matrix.

  p′ = M₃ₓ₃ p + t

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Matrix product

  Multiplies two 4 × 4 matrices.

  C = A B

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Small linear system

  Solves A x = b for a small dense system.

  Gaussian elimination with partial pivoting, then back substitution

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Number shown with up to four decimals

  Formats a length or number for display.

  text = value rounded to dp decimals (4 by default); |value| < 10⁻⁹ shown as 0

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Triangle buffer for drawing

  Copies the part's mesh into the vertex buffer the GPU draws.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• Result colors on the part

  Spreads per-vertex or per-corner result colors onto the drawn triangles.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• What is under the pointer

  Finds the face, edge or plane under the pointer.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• Grid size and origin

  The number of cubes along each axis and where the grid starts.

  n = max(1, ceil(size ÷ h − 10⁻⁶)); origin = box center − n h ÷ 2

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Bounding box of the mesh

  The smallest box around the part's surface triangles.

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Study size estimate

  The rough element count, node count and memory shown before a voxel study is meshed.

  elements ≈ round(volume ÷ h³)

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Where a row's ray crosses the surface

  The x positions at which the ray through a row of cube centers enters and leaves the part.

  x = (w₀ xₐ + w₁ x_b + w₂ x_c) ÷ (w₀ + w₁ + w₂), w from the y/z projection

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Cubes between two crossings

  Marks the cubes of a row whose centers lie between an entry and an exit crossing.

  i from ceil((x_in − origin) ÷ h − 0.5) to floor((x_out − origin) ÷ h − 0.5)

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Position of a grid node

  The coordinates of a grid node from its index.

  p = origin + (i, j, k) × h

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Distance from a point to a triangle

  The squared distance from a grid node or cube center to a surface triangle.

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Faces nearest a point

  Which of the part's faces a grid node lies on, with each face's normal there.

  faces with d² ≤ max(d_min² + (10⁻³ h)², (d_min + 10⁻³ h)²)

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Tributary area of surface nodes

  The share of a picked face's area carried by each grid node, used to spread loads.

  each exposed cube face with all four corners on the picked faces gives h² ÷ 4 to each corner; area = Σ h²

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Mean outward normal at picked nodes

  The direction used for pressure and normal force at each node of a picked face.

  n = Σ face normals ÷ |Σ face normals|

  Voxel grid of a part voxel.js

  • PartCraft
  32-bit in part

• Cubes near picked faces

  Marks the cubes whose centers are within a distance of picked faces, for preserve and obstacle regions.

  kept when the distance from the cube center to the faces' triangles ≤ dist

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Section outline at its anchor

  The section's outline moved so the chosen anchor point sits on the member's line, with its width, height and radius.

  w = u_max − u_min; h = v_max − v_min; R = max |point|

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• Member orientation

  The across and depth directions of a member's section about its line.

  v = up − (up·d) d, normalized; u = v × d; then rotated about d

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• When line ends meet

  The distance within which two line ends count as one joint.

  tol = 10⁻⁶ × max(1, largest |end point|, longest member)

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• How far a member runs past a corner

  The extra length a raw member needs so a miter cut still has material.

  ext = min(50 R, R ÷ max(tan(angle ÷ 2), 0.02)) + R

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• Miter cut plane

  The plane halfway between two members that both are cut on.

  n = (d_in + d_out) ÷ |d_in + d_out|; offset = n·P

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• Butt cut planes

  Where the through member ends and where the other is cut square against it.

  reach = max |(u U + v V)·d| over the outline; offsets = ∓(d·P) ∓ reach

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• Gusset plate

  The triangular plate in a corner and the angle listed for it.

  corners (0, 0), (s, 0), (s d_out·x, s d_out·y); angle = acos(x·d_out)

  Structural members and their cut list weldment.js

  • PartCraft
  64-bit

• Cut length and end angles of a member

  The length of each finished piece along its line and the angle of each end cut, as stored in the cut list.

  length = round((max s − min s) × 10) ÷ 10, s = (vertex − a)·d; angle = round(acos(|cut·d|) × 180 ÷ π × 100) ÷ 100

  Structural members and their cut list weldment.js

  • PartCraft
  32-bit in part

• Cut list rows

  Groups identical pieces and counts them.

  same piece when section, length to one decimal, both angles and both end kinds match

  Structural members and their cut list weldment.js

  • PartCraft
  32-bit in part

• Cut list CSV

  Writes the cut list as CSV text.

  member: length with one decimal; cap: w × h with one decimal

  Structural members and their cut list weldment.js

  • PartCraft
  32-bit in part

• Cage move from the gizmo

  The transform applied to the selected cage vertices while the gizmo is dragged.

  M = W_now × W_start⁻¹

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Point for a new face

  The position stored when a face is drawn by clicks.

  existing vertex within 9 px, else ray ∩ plane y = 0 rounded to 0.001

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Direction of an edge extrusion

  The direction and distance open edges are extruded by.

  d = Σ (edge middle − face center); offset = d ÷ |d| × distance

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Tolerance for joining a new face

  The distance within which a new face's points join existing cage vertices.

  tol = 10⁻⁶ × max(1, size setting)

  Freeform cage edits from the workspace ws-form.js

  • PartCraft
  64-bit

• Picked point as an anchor

  The anchor position in the part's own axes and the face normal at the pick.

  p = M⁻¹ p_pick, each coordinate rounded to 6 decimals; n = (b − a) × (c − a) ÷ length

  Anchor points of manual annotations ws-illustrate.js

  • PartCraft
  64-bit

• Is a face square to the tool axis

  Decides whether a picked flat or round face can be machined from the setup's tool axis.

  accepted when ||n·axis| − 1| ≤ 10⁻⁹

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Matching a pick to one already stored

  Decides whether a clicked face is already in the operation's geometry, to add or remove it.

  same when names agree and |at − ref| < 10⁻³

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Numbers shown in the panel

  Formats stock sizes, feeds, gouge depth and remaining stock for the panel.

  text = value rounded to dp decimals (3 by default), trailing zeros dropped

  Machining picks and readouts ws-manufacture.js

  • PartCraft
  64-bit

• Part mesh for scripts

  The open part as a triangle mesh a script can compute on through part().

  Script workspace: the part handed to scripts and picked values ws-script.js

  • PartCraft
  32-bit in part

• Picked point and nearest node

  The coordinates of the clicked point and the mesh node closest to it.

  node = the hit triangle's corner with the least √(Δx² + Δy² + Δz²) to the hit point

  Script workspace: the part handed to scripts and picked values ws-script.js

  • PartCraft
  32-bit in part