Calculations

Calculations

• Piece list on the assembly sheet

  One line per piece with its size and volume, then the joints and the pins.

  size shown = round(mm × 10) ÷ 10; volume shown = round(mm³ ÷ 1000 × 10) ÷ 10 cm³; pin length = dowel length − clearance

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part

• Normal of a slanting cut

  The normal of a cut across one axis, turned by an angle about another axis.

  n = axis·cos r + (about × axis)·sin r, r = degrees × π ÷ 180

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Cut places typed by the person

  Turns the typed distances from the low end of the model into cut planes.

  point = middle of the box, with axis coordinate lo + v; d = n·point

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Connector spot picked by hand

  Where the ray under a click crosses the cut plane between two pieces.

  t = (d − n·o) ÷ (n·dir); p = o + dir × t

  Splitting a model into printable pieces print-split-ui.js

  • ModelCraft
  • PartCraft
  64-bit

• Binary STL

  Writes each triangle's unit normal and three corners.

  normal = n ÷ |n| from the winding; every number written with setFloat32

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  32-bit in part

• A coordinate as 3MF text

  Formats one coordinate for the 3MF file.

  0 if |v| < 10⁻¹²; else v rounded to 10 significant digits

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• 3MF vertex list

  Writes every corner of the mesh as a vertex element.

  STL and 3MF files for the printer print3d.js

  • ModelCraft
  • PartCraft
  64-bit

• Letter outlines from strokes

  The outline of the text: each stroke segment as a rectangle of the pen's width, each stroke point as a disc.

  r = height × weight ÷ 2; rectangle corners = a ∓ r·n̂, b ∓ r·n̂; disc corner = c + r·[cos(i·π ÷ 6), sin(i·π ÷ 6)]; line k at y − 1.6·k·height

  3D text text3d.js

  • ModelCraft
  64-bit

• Letters as faces or solids

  The component the app places: flat faces, or a closed solid from z = 0 to the depth.

  g6(v) = round(v × 10⁶) ÷ 10⁶ on every coordinate

  3D text text3d.js

  • ModelCraft
  32-bit in part

• The four strongest bones per vertex

  Keeps each vertex's four largest weights and scales them to add up to 1.

  w_q = w_q ÷ Σ of the kept weights

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Linear blend skinning

  Moves each vertex to the weighted sum of its positions under its bones' matrices.

  p′ = Σ wᵢ × (Mᵢ × p)

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Dual quaternion skinning

  Blends the bones' dual quaternions by weight, makes the result unit and moves the vertex with it.

  (r, d) = Σ ±wᵢ (rᵢ, dᵢ); both divided by |r|; p′ = r p r⁻¹ + 2 d r⁻¹

  Skin weights and skinning anim/rig.js

  • ModelCraft
  32-bit in part

• Distance from a point to a bone

  The distance from a vertex to the segment from a bone's head to its tail.

  k = clamp(((p − a) · (b − a)) ÷ |b − a|², 0, 1); d = |a + (b − a) k − p|

  Skin weights and skinning anim/weights.js

  • ModelCraft
  64-bit

• Edge weights of the mesh

  For each vertex, its neighbors and a weight for each edge, used to spread heat and to smooth.

  edge weight = 1 ÷ max(1e-12, length²)

  Skin weights and skinning anim/weights.js

  • ModelCraft
  64-bit

• Automatic weights

  Each bone's weight on each vertex, by bone heat or by distance, with faint weights dropped and each vertex's weights adding up to 1.

  heat: solve (L + H) w = H p per bone, H = 1 ÷ nearest distance²; envelope: w = 1 ÷ d⁴ for bones within twice the nearest distance; drop w < 0.01; divide by the sum

  Skin weights and skinning anim/weights.js

  • ModelCraft
  32-bit in part

• One dab of the weight brush

  Raises, lowers, sets or smooths a bone's weight on the vertices under the brush, and scales the other bones' weights to share what is left.

  falloff f = (1 − d² ÷ r²)² × strength; add: w + f (1 − w); subtract: w − f w; others scaled by max(0, 1 − w) ÷ their sum

  Skin weights and skinning anim/weights.js

  • ModelCraft
  32-bit in part

• Weights from packed joint layers

  Turns the four joint numbers and weights per vertex read from a file into one weight layer per bone.

  weight of bone b on vertex v = Σ of the packed weights that name b

  Skin weights and skinning anim/weights.js

  • ModelCraft
  32-bit in part

• Skin weights read from an FBX file

  Keeps each control point's four strongest clusters and scales their weights to add up to 1.

  w = w ÷ Σ of the four kept weights

  Skin weights and skinning io/fbx-import.js

  • ModelCraft
  32-bit in part

• Morph weights and the skin

  A primitive's positions and normals as the scene shows them: morph targets added by their weights, then moved by the skin's joints.

  morph: p += w × target; skin: p′ = Σ wᵢ (Jᵢ × IBMᵢ × p) ÷ Σ wᵢ

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  32-bit in part

• Welding vertices

  Makes one mesh vertex for every distinct point of a primitive and swaps the axes to Z up.

  same vertex when x, y and z (and joints and weights) are equal as text; stored as (x, −z, y)

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Triangles with no area

  Drops triangles that repeat a vertex or whose corners lie on one line.

  kept when |(b − a) × (c − a)| > 1e-18

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Sharp edges

  Marks an edge sharp where the file's normals break across it, or where the two faces bend.

  with normals: sharp unless n₁ · n₂ > 0.9995 at both ends; without: sharp when the face normals' dot product < 0.99996

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Node matrix into Z up

  Changes a node's matrix from Y up to Z up.

  M′ = C × M × C⁻¹

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Instanced placements

  One placement per instance from the file's per-instance translation, rotation and scale.

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Texture coordinate transform

  Applies a texture's offset, rotation and scale to a UV pair.

  u′ = cos r × u s_u + sin r × v s_v + o_u; v′ = −sin r × u s_u + cos r × v s_v + o_v

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Lamps

  A lamp's position, direction, brightness and cone from a glTF light.

  lumens = candela × 4π; spot angle (°) = 2 × outer cone × 180 ÷ π

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Cameras

  A saved view's eye, target and field of view from a glTF camera.

  target = eye + 10 × forward; field of view (°) = yfov × 180 ÷ π; parallel view height = 2 × |ymag|

  glTF import: meshes, skins and placements io/gltf-import.js

  • ModelCraft
  64-bit

• Reading one number

  Reads a value of the type the header names, from text or from binary.

  PLY import and export io/ply.js

  • ModelCraft
  64-bit

• Imported positions

  Stores each vertex's x, y and z.

  PLY import and export io/ply.js

  • ModelCraft
  64-bit

• Points and normals in world coordinates

  Carries a point and a normal through a placement's matrix for export.

  p′ = M × p; n′ = (M × n without translation), made unit

  PLY import and export io/ply.js

  • ModelCraft
  64-bit

• Exported positions

  Writes each vertex's world position and normal into the file.

  PLY import and export io/ply.js

  • ModelCraft
  32-bit in part

• Shared vertices on export

  Gives corners with the same position, normal, color and UV one vertex.

  key = x, y, z as numbers + normal to 5 decimals + color + UV

  PLY import and export io/ply.js

  • ModelCraft
  64-bit

• Reading a binary file

  Reads each triangle's nine coordinates and its color word.

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Reading a text file

  Parses every 'vertex x y z' line.

  STL import and export io/stl.js

  • ModelCraft
  32-bit in part

• Unit factors

  The factor from the file's unit to meters.

  mm 0.001; cm 0.01; m 1; inch 0.0254; foot 0.3048

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Welding corners and applying the unit

  Makes one vertex for every distinct corner and scales it to meters.

  same vertex when x, y and z are equal as text; stored as (x, y, z) × unit factor

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Triangles with no area

  Drops triangles whose corners lie on one line.

  dropped when |(b − a) × (c − a)| < 1e-18

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Exported triangles

  Writes each triangle's normal and corners in millimeters.

  n = (b − a) × (c − a), made unit; corner (mm) = world corner (m) × 1000

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Proportional editing weights

  For every vertex, how much of the selected vertices' move it follows: 1 for the selected ones, falling to 0 at the radius.

  t = distance ÷ radius; smooth: x²(3 − 2x) with x = 1 − t; sphere: √(1 − t²); root: √x; linear: x; sharp: x²; constant: 1

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Set a crease

  Stores the crease (0 smooth to 1 sharp) on each listed edge for subdivision.

  v = clamp(value, 0, 1)

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Crease shared by edges

  The crease the selected edges have in common, shown as the starting value when a crease is edited.

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Applying a move with proportional weights

  Writes each affected vertex's new position while a grab, rotate or scale is in progress.

  q = T(p); r = p + (q − p)·w; stored position = M⁻¹·r

  Proportional editing weights and edge creases mesh/edit-tool.js

  • ModelCraft
  32-bit in part

• Crease from the pointer or the wheel

  Turns the pointer's distance from the selection's center, or a wheel step, into a crease between 0 and 1.

  pointer: crease = clamp(round(px ÷ 2) ÷ 100, 0, 1); wheel: crease = clamp(round((crease ± 0.1) × 10) ÷ 10, 0, 1)

  Proportional editing weights and edge creases mesh/edit-tool.js

  • ModelCraft
  32-bit in part

• Boolean of two meshes

  Runs the union, intersection or difference and copies the result's vertex positions.

  Boolean modifier mesh/modifiers.js

  • ModelCraft
  32-bit in part

• Mesh from triangles

  Builds a mesh from triangle positions and indices, welds coincident vertices and marks sharp edges; used after a boolean and after decimate.

  weld within 1e-9; sharp when n₁·n₂ < cos(30°)

  Boolean modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Squared distance to a triangle

  The squared distance from a point to the nearest point of a triangle.

  q = nearest point by region (vertex, edge or interior); d² = |p − q|²

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  64-bit

• Distance samples near the surface

  For every grid point within two voxels of a triangle, the distance to the nearest triangle.

  g = voxel ÷ 2; d(i, j, k) = √(min over triangles of d²), capped at 2·voxel

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  32-bit in part

• Inside or outside

  Marks each grid point as inside or outside the mesh by counting the faces a ray along +x crosses.

  crossing when w₀, w₁, w₂ share a sign; x = (w₀·x_A + w₁·x_B + w₂·x_C) ÷ (w₀ + w₁ + w₂); inside when the winding count before x is above 0

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  64-bit

• Level set of the samples

  The distance at any point by blending the eight samples round it, handed to the library that builds the new mesh.

  f(x, y, z) = −trilinear blend of the 8 samples; new mesh = level set of f at 0 with edge length = voxel

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  32-bit in part

• Put vertices on a surface

  Moves each listed vertex to the nearest point of a target surface.

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  64-bit