Calculations

Calculations

• 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

• Vertex data of a primitive

  Writes a primitive's positions, normals, texture coordinates, colors and indices.

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Bounds of the positions

  The smallest and largest x, y and z written with the positions.

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Shared vertices

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

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

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Placement matrices

  Writes a placement's matrix on its node.

  entry = 0 when |entry| < 1e-15, else the entry

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Lamps

  A lamp's intensity and cone as glTF writes them.

  candela = lumens ÷ (4π); outer cone = min(π ÷ 2, angle × π ÷ 360)

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Saved views as cameras

  A saved view's field of view or height as a glTF camera.

  yfov = fov × π ÷ 180; xmag = ymag = height ÷ 2

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Node matrix for a lamp or camera

  A matrix at a point whose −Z looks along a direction with +Y as near world up as it can be.

  f = dir made unit; r = (f × up) made unit; u = r × f

  glTF export io/gltf-write.js

  • ModelCraft
  64-bit

• Reading a vertex

  Parses the three coordinates of a 'v' line.

  OBJ import and export io/obj.js

  • ModelCraft
  64-bit

• Points into Z up

  Copies an object's points into the model's coordinates.

  (x, y, z) → (x, −z, y)

  OBJ import and export io/obj.js

  • ModelCraft
  64-bit

• Sharp edges

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

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

  OBJ import and export io/obj.js

  • ModelCraft
  64-bit

• Number format on export

  Prints a number for the exported file.

  text = the number rounded to 6 decimals, trailing zeros removed

  OBJ import and export io/obj.js

  • ModelCraft
  64-bit

• Exported points

  Carries a mesh point to world coordinates and Y up for the file.

  p′ = placement matrix × p; (x, y, z) → (x, z, −y)

  OBJ import and export io/obj.js

  • ModelCraft
  64-bit

• Texture coordinates with a map's scale and offset

  Applies a material's map scale and offset to a texture coordinate on import.

  u′ = u × scale_u + offset_u; v′ = v × scale_v + offset_v

  OBJ import and export io/obj.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

• Rounding of every parameter

  Rounds each number of a converted material.

  v → the number nearest v with 7 significant digits

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Film thickness and dispersion

  A thin film's thickness in micrometers and a glass's Abbe number.

  thickness (µm) = maximum thickness (nm) ÷ 1000; Abbe number = 20 ÷ dispersion

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Emission strength

  Splits an emissive color into a color with a peak of 1 and a luminance.

  peak = largest channel; color = channels ÷ peak; luminance = peak × strength

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Shininess to roughness

  A roughness from a Blinn–Phong shininess exponent.

  roughness = (2 ÷ (s + 2))^¼, clamped to [0, 1]

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Knot span

  Finds the knot interval that holds a parameter.

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• de Boor's algorithm

  A point of a B-spline from its control points and knots.

  a = (t − k₀) ÷ (k₁ − k₀); d_j = (1 − a) d_(j−1) + a d_j, repeated degree times

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Rational point

  Divides a homogeneous point by its weight.

  (x, y, z) = (X, Y, Z) ÷ w

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a NURBS surface

  A surface point from its control net.

  de Boor along v for each active row, then along u, then the division by the weight

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a curve

  A point on a line, arc, polyline, NURBS curve or chain of curves at a parameter.

  arc: a = angle mapped from the parameter; p = o + r cos a × x + r sin a × y

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• A curve as a polyline

  Cuts a curve into chords that stay within a tolerance of it, keeping every knot and corner.

  a piece is halved while its middle or quarter points lie farther than tol from the chord, at most 10 times

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Rotation about an axis

  Turns a point about a line by an angle, for surfaces of revolution.

  p′ = o + v cos a + (z × v) sin a + z (z · v)(1 − cos a)

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a surface

  A point on a plane, NURBS surface, surface of revolution or sum surface.

  plane: o + a x + b y; revolution: the profile point turned about the axis; sum: p(u) + q(v) + base

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Surface normal

  The unit normal of a surface at a point.

  ∂u ≈ S(u + h, v) − S(u − h, v); ∂v likewise; n = ∂u × ∂v, made unit; h = 1e-5 × parameter range

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• A saved mesh as triangles

  Copies a mesh record's points and splits its quads into triangles.

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Joining points that coincide

  Makes one vertex of points that sit at the same place, so the faces of a solid close up.

  q = max(diagonal × 1e-9, 1e-12); same vertex when round(x ÷ q), round(y ÷ q) and round(z ÷ q) are equal

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Sampling a shared edge

  Samples an edge once on its 3D curve, for both faces that share it.

  fraction s = (t − t₀) ÷ (t₁ − t₀) along the edge's own direction

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Scaling the parameter plane

  Scales u and v so a step in each is about as long on the surface, before triangulating.

  L_u, L_v = lengths of the two mid-lines measured with 8 chords; k_u = L_u ÷ Δu; k_v = L_v ÷ Δv

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Halving an interior edge

  Decides whether a triangle edge needs a new point at its middle, and adds it.

  halve when the edge is longer than 2 tol and (|S(mid) − chord middle| > tol or n_a · n_c < cos 20°)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Grid step from the surface's bend

  How far apart interior points are placed, from how much the surface bends.

  k = angle between neighboring normals ÷ their distance (75th percentile of a 9 × 9 grid); h = min(√(8 tol ÷ k), 0.349 ÷ k), not below max(L) ÷ 400 or √(L_u L_v ÷ 40000)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Grid points inside a face

  Picks the grid points that lie inside the face's loops and not too near a side.

  inside by counting crossings of the row (even-odd); kept when at least half a grid step from every side

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Extrusions

  Walls along the path from each profile, and caps from the closed profiles with their holes.

  frame: T = path direction, X = up × T, Y = T × X; point = origin + x X + y Y; profile closed when its ends are within max(tol, 1e-9)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Tessellation tolerance

  The distance triangles may stand off a surface, per object.

  base = given tolerance, else the file's, else 0.01; tol = max(base, 0.0015 × diagonal of the object's rough box)

  3DM surfaces into triangles io/3dm/scene.js

  • ModelCraft
  • PartCraft
  64-bit