Calculations

Calculations

• 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

• One sample of a property

  Reads one stored sample of a scalar or array property into a typed array.

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• 16-bit float to a number

  Decodes a half-precision float.

  subnormal: ± m × 2⁻²⁴; normal: ± (1 + m ÷ 1024) × 2^(e − 15)

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• Which time sample is read

  The index of the stored sample nearest a given time.

  time of sample i = times[i mod n] + floor(i ÷ n) × time per cycle; nearest by |time − t|

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• An object's transform

  Builds an object's matrix from its list of scale, translate, rotate and matrix operations.

  M = op₁ × op₂ × … (the first operation outermost)

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• Rotation about an axis

  The matrix of a turn by an angle in degrees about an axis.

  a = deg × π ÷ 180; Rodrigues' formula with the axis made unit

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• Mesh points into Z up

  Copies the file's points into the model's coordinates.

  (x, y, z) → (x, −z, y)

  Alembic import io/abc.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

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• Which way the faces turn

  Decides whether the file's faces must be turned round, by comparing each face's own normal with the file's normals.

  Newell normal n = Σ (yₐ − y_b)(zₐ + z_b), …; agree when n · Σ file normals > 0

  Alembic import io/abc.js

  • ModelCraft
  64-bit

• Axes and unit

  The matrix that takes the file's coordinates to Z up and meters.

  k = UnitScaleFactor × 0.01; columns = ± k along the model axis each file axis maps to

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Node matrices into the model's axes

  Carries a node's matrix from the file's axes and unit into the model's.

  M′ = A × M × A⁻¹

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Euler angles to a matrix

  The rotation matrix of three angles in degrees in the file's rotation order.

  XYZ order: R = Rz × Ry × Rx, angles × π ÷ 180

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• An object's own matrix

  Builds a model's local matrix from its translation, offsets, pivots, pre-, own and post-rotation, and scaling.

  M = T × Roff × Rp × Rpre × R × Rpost⁻¹ × Rp⁻¹ × Soff × Sp × S × Sp⁻¹

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Mesh points into the model

  Carries each control point through the axis and unit matrix.

  p′ = A × p

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Lamps

  A lamp's position, direction and brightness from an FBX light.

  lumens = Intensity × 8; direction = the node's −Y, made unit

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Cameras

  A saved view's eye, target and field of view from an FBX camera.

  target = eye + 10 × forward; parallel view height = OrthoZoom × 10

  FBX import: axes, units and transforms io/fbx-import.js

  • ModelCraft
  64-bit

• Reading an accessor

  Reads one accessor's numbers, with its sparse part, into a typed array.

  normalized: v ÷ 127, 255, 32767 or 65535 (signed values not below −1)

  glTF import: reading numbers io/gltf-read.js

  • ModelCraft
  64-bit

• Positions

  Takes a primitive's positions as 32-bit floats and checks each is a sane number.

  glTF import: reading numbers io/gltf-read.js

  • ModelCraft
  64-bit

• Matrix from translation, rotation and scale

  A node's 4×4 matrix from its translation, quaternion and scale.

  quaternion made unit; rotation entries 1 − 2(y² + z²), 2(xy + zw), …, each column times its scale

  glTF import: reading numbers io/gltf-read.js

  • ModelCraft
  64-bit

• A node's local matrix

  Uses the node's 16-number matrix as given, or builds it from translation, rotation and scale.

  glTF import: reading numbers io/gltf-read.js

  • ModelCraft
  64-bit

• 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

• 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