Calculations

Calculations

• Rotation matrix to quaternion

  The unit quaternion of a rotation given by its three axes.

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Placement matrix from a body state

  The placement's world matrix for a simulated position and turn.

  axes = q applied to X, Y, Z, times the placement's scale; origin = body center − rotated, scaled center offset

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Mass and inertia

  The inverse mass and inverse moments of inertia of a box or a sphere.

  sphere: I = 0.4 m r²; box: I_x = m (b² + c²) ÷ 12, and likewise for y and z

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Depth of a point inside a box

  How far a point is inside a box and the outward normal of the nearest face.

  local = q⁻¹ (p − center); depth = min over axes of (half + margin − |local|) − margin

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Positional correction

  Pushes two bodies apart along a normal by a given distance, sharing it by their mass and inertia.

  w = Σ (1 ÷ m + (r × n) · I⁻¹ (r × n)); λ = c ÷ w; x += n λ ÷ m; q += ½ (Δω, 0) q, made unit

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Step size and frame count

  How many frames are recorded and how long one substep is.

  frames = max(1, round((to − from) × fps) + 1); h = 1 ÷ fps ÷ substeps

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Integration of one substep

  Advances each moving body's speed, position and turn by one substep under gravity.

  v += g h; x += v h; q += ½ h (ω, 0) q, made unit

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Contacts and static friction

  Pushes each touching point out of the surface and holds it still where friction can.

  μ = √(μ_a μ_b); held when slide < μ |λ| × 1.5 + 1e-9

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Velocities from positions

  Works each body's speed and spin back from how far it moved in the substep.

  v = (x − x_prev) ÷ h; ω = 2 ÷ h × vector part of (q × q_prev⁻¹)

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Dynamic friction and bounce

  Slows sliding at each contact and sets the speed a body leaves a surface with.

  friction impulse = min(μ |λ| ÷ h, sliding speed ÷ inverse mass); leaving speed = −e × min(incoming normal speed, 0); e = 0 when |incoming| < 2 |g · n| h

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• Settled test

  Says whether every body has come to rest at the end of the run.

  settled when every |v| < 0.02 and every |ω| < 0.05

  Rigid body simulation anim/physics.js

  • ModelCraft
  64-bit

• 4×4 matrix product

  The product of two column-major 4×4 matrices.

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• 4×4 matrix inverse

  The inverse of a 4×4 matrix by cofactors.

  M⁻¹ = adjugate(M) ÷ det(M)

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• A bone's rest frame

  The matrix from a bone's own frame to the mesh: Y from head to tail, Z as near the mesh's +Z as it can be, turned by the roll.

  y = (tail − head) made unit; z = (up − y (up · y)) made unit; x = y × z; roll turns x and z about y

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• Posed bone and skin matrices

  Each bone's posed frame and the matrix that moves rest points to the pose.

  W = W_parent × R_parent⁻¹ × R × local; skin = W × R⁻¹

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• Skin matrices as dual quaternions

  Each bone's turn and move as a unit dual quaternion, for dual-quaternion skinning.

  dual part = ½ × (t, 0) × q

  Bone frames and pose matrices anim/rig.js

  • ModelCraft
  64-bit

• A bone's pose from a drag

  The location or rotation a bone gets when it is dragged in the view.

  turn: angle swept round the bone's head on screen, about the view axis carried into the bone's frame; move: pixels × pixel size, along the camera's right and up, carried into the bone's frame

  Bone frames and pose matrices anim/rig-view.js

  • ModelCraft
  64-bit

• 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

• 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