Calculations

Calculations

• 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

• 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

• 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

• 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

• 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

• Rough box of an object

  A box round an object from its vertices, curve break points and control points, used only to set the tolerance.

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

  • ModelCraft
  • PartCraft
  64-bit

• Preset curves

  The control points of a new curve: an S, a Bézier circle, a NURBS circle or a straight path.

  Bézier circle: handle length = r × 0.5522847498; NURBS circle: 8 points on an octagon of radius r ÷ 0.9254

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Automatic handles

  Places a Bézier point's handles when they are automatic or vector.

  auto: along (next − previous) made unit, a third of each neighbor's distance; vector: a third of the way to each neighbor

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Point on a cubic Bézier segment

  A point of a segment at a fraction t.

  p = (1 − t)³ a + 3 (1 − t)² t b + 3 (1 − t) t² c + t³ d

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Point on a NURBS spline

  A point of a NURBS spline at a parameter.

  de Boor on (x w, y w, z w, w), then (x, y, z) ÷ w

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Sampling a spline

  Points along a spline with tangents, the fraction of the length at each, and the radius and tilt there.

  resolution samples per segment; tangent = (next − previous) made unit; s = Σ chord lengths so far ÷ total

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Frames along the curve

  A side and an up direction at every sample that twist as little as possible, with the tilt applied.

  double reflection from one sample to the next; closed curve: the leftover angle is undone in proportion to s

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Taper along the curve

  The scale of the cross-section at a fraction of the length.

  Catmull-Rom through the taper values, not below 0

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Cross-section

  The outline swept along the curve: a circle, a stadium or a flat ribbon.

  circle: 4 (resolution + 1) points at radius depth; stadium: two half circles offset by ± extrude

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Sweeping the section

  Places the cross-section at every sample and joins the rings into faces.

  point = sample + side × (−x k) + up × (y k); k = radius × taper

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Offsetting a loop

  Moves each vertex of a flat outline outward by a distance, with mitered corners.

  m = (n₁ + n₂) made unit; p′ = p + m × d ÷ max(0.25, m · n₁)

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Filled shapes

  Sorts closed flat loops into outlines and holes and builds the top, bottom and edge of the filled shape.

  nesting: a loop inside an even number of others is an outline; direction from the sign of the shoelace area

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Adding a point in a segment

  Splits a segment at its middle; a Bézier segment keeps its shape.

  de Casteljau at t = ½

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Moving, scaling and turning points

  The new place of each selected point and handle while it is dragged.

  grab: p + (M⁻¹ w₁ − M⁻¹ w₀); scale: c + (p − c) × |a₁| ÷ max(1, |a₀|); turn: Rodrigues' formula about the view axis through c

  Curve objects mesh/curve-tool.js

  • ModelCraft
  64-bit

• A new point at the pointer

  Where a point added after the end of a spline goes.

  world point = pick ray ∩ plane through the last point; local = M⁻¹ × world; flat curve: z = 0

  Curve objects mesh/curve-tool.js

  • ModelCraft
  64-bit

• Vertex quadrics

  Adds each triangle's plane to the error quadric of its three vertices.

  n = unit((b − a) × (c − a)), d = −n·a; Q(v) += [a², ab, ac, ad, b², bc, bd, c², cd, d²]

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Border planes

  Adds a plane square to the face through each open border edge, so borders stay in place.

  n = unit((B − A) × face normal), d = −n·A; added four times with n and d multiplied by 10

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit