Calculations

Calculations

• Mean radiance of each pixel

  The picture as the average of its samples.

  color = Σ ÷ n

  Rendered picture: sample sums and color tables compute/cpu-render.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Color through a .cube table

  A color after the table's 1D and 3D look-ups.

  x = clamp((r − lo) ÷ (hi − lo)); trilinear: c = c₀ + (c₁ − c₀) × d along each axis in turn

  Rendered picture: sample sums and color tables exr/cube.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Table rows read from a .cube file

  The entries of a color table as they are held in memory.

  Rendered picture: sample sums and color tables exr/cube.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Numbers written to a .cube file

  A table entry as text in a saved color table.

  text = Number(v.toPrecision(7))

  Rendered picture: sample sums and color tables exr/cube.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Filmic tone curve

  Maps scene brightness to screen brightness for the filmic view transform.

  h(x) = (x (0.15 x + 0.05) + 0.004) ÷ (x (0.15 x + 0.5) + 0.06) − 0.02 ÷ 0.3; output = h(2 c) ÷ h(11.2)

  Real-time view and render bench rt/rt-post.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Baked probe light

  Collects the bounce light at each probe and removes the sky's own part.

  shᵢ,ₖ = bakedᵢ,ₖ − skyₖ (k = 0…11)

  Real-time view and render bench rt/rt-gi.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Mean radiance per pixel

  Averages the accumulated samples of each pixel for the bench and the parity comparison.

  mean = Σ samples ÷ n

  Real-time view and render bench render/bench-kernels.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Scene for the standalone viewer

  Packs the drawn scene into the viewer's file.

  Real-time view and render bench rt/rt-export.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Placements from the modifier stack

  The list of placement matrices for the source, first the original.

  stepₖ = (hiₖ − loₖ) × relₖ + absₖ; Mᵢ = translate(i × step) × M

  Group modifiers: scale and array placements group-mods.js

  • ModelCraft
  64-bit

• Number of copies

  How many placements the enabled Array items make together.

  copies = Π countᵢ

  Group modifiers: scale and array placements group-mods.js

  • ModelCraft
  Exact

• Size of a loose piece

  Decides whether a separate piece is small enough to remove.

  max(Δx, Δy, Δz) < minSize

  3D-print mesh repair print-repair.js

  • ModelCraft
  64-bit

• Middle point of a filled hole

  The new vertex a hole's fan of triangles meets at.

  c = Σ pᵢ ÷ n

  3D-print mesh repair print-repair.js

  • ModelCraft
  64-bit

• Signed volume and inside-out pieces

  Finds closed pieces that face inward and turns them outward.

  V = Σ a · (b × c) ÷ 6; turned when V < 0 and the piece is not inside another

  3D-print mesh repair print-repair.js

  • ModelCraft
  64-bit

• Ray and triangle

  Where a ray meets a triangle, for the inside test.

  det = e₁ · (d × e₂); u = (s · h) ÷ det; v = (d · q) ÷ det; t = (e₂ · q) ÷ det

  3D-print mesh repair print-repair.js

  • ModelCraft
  64-bit

• Delaunay triangulation

  Triangles over a set of points in plan.

  D = 2 (aₓ(b_y − c_y) + bₓ(c_y − a_y) + cₓ(a_y − b_y)); center = (Σ |p|² × Δ) ÷ D; inside when |p − center|² < r² (1 − 10⁻¹²)

  Terrain from contours and terrain grids terrain.js

  • ModelCraft
  64-bit

• Terrain from contour edges

  Collects the contour points, merges close ones and triangulates them.

  key = round(100 x), round(100 y); thinning step = n ÷ 8000

  Terrain from contours and terrain grids terrain.js

  • ModelCraft
  64-bit

• Terrain grid

  A flat grid of triangles of a given width, depth and spacing.

  nₓ = round(width ÷ spacing), 1…200; x = x₀ + width × i ÷ nₓ

  Terrain from contours and terrain grids terrain.js

  • ModelCraft
  64-bit

• Entourage parts into the model

  Converts each part's triangles from millimeters to meters.

  p (m) = p (mm) ÷ 1000

  Starter library: plants and entourage sizes starter.js

  • ModelCraft
  64-bit

• Plant scaled to its listed size

  The scale that makes a plant model as wide and tall as the species' data says.

  s_xy = 1000 × spread ÷ max(Δx, Δy, 1); s_z = 1000 × height ÷ max(z_max, 1)

  Starter library: plants and entourage sizes starter.js

  • ModelCraft
  64-bit

• Unit factor

  Chooses the factor that turns the file's unit into meters.

  k = meters per file unit, or 0.001 when that is missing, zero or not finite

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Mesh points in meters

  Scales a mesh's points and builds the model's mesh from them.

  p′ = p × k

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Block placement matrix

  Turns a block reference's matrix into the model's placement.

  m[c × 4 + r] = x[r × 4 + c]; translation × k

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Curve points in meters

  Scales each curve's points and drops segments of no length.

  a′ = a × k; kept when a′ ≠ b′

  3DM import: units, placements and mesh points io/3dm-import.js

  • ModelCraft
  • PartCraft
  64-bit

• Mesh record: 32-bit and 64-bit points

  Reads a mesh's points from the file and chooses between its 32-bit and 64-bit copies.

  use the 64-bit points when |aᵢ − vᵢ| ≤ |vᵢ| × 10⁻⁶ + 10⁻⁶ × max(1, |aᵢ|) for every coordinate

  3DM import: units, placements and mesh points io/3dm/objects.js

  • ModelCraft
  • PartCraft
  64-bit

• Orientation of three points

  Twice the signed area of a triangle: which way it turns.

  (b − a) × (c − a) = (bₓ − aₓ)(c_y − a_y) − (b_y − a_y)(cₓ − aₓ)

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Point inside a circumcircle

  Whether a fourth point lies inside the circle through three others.

  determinant of the rows (pₓ − dₓ, p_y − d_y, |p − d|²) for p = a, b, c; inside when it is positive

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Triangle that holds a point

  Walks from the last triangle toward the point, then searches all triangles if the walk fails.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Point insertion

  Splits the triangle a new point falls in, or the side it falls on.

  ε = 10⁻⁶ × orient(a, b, c); on a side when a sub-area ≤ ε

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Edge flip

  Flips an edge while the opposite corner lies inside the circumcircle.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Flips with no new points

  Makes an existing triangulation Delaunay by flipping edges.

  Triangles inside a trimmed face io/3dm/cdt.js

  • ModelCraft
  • PartCraft
  64-bit

• Welded corners in meters

  Merges corners that coincide and converts them from millimeters to meters.

  key = round(x ÷ q), round(y ÷ q), round(z ÷ q); p (m) = p (mm) × 0.001

  STEP import: welding and meters io/step-import.js

  • ModelCraft
  64-bit

• Single values by type

  Reads one property value with the width the file states.

  FBX reader: number types as stored io/fbx-parse.js

  • ModelCraft
  64-bit

• 64-bit whole numbers

  Reads a signed 64-bit whole number from two 32-bit halves.

  v = hi × 2³² + lo when −2²¹ ≤ hi < 2²¹; otherwise decimal text

  FBX reader: number types as stored io/fbx-parse.js

  • ModelCraft
  Exact

• Arrays by type

  Reads an array of floats, doubles or whole numbers.

  FBX reader: number types as stored io/fbx-parse.js

  • ModelCraft
  64-bit

• Triangles of a referenced object

  The target's triangles in the modified mesh's space, for booleans and similar modifiers.

  q = (toLocal × placement) × p

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Cage point displacements

  How far each cage point has been moved from its place on the even grid.

  rest = min + (max − min) × i ÷ (n − 1); disp = M × now − M × rest

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Cage around a mesh

  A box of grid points around a mesh's bounds.

  d = pad × max side; min′ = min − d; max′ = max + d; p = min′ + (max′ − min′) × i ÷ (n − 1)

  Modifier references and lattice cages mesh/mod-eval.js

  • ModelCraft
  64-bit

• Face points

  The new point in the middle of each face.

  F = Σ corners ÷ n

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Edge points

  The new point on each edge.

  m = (a + b) ÷ 2; e = (a + b + F₁ + F₂) ÷ 4; point = e + (m − e) × s for 0 ≤ s < 1

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Vertex points

  Where each old vertex moves to.

  smooth = (F̄ + 2 R̄ + (n − 3) V) ÷ n; crease = (6 V + a + b) ÷ 8; k = min(1, (sₐ + s_b) ÷ 2); point = smooth + (crease − smooth) × k

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Edge sharpness per level

  The sharpness each new edge inherits.

  s′ = max(0, s − 1); open edge: ∞; crease layer = min(1, s′ ÷ 4)

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  32-bit in part

• Texture coordinates and other corner layers

  Interpolates per-corner values onto the new quads.

  corner: own value; edge: mean of two; face: mean of the face's corners

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Limit positions

  Where each vertex of an all-quad mesh lies on the limit surface.

  smooth: (n² V + 4 Σ E + Σ F) ÷ (n (n + 5)); crease or border: (4 V + a + b) ÷ 6

  Subdivision surfaces mesh/subdiv.js

  • ModelCraft
  64-bit

• Multiplicity of a knot

  How many times a value occurs in a knot vector.

  count of |Uᵢ − u| ≤ tol × (U_last − U_first)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Distinct knots

  The different knot values with their multiplicities.

  a knot joins the previous one when |x − u_prev| ≤ tol × range

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Clamped uniform knot vector

  A knot vector with evenly spaced interior knots.

  uᵢ = a + (b − a) × i ÷ (inner + 1)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Knot vector check

  Checks a knot vector read from a file: count, order, clamping and multiplicities.

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Greville abscissae

  One parameter value per control point.

  gᵢ = (Uᵢ₊₁ + … + Uᵢ₊ₚ) ÷ p

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  64-bit

• Binomial coefficients

  The number of ways to choose k of n.

  C(n, k) = C(n − 1, k − 1) + C(n − 1, k)

  Knot vectors nurbs/basis.js

  • ModelCraft
  • PartCraft
  Exact

• A number as IGES text

  Writes one real number into the file.

  whole numbers below 10¹⁵: digits and a point; otherwise String(x) with E → D

  IGES files of NURBS curves and surfaces nurbs/iges.js

  • ModelCraft
  • PartCraft
  64-bit