Calculations

Calculations

• Radii and cone angles

  The radius of a cylinder, cone, sphere or torus in millimeters and a cone's half angle in radians.

  r × unit in mm; α × angle unit, accepted when 0 < |α| < π ÷ 2

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Knot vectors

  The full knot vector of a B-spline from the file's knots and multiplicities, or the one a uniform, quasi-uniform or Bézier form implies.

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  Exact

• Model size and chord tolerance

  The size of the model and the distance a triangle may stray from a curved face.

  size = diagonal of the box around every point; tol = max(4e-4 × size, 1e-4 mm)

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Closing a loop of edges

  Drops the last point of an edge loop when it repeats the first.

  |first − last| ≤ 0.01 × tol

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Placement matrices

  A placement as a 4 × 4 matrix, the product of two and the inverse of one.

  M⁻¹ = [Rᵀ, −Rᵀ t]

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• A component placed in its assembly

  The transform that carries a component's coordinates into the assembly that uses it.

  T = M(A₂) × M(A₁)⁻¹

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Placed vertex positions

  Every vertex of a solid moved to its place and stored for the app.

  p′ = R p + t

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Faceted faces

  Triangles for a flat face given as loops of points.

  n = Σ ((yᵢ − yⱼ)(zᵢ + zⱼ), (zᵢ − zⱼ)(xᵢ + xⱼ), (xᵢ − xⱼ)(yᵢ + yⱼ))

  STEP import: units, points and assembly placement step/read.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Points along an edge

  The points of an edge from its start vertex to its end vertex along its curve.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• How finely a curve is divided

  The parameters at which an edge curve is sampled.

  circle step = min(π ÷ 24, 2 acos(1 − tol ÷ r)); else halve while the middle is more than tol from the chord or the turn exceeds π ÷ 24

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Turn and crossing tests

  Whether three points turn left, right or run straight, and whether two segments cross.

  x = (a − o) × (b − o); left when x > 1e-9 × |ab| × |bc|, right when x < −1e-9 × |ab| × |bc|

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Holes joined to the outer ring

  One ring made from a face's outer loop and its holes by bridges, ready for ear clipping.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Ear clipping

  Triangles of a simple polygon in the plane.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• A loop in the surface's (u, v)

  The parameters of each boundary point, with periodic directions unwrapped and poles split.

  uᵢ ← uᵢ + round((uⱼ − uᵢ) ÷ period) × period

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Scaled layout and first triangles

  The (u, v) points scaled to lengths, the loops given their turning sense, and the first triangulation.

  M = (u × s_u, v × s_v); loop sense = sign of Σ (xᵢ yⱼ − xⱼ yᵢ)

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Apex of a cone

  The parameter at which a cone's radius reaches zero, for a face that closes at the apex.

  R = |S(0, 0) − S(π, 0)| ÷ 2; k = |S(0, 1) − S(π, 1)| ÷ 2 − R; v_apex = −R ÷ k

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Interior grid spacing

  How far apart interior points are placed on each kind of surface.

  angle step = min(π ÷ 24, 2 acos(1 − tol ÷ r))

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Interior points inserted

  Grid points added inside a face and joined into the triangulation.

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• In-circle test

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

  det of rows (a − d, |a − d|²), (b − d, |b − d|²), (c − d, |c − d|²)

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Edge flips

  Inside edges flipped toward a Delaunay triangulation in the scaled (u, v).

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Refinement to the chord tolerance

  Inside edges split where the surface is farther than the tolerance from the triangles.

  split when |S(mid uv) − (P_a + P_b) ÷ 2| > tol, or when a triangle's center is more than tol from its plane

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Facing and self-check

  The 3D position of every point, the triangles turned to face outward, and a check of how far the mesh is from the surface.

  worst = max |S(center uv) − center of triangle| over up to 64 triangles; noted when worst > 25 × tol

  STEP import: triangles for curved faces step/tess.js

  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Distance between two points

  The length a headset measurement reports.

  d = √(Δx² + Δy² + Δz²)

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Snap to a corner

  The corner of the hit triangle nearest the hit, when it is within the snap radius.

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Model to headset transform

  The matrix that places the model in the headset's space.

  M = T(anchor) · R_y(yaw) · S(scale) · T(−at)

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Inverse of a rig or pose

  The inverse of a rigid motion with one uniform scale.

  M⁻¹ = [Rᵀ ÷ s², −(Rᵀ ÷ s²) t], s² = squared length of the first column

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Section plane

  The plane of a section cut through what was pointed at.

  n = normal toward the eye; p = hit − n × depth; d = n·p

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Issue camera

  The viewpoint, direction and up vector saved with an issue, in the app's coordinates.

  to app = M_up⁻¹ · M_rig⁻¹; eye = to app × head position

  Headset review: measuring, section cuts and issue cameras xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• The point being pointed at

  The model point, surface normal and triangle under a controller's ray.

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

  • BimCraft
  • ModelCraft
  32-bit in part

• A measurement in the headset

  The length shown on the label between two picked points.

  d = |q − p|, shown through the app's length format

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Where an issue is placed

  The point and camera handed to the app with a new issue.

  Headset review: measuring, section cuts and issue cameras xr/xr-session.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Two-point fit

  The turn and move that carry two model points onto two site points, with the remaining error.

  yaw = atan2(Δw) − atan2(Δm); t = mid(w) − R mid(m); residual = max(|R m₁ + t − w₁|, |R m₂ + t − w₂|); scale off = |Δw| ÷ |Δm| − 1

  AR on site: lining the model up with the building xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• A saved alignment applied

  A model point moved to the site by the alignment kept in the file.

  (x′, z′) = (x cos yaw − z sin yaw + t_x, x sin yaw + z cos yaw + t_z)

  AR on site: lining the model up with the building xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Re-check on a later visit

  How far the first reference point is from where the saved alignment puts it, and what to do about it.

  offset = √(Δx² + Δz²); keep < 0.05 m ≤ move < 0.5 m ≤ align again

  AR on site: lining the model up with the building xr/xr-core.js

  • BimCraft
  • ModelCraft
  64-bit

• Placing a reference point

  Takes the site point under the phone's center mark and runs the fit or the re-check with it.

  t_y = (w₁_y + w₂_y) ÷ 2 − (m₁_y + m₂_y) ÷ 2

  AR on site: lining the model up with the building xr/xr-session.js

  • BimCraft
  • ModelCraft
  64-bit

• Camera jitter per pass

  Moves the camera by less than a pixel each pass so that the sum of passes is anti-aliased.

  j_x = (h₂(i) − ½) × 2 ÷ w; rows 0 and 1 of the projection += j × row 3

  Rendered pictures render/render-core.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Section cut for the path tracer

  Clips every triangle of the scene against the section planes before the picture is traced.

  new corner = a + (b − a) × d_a ÷ (d_a − d_b)

  Rendered pictures render/render-pt.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Tiling fractal noise

  The noise value behind the library's brick, wood, stone and concrete textures.

  Σ ampₒ × noise(u fₒ, v fₒ) ÷ Σ ampₒ, f doubling and amp × gain per octave

  Rendered pictures render/textures.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Normal map bytes

  Turns a baked material's normal layer into the texture three.js reads.

  x = (byte − 128) ÷ 127; z = √(1 − x² − y²); out = (x ÷ 2 + ½) × 255

  Rendered pictures render/three-mats.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Average and largest sample count

  The two numbers the render dialog states after an adaptive render.

  avg = round(10 × Σ nᵢ ÷ (w × h)) ÷ 10; max = largest nᵢ

  Samples per pixel reported after a render render/render-pt.js

  • BimCraft
  • ModelCraft
  32-bit in part

• Probe light as spherical harmonics

  Sums a small traced panorama into the four coefficients per color the view shades with.

  c = Σ L × Y × dΩ, dΩ = (2π ÷ w)(π ÷ h) sin θ

  Real-time view rt/rt-probes.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Cube faces to a 360° picture

  Builds the equirectangular panorama from six rendered faces.

  direction from (longitude, latitude); face = largest d·a; color = bilinear mix of four pixels

  Real-time view rt/rt-pano.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Display chain for one pixel

  Turns a linear color into the color shown: white balance, exposure, highlight knee, contrast, view transform.

  c ← max(0, WB c) × 2^stops; c ← 0.18 × (c ÷ 0.18)^(1 + contrast)

  Real-time view rt/rt-look.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Standing and falling in walk mode

  Keeps the walker's eye at eye height above the floor below, stepping up or falling as the floor changes.

  v ← v − 9.81 Δt; height ← max(floor + eye height, height + v Δt)

  Real-time view rt/rt-walk.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Target area and the count asked for

  The area of the faces a scatter set covers and the number of items its density asks for.

  area = Σ ½ |e₁ × e₂|; asked = min(2,000,000, round(area × density)); paint: Σ max(1, round(π r² × density))

  Scattered plants, rocks and people: how many and where rt/rt-scatter.js

  • BimCraft
  • ModelCraft
  64-bit

• A random point on the target

  Picks a triangle in proportion to its area and a point inside it.

  P = A + a (B − A) + b (C − A), (a, b) reflected when a + b > 1

  Scattered plants, rocks and people: how many and where rt/rt-scatter.js

  • BimCraft
  • ModelCraft
  64-bit

• Dropping a point onto the target

  Where a vertical ray from a brush dab meets a target triangle.

  t = (e₂·q) ÷ det, det = e₁·(d × e₂)

  Scattered plants, rocks and people: how many and where rt/rt-scatter.js

  • BimCraft
  • ModelCraft
  64-bit

• Slope and height limits

  Drops a candidate that stands on too steep a face or outside the height range.

  keep when n·up ≥ cos(slope) − 1e-9 and min ≤ p·up ≤ max

  Scattered plants, rocks and people: how many and where rt/rt-scatter.js

  • BimCraft
  • ModelCraft
  64-bit

• Values to texture bytes

  Stores a value from 0 to 1, or a normal component from −1 to 1, as one byte of a texture layer.

  byte = ⌊255 x + ½⌋; normal byte = ⌊127 x + 128.5⌋

  Material graphs baked to textures shading/bake.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Tiling value noise

  The smooth noise that procedural textures are built from.

  a + (b − a) f_x + (c − a) f_y + (a − b − c + d) f_x f_y, f = t² (3 − 2t)

  Material graphs baked to textures shading/lib.js

  • BimCraft
  • ModelCraft
  32-bit, display only