Calculations

Calculations

• Drawing coordinates to scene coordinates

  The positions sent to the GPU: centered on the drawing and scaled to about 100 units across.

  scene = (p − center) × 100 ÷ largest extent

  DWG and DXF views on screen dwg-view3d.js

  • BimCraft
  32-bit, display only

• Nearest line to the pointer

  The distance in pixels from the pointer to a line, for picking.

  t = clamp(((p − a) · (b − a)) ÷ |b − a|², 0, 1); d = |a + t·(b − a) − p|

  DWG and DXF views on screen dwg-view3d.js

  • BimCraft
  32-bit, display only

• Zoom to the drawing

  The zoom and center that show the whole drawing or sheet.

  zoom = min(W ÷ w, H ÷ h) × 0.92; center = middle of the bounds

  DWG and DXF views on screen dwg-view.js

  • BimCraft
  32-bit, display only

• Feature edges of one element

  The edges of an element where its faces meet at more than the feature angle, plus its free edges.

  edge drawn when n₁ · n₂ ≤ cos(angle), or when only one triangle uses it

  Element outlines in the IFC view ifc-vis.js

  • BimCraft
  32-bit, display only

• Edge segments of a batch

  Collects every element's edges into one array of line segments with a run per element.

  Element outlines in the IFC view ifc-vis.js

  • BimCraft
  32-bit, display only

• Cutting a story's elements

  The cut segments of every element the cut plane passes through, and the edges of what lies below or above.

  Plan view of elements drawn from their mesh model-plan.js

  • BimCraft
  32-bit, display only

• Cutting one triangle

  The segment where a triangle crosses the cut plane.

  t = (h − zᵢ) ÷ (zⱼ − zᵢ); point = pᵢ + t (pⱼ − pᵢ)

  Plan view of elements drawn from their mesh model-plan.js

  • BimCraft
  32-bit, display only

• Joining cut segments into loops

  Closed outlines from the cut segments, for the fill.

  Plan view of elements drawn from their mesh model-plan.js

  • BimCraft
  32-bit, display only

• Sun direction for a rendering

  The direction to the sun in the 3D view's axes.

  model = east × dₓ + north × d_y, with d_z kept; view = (mₓ, m_z, −m_y), normalized

  Rendering: sun direction, lamps and painted faces render-bim.js

  • BimCraft
  32-bit, display only

• Lamps from light fixtures

  A lamp for each light fixture: where it is, how bright and how wide.

  lumens = stored value, else watts × 80, else 800; position = middle of the fixture's box, just under it

  Rendering: sun direction, lamps and painted faces render-bim.js

  • BimCraft
  32-bit, display only

• Which side of an element a face is

  Names the axis of the element's placement that a face normal runs along, to pick the material painted on that side.

  axis = the placement column with the largest |n · column|

  Rendering: sun direction, lamps and painted faces paint-bim.js

  • BimCraft
  32-bit, display only

• Planes of a section cell

  The outward normal and a point of each plane that bounds a section cell, with tilted faces and extra planes.

  n = unit(n₀ + tan a × e_u + tan b × e_v); extra: n = (cos el cos az, cos el sin az, sin el)

  Section volumes in the 3D view section-volume.js

  • BimCraft
  32-bit, display only

• Corners and edges of a section cell

  The corners, faces and edges of the convex cell its planes bound, drawn as the cell's frame.

  corner = (d_a (n_b × n_c) + d_b (n_c × n_a) + d_c (n_a × n_b)) ÷ (n_a · (n_b × n_c))

  Section volumes in the 3D view section-volume.js

  • BimCraft
  32-bit, display only

• A plane through a point as an extra plane

  Turns a plane given by a normal and a point into the azimuth, elevation and offset a section cell keeps.

  az = atan2(u_y, u_x); el = asin(clamp(u_z, −1, 1)); off = u · (p − center)

  Section volumes in the 3D view section-volume.js

  • BimCraft
  32-bit, display only

• Clipping planes of a section box

  Turns each section cell's planes into the viewer's clipping planes and draws the cell's frame a hair inside it.

  viewer normal = turn(−n_x, −n_z, n_y); frame inset = Σ sizes × 1e-5 + 1e-4

  3D view drawing viewer3d.js

  • BimCraft
  32-bit, display only

• Section frame turned with the building

  Rotates a viewer point about the vertical through the model's center into the section frame.

  x′ = c_x + x cos a + z sin a; z′ = c_z − x sin a + z cos a

  3D view drawing viewer3d.js

  • BimCraft
  32-bit, display only

• Model bounds in the section frame

  The bounds of every drawn vertex in the turned section frame.

  box = min and max of turned vertex positions

  3D view drawing viewer3d.js

  • BimCraft
  32-bit, display only

• Child boxes of a node

  Stores the boxes of a node's four children as 8-bit offsets from the node's corner.

  scale = 2^e, the smallest with 255 × 2^e ≥ extent; lo = floor((box − origin) ÷ scale), hi = ceil(…)

  Ray casting for rendering and the render bench compute/bvh-wide.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Ray through the hierarchy

  Walks the hierarchy along a ray and returns the nearest hit distance.

  per axis: t = (box − origin) × (1 ÷ direction); hit if max of the near values ≤ min of the far values × (1 + 4×10⁻⁷)

  Ray casting for rendering and the render bench compute/bvh-wide.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Preview shading of a tile

  Shades each pixel of a tile from the surface it sees, lit from the eye.

  value = 0.08 + 0.72 × |n · d| ÷ |n|

  Ray casting for rendering and the render bench compute/preview-kernel.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Camera rays for one tile

  Casts the camera rays of a tile and counts the rays, the hits and the sum of the hit distances, for timing.

  direction = normalize(camera matrix · ((2 (x + jitter) ÷ W − 1) ÷ f_x, (1 − 2 (y + jitter) ÷ H) ÷ f_y, −1))

  Ray casting for rendering and the render bench compute/raycast-job.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Triangles in world space

  Every mesh triangle moved by its placement, cut by the section planes and stored for the kernels.

  Path tracer scene, light and sky buffers render/pt-pack.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Derived material values

  The relative refractive index, coat Fresnel average, extinction and dispersion term of a material record.

  μt = −ln(color) ÷ depth; B = (η − 1) ÷ ((abbe ÷ scale) × (1 ÷ 0.4861² − 1 ÷ 0.6563²))

  Path tracer scene, light and sky buffers render/pt-pack.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Physical sky color

  The sky's color in a direction for a sun position and haze.

  Path tracer scene, light and sky buffers render/pt-sky.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Light profile grid

  A lamp's IES light distribution as a grid of direction multipliers.

  multiplier = candela(θ, φ) × 4π ÷ flux

  Path tracer scene, light and sky buffers render/pt-ies.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Adding samples to a pixel

  The sum of sample colors and the sample count of every pixel in the reference tracer.

  Path tracer kernels render/pt-ref.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Fresnel reflectance of glass

  The share of light a clear surface reflects at an angle.

  F = (r∥² + r⊥²) ÷ 2

  Path tracer kernels render/pt-ggx.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Sample position

  The random-like number in [0, 1) used for one dimension of one sample of one pixel.

  Path tracer kernels render/pt-sampler.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Adaptive sampling test

  Decides when a pixel has had enough samples.

  v = max(0, (S₂ − S₁² ÷ J) ÷ (J − 1)); stop when √(v ÷ J) ≤ noise × (μ + 0.02)

  Path tracer kernels render/pt-adapt.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Lamp strength

  A lamp's lumens turned into the strength the renderers use.

  point or spot: Φ ÷ 4π × 0.02; panel: Φ ÷ π × 0.02

  Fast render lights and texture placement render/looks.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Sun direction for a pass

  A direction inside a small cone around the sun, one per render pass, for soft shadow edges.

  r = √h₅ × tan(degrees × π ÷ 180); a = 2π h₇

  Fast render lights and texture placement render/looks.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Texture coordinates of a point

  Where a point of a surface falls on its material's texture, in meters along and up the surface.

  u = p · (up × n) ÷ |up × n|; v = p · (n × u axis)

  Fast render lights and texture placement render/materials.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• 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

• 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

• 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

• Ambient occlusion of a texel

  The share of directions above a surface point that are not blocked nearby.

  open ÷ rays, directions (√u₁ cos 2πu₂, √u₁ sin 2πu₂, √(1 − u₁)) about the normal

  Material graphs baked to textures shading/texbake-job.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Decal geometry

  The triangles under a decal, clipped to its box and lifted off the surface.

  corner = o + x u + y v + z n + 0.0005 × face normal

  Material graphs baked to textures shading/decal.js

  • BimCraft
  • ModelCraft
  32-bit, display only

• Affine through three points

  The affine transform that carries three source points onto three destination points.

  det = (u₁ − u₀)(v₂ − v₀) − (u₂ − u₀)(v₁ − v₀); a = ((x₁ − x₀)(v₂ − v₀) − (x₂ − x₀)(v₁ − v₀)) ÷ det

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Cells of a warped picture

  The cells a picture is cut into so that each can be drawn with one affine transform.

  split while max |affine(p) − exact(p)| > tol over the fourth corner, the middle and the four edge midpoints

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Tile level for the screen

  Which level of a tile grid is drawn at the current zoom.

  the coarsest level with res ≤ 1.5 × screen resolution

  Map pictures warped onto the plan basemap.js

  • CadCraft
  • BimCraft
  • CivilCraft
  32-bit, display only

• Warp plan for a picture or tile

  How finely a map image is cut into patches so that it lands in the right place on screen.

  tolerance = 0.25 ÷ zoom project units, rounded to a half power of 2

  Map layers under the drawing maplayers.js

  • CadCraft
  • BimCraft
  32-bit, display only

• Drawing the patches

  Each patch of a map image drawn on the canvas.

  Map layers under the drawing maplayers.js

  • CadCraft
  • BimCraft
  32-bit, display only

• Drawing coordinates to screen position

  Where a drawing point is on screen, in CSS pixels of the active pane.

  sx = (x − cx)·zoom + w ÷ 2; sy = h ÷ 2 − (y − cy)·zoom (with a turn: the difference is rotated first)

  2D drawing on screen view.js

  • CadCraft
  32-bit, display only