Calculations

Calculations

• Is the item big enough

  Refuses a hatch with fewer than 3 corners or under 100 mm², and a line under 10 mm.

  L = Σ √(Δx² + Δy²); refuse when L < 10 mm or |area| < 100 mm²

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• Move drafting items

  Every point of the picked items moved by one offset.

  (x, y) → (x + dx, y + dy)

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• Is a point inside the region

  Whether a point lies inside the outline and outside its holes.

  even-odd count of edges crossed by a ray toward +x

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• Clip a stroke to the region

  The pieces of a straight stroke that lie inside the region.

  t = ((c − a) × e) ÷ (d × e), u = ((c − a) × d) ÷ (d × e); keep 0 < t < 1, 0 ≤ u ≤ 1

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• Strokes of a hatch pattern

  The lines, dots or zigzags of a pattern inside a region, at its scale.

  line k of a family: n · p = ⌊(n_min − offset) ÷ gap⌋ × gap + k × gap, gap = pattern gap × scale

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• Pattern scale

  The scale of a hatch pattern as it is kept.

  scale = min(50, max(0.1, v)), or 1 when v is not a positive number

  Drafting lines and hatch regions drafting.js

  • BimCraft
  64-bit

• A typed point

  The next point of a drafting line from typed coordinates, a typed offset, a typed length and angle, or a typed length along the pointer's direction.

  polar: (x + L·cos(a·π ÷ 180), y + L·sin(a·π ÷ 180)); length only: last + L × (to − last) ÷ |to − last|

  Drafting lines and hatch regions drafting-ui.js

  • BimCraft
  64-bit

• Combine two transforms

  The product of two plan transforms, each six numbers.

  [a b c d e f]: x′ = a·x + c·y + e, y′ = b·x + d·y + f

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Block insert transform

  Where the content of a block lands: insertion point, rotation, scale, base point and its place in a row and column array.

  M = T(insert) · R(rotation) · S(sx, sy) · T(−base + (col × colSpacing ÷ sx, row × rowSpacing ÷ sy))

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Tilted block seen from above

  The plan transform of an insert whose extrusion direction is neither up nor down.

  Ax = unit(Wz × N) (or Wy × N when |Nx| and |Ny| < 1/64); Ay = unit(N × Ax); offset = N·z

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Downward extrusion

  Mirrors an entity in X when its extrusion direction points down.

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Transform points

  A list of points moved through a transform into the space's coordinates.

  x′ = a·x + c·y + e; y′ = b·x + d·y + f

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Segments in an arc

  How many straight segments stand in for an arc.

  n = max(8, min(128, ⌈|sweep| ÷ 2π × 72⌉))

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Points along an arc or circle

  An arc or circle as points on it.

  pᵢ = (cx + r·cos aᵢ, cy + r·sin aᵢ), aᵢ = a₀ ± sweep × i ÷ n

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Points along an ellipse

  An ellipse or elliptical arc as points on it.

  p(t) = c + R(rot) · (L·cos t, L·ratio·sin t), L = |major axis|, rot = atan2(my, mx)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Polyline with bulge arcs

  A polyline whose segments may be arcs, as points.

  θ = 4·atan(bulge); r = chord ÷ (2·sin(θ ÷ 2)); center = chord middle + normal × r·cos(θ ÷ 2)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Points along a spline

  A spline as points: its fit points, or samples of the B-spline from its control points.

  de Boor: dⱼ = (1 − a)·dⱼ₋₁ + a·dⱼ, a = (u − kᵢ) ÷ (kᵢ₊ₚ₋ᵣ₊₁ − kᵢ); point = (Σ w·P) ÷ Σ w

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• What a sheet viewport looks at

  The model point at the middle of a viewport on a sheet.

  vx = target_x + cos(tw)·cx + sin(tw)·cy; vy = target_y − sin(tw)·cx + cos(tw)·cy

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• A viewport's window on the model

  The box in model space that a viewport shows, used to say when it looks at nothing.

  half width = w ÷ 2 × viewHeight ÷ h; half height = viewHeight ÷ 2; corners turned by the twist

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Picture placement

  The transform from a picture's pixels to drawing coordinates.

  world = P + u·px − v·py + v·h (pixel y runs down; P is the lower left corner)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Lines of a paragraph of text

  Where each line of a multi-line text starts.

  pitch = h × 1.667 × line spacing; line i at insert + (top − pitch × i) along the up direction

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Text position, height and angle

  One line of text carried through the entity's transform.

  H = h × |second column of M|; angle = atan2(m₁, m₀) ± rotation (− when M mirrors)

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Paper unit of a sheet

  Whether a sheet is in inches or millimeters.

  unit = mm when the sheet is wider than 100 units, else inches

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Line weight in millimeters

  A stored line weight, in hundredths of a millimeter, as millimeters.

  mm = v ÷ 100 for 0 ≤ v ≤ 211, else 0.25

  DWG and DXF drawings compiled for the plan view dwg-compile.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Pointer position in drawing coordinates

  The drawing coordinates under the pointer, for the coordinate readout.

  x = cx + (px − W ÷ 2) ÷ zoom; y = cy − (py − H ÷ 2) ÷ zoom

  DWG and DXF drawings compiled for the plan view dwg-view.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Model to sheet through a viewport

  The transform that carries a model point onto the sheet through one viewport.

  k = h ÷ viewHeight; paper = center + k · R(twist) · (model − view point)

  DWG and DXF drawings compiled for the plan view dwg-view.js

  • BimCraft
  • TakeoffCraft
  64-bit

• Millimeters per paper unit

  The size of one paper unit of a sheet in millimeters, for line weights.

  1 for millimeters, 25.4 for inches

  DWG and DXF drawings compiled for the plan view dwg-view.js

  • BimCraft
  • TakeoffCraft
  Exact

• Combine two 3D transforms

  The product of two 3 × 4 transforms.

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Transform a point

  A point carried through a 3 × 4 transform.

  x′ = m₀·x + m₁·y + m₂·z + m₃ (and the same for y′, z′)

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Extrusion direction

  The unit extrusion direction of an entity, or straight up when it has none.

  N = (x, y, z) ÷ √(x² + y² + z²)

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Object coordinate system

  The transform from an entity's own plane to world coordinates.

  Ax = unit(Wz × N) (or Wy × N when |Nx| and |Ny| < 1/64); Ay = unit(N × Ax); columns Ax, Ay, N

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Thickness vector

  The offset between the bottom and top of an entity that has thickness.

  up = M · (N × thickness), without the translation

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Block insert transform in 3D

  Where the content of a block lands in 3D, with rows and columns.

  M = OCS · T(insert) · Rz(rotation) · T(col × colSpacing, row × rowSpacing, 0) · S(sx, sy, sz) · T(−base)

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Ellipse in 3D

  An ellipse or elliptical arc as 3D points.

  minor = (N × major) × ratio; p(t) = c + cos t · major + sin t · minor

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Half width of a wide polyline

  The offset to each side of a polyline that has a start and end width.

  w(k) = (sw + (ew − sw) × length so far ÷ segment length) ÷ 2, along the normal of the path at k

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Spline in 3D

  A spline as 3D points: its fit points, or samples of the B-spline from its control points.

  de Boor on (w·x, w·y, w·z, w); point = (Σ w·P) ÷ Σ w

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Extents of the drawing

  The smallest box round everything drawn, in drawing units.

  DWG and DXF drawings in 3D dwg-compile3d.js

  • BimCraft
  64-bit

• Section plane offset

  Where a section plane cuts the drawing, from a typed offset in drawing units.

  at = lo + clamp(offset × s, 0, hi − lo); offset shown = (at − lo) ÷ s, s = 100 ÷ largest extent

  DWG and DXF drawings in 3D dwg-view3d.js

  • BimCraft
  64-bit

• Section range

  The largest offset a section plane can take along an axis, in drawing units.

  size = (max − min) ÷ s

  DWG and DXF drawings in 3D dwg-view3d.js

  • BimCraft
  64-bit

• 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

• Viewport windows on the model

  For each sheet viewport, the half width and half height of the model area it shows.

  k = h ÷ viewHeight; half width = w ÷ 2 ÷ k; half height = h ÷ 2 ÷ k

  Equipment list from drawing tags dwg-equipment.js

  • BimCraft
  64-bit

• Sheets that show a point

  Which sheets show a given model point.

  u = dx·cos − dy·sin, v = dx·sin + dy·cos; inside when |u| ≤ half width and |v| ≤ half height

  Equipment list from drawing tags dwg-equipment.js

  • BimCraft
  64-bit

• Quantity of a row

  The scheduled quantity of an item tagged in several places.

  qty = Σ tag quantities

  Equipment list from drawing tags dwg-equipment.js

  • BimCraft
  64-bit

• Heat capacity of air per volume

  The energy to warm one cubic meter of air by one kelvin, in watt-hours.

  ρ·c = 1.2 kg/m³ × 1005 J/kgK ÷ 3600 = 0.335 Wh/m³K

  Heating and cooling loads per room energy-bim.js

  • BimCraft
  64-bit

• Does a ray reach a box

  Whether a ray passes through an element's bounding box within its reach.

  Heating and cooling loads per room energy-bim.js

  • BimCraft
  64-bit

• Where a ray crosses triangles

  Every distance along a ray at which it crosses a triangle of an element.

  det = e₁ · (d × e₂); u = (t · (d × e₂)) ÷ det; v = (d · (t × e₁)) ÷ det; distance = (e₂ · (t × e₁)) ÷ det

  Heating and cooling loads per room energy-bim.js

  • BimCraft
  64-bit

• Is a point inside a room

  Whether a point lies inside the closed shell of a space.

  inside when a slanted ray crosses the shell an odd number of times

  Heating and cooling loads per room energy-bim.js

  • BimCraft
  64-bit

• Room volume

  The volume enclosed by a room's triangles.

  V = |Σ a · (b × c)| ÷ 6 over the triangles

  Heating and cooling loads per room energy-bim.js

  • BimCraft
  64-bit