Calculations

Calculations

• Constraint equations

  The residual of each constraint: how far the current geometry is from meeting it.

  coincident: P₀ − P₁; distance: |P₁ − P₀| − d; parallel: (u × v) ÷ (|u|·|v|); tangent: distance(center, line) − r; angle: atan2(u × v, u·v) − θ·π ÷ 180

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Circle of a polyline arc segment

  The center and radius of a polyline's arc segment as the solver sees it.

  θ = 4·atan(b); r = |c ÷ (2·sin(θ ÷ 2))|; h = ± r·cos(θ ÷ 2); center = chord midpoint + normal × h

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Derivatives

  How each equation changes with each coordinate it depends on.

  ∂f/∂x ≈ (f(x + h) − f(x − h)) ÷ 2h, h = 1e-6 × max(1, |x|)

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Cholesky solve

  Solves the damped normal equations for one step.

  A = L·Lᵀ; L·y = b; Lᵀ·z = y

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Newton iteration

  Repeats damped least-norm steps until every equation is met.

  (J·Jᵀ + μ·max(scale, 1e-12)·I)·y = −r; dx = Jᵀ·y; x ← x + α·dx, α = 1, ½, ¼ …

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Writing the solution back

  The changed objects after a solve, and the remaining residual.

  residual = √(Σ rᵢ²)

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Radius sign and arc angles

  Puts solved values back into an object in its stored form.

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Degrees of freedom and redundant constraints

  How many independent constraints there are, which ones add nothing, which conflict and which objects are fully fixed.

  v = row ÷ |row|; v ← v − (v·b)·b for each kept row b, twice; redundant when |v| < 1e-7; degrees of freedom = variables − rank

  2D constraint solver gsolve.js

  • CadCraft
  64-bit

• Arcs as straight pieces

  The points that stand in for an arc or circle in a hatch boundary.

  n = max(4, ⌈|sweep| ÷ (2π ÷ 128)⌉); point = c + r·(cos a, sin a)

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• A closed object as a boundary

  The boundary loop of a circle, a closed polyline or a full ellipse picked as an object.

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Boundary tolerance

  The distance within which boundary pieces count as touching.

  tol = 1e-9 × max(width, height, 1)

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Where two boundary pieces cross

  The parameters at which two segments cross or touch, so each can be split there.

  u = ((c − a) × q) ÷ (r × q); v = ((c − a) × r) ÷ (r × q)

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Merging boundary corners

  The corner points of the boundary, with points that nearly coincide made one.

  snap = 1000 × tol; split point = start + (end − start)·t

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Faces and their areas

  Every closed region the boundary pieces make, with the signed area used to choose among them.

  area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Point inside a face

  Whether the picked point, or an island, lies inside a face.

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Pattern scale and angle

  The pattern's line families at the hatch's scale, angle and origin.

  k = scale ÷ (1 metric, 25.4 inch); origin = R(angle)·(x, y)·k; spacing and dashes × k

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Pattern lines clipped to the boundary

  The dashes of every pattern line that fall inside the hatch.

  line k origin = O + k·(dx·u + dy·n); crossing: t = d₁ ÷ (d₁ − d₂) along an edge

  Hatch boundaries and pattern lines hatch.js

  • CadCraft
  64-bit

• Rounding of stored values

  The two rounding helpers every stored number of an item goes through.

  r4(v) = round(v × 10000) ÷ 10000; r2(v) = round(v × 100) ÷ 100

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Block to millimeters

  The transform from a block's own coordinates and units to the item's millimeters.

  M = [mm, 0, 0, mm, −base.x·mm, −base.y·mm]; scale of a transform = √|det|; angle = r4(atan2(y, x) in degrees)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Lines, circles and arcs of a block

  A block's lines, circles and arcs as plan entities in millimeters.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Nested block references

  The transform of a block reference inside the block, with its rows and columns.

  local = T(insert) · R(rotation) · S(sx, sy) · T(−base + column·spacing ÷ sx, row·spacing ÷ sy)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Mesh body from a block's 3D faces

  The item's 3D body and height when the block holds 3D faces.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Arcs as points, 5° steps

  The points that replace a circle or arc that cannot stay an arc.

  n = max(8, ⌈sweep ÷ 5°⌉); point = c + r·(cos a, sin a)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Splines as points

  The points that replace a block's spline in the item's plan.

  de Boor recurrence at u = lo + (hi − lo)·s ÷ n, n = max(16, 8 × control points)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Arc points for bounds

  The points of an arc used to find a plan's bounds.

  n = max(4, ⌈sweep ÷ 10°⌉); point = c + r·(cos a, sin a)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Bounds of a plan

  The box round an item's plan, from which its width and depth are taken.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Box body on a plan

  A box body standing on the plan's bounds with a given height.

  w = max x − min x; d = max y − min y; each of x, y, z, w, d, h rounded to 4 decimals

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Item from a mesh file

  A mesh file's triangles as an item body in millimeters, with its footprint and sizes.

  p = file point × unit in mm; Y-up: (x, y, z) → (x, −z, y); stored = r4(p − (center x, max y, min z))

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• STL corners

  The corner points of an STL file's triangles.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• glTF accessor values

  The numbers of a glTF position or index accessor.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• glTF node transforms

  The matrix that places each glTF node's mesh.

  M = T · R(q) · S, R from the quaternion (qx, qy, qz, qw)

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Convex hull footprint

  The outline round a mesh item's points, used as its plan.

  monotone chain; turn = (a − o) × (b − o), popped when ≤ 0

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Box of a body

  The box a body fills: a box body itself, or the extent of a mesh body.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Box item from width, depth and height

  The rectangle plan and box body of an item given only by its sizes.

  x from −w ÷ 2 to +w ÷ 2; y from −d to 0; label height = min(0.18 d, 150), reduced to fit 90% of w

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Sizes from attributes

  Width, depth and height read from a block's attributes.

  Library item geometry from blocks, meshes and tables libitem.js

  • CadCraft
  • BimCraft
  64-bit

• Meters per drawing unit

  The table that converts drawing units to the 3D model's meters.

  inch 0.0254; foot 0.3048; yard 0.9144; mile 1609.344; no unit 0.001

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  Exact

• Outline of a closed 2D object

  The ring of points a solid is extruded from, and its area.

  circle: 48 points at radius r; arc: steps of at most 7.5°; area = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Extruded solid

  A solid made by pushing a ring up or down by a height.

  local ring = ring − (min x, min y); placed at (min x, min y, z₀)

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Cylinder

  A cylinder from a center, radius and height.

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Volume of each solid

  The volume listed for every closed solid.

  volume = |V of the definition| × |det of the placement's 3 × 3 part|

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Changing the drawing's units

  Scales the 3D model so it stays the same size on the drawing when the units change.

  ratio = m per new unit ÷ m per old unit; every vertex and placement offset × ratio

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• 3D faces written to a file

  The model's faces as 3D faces in drawing units for DWG and DXF.

  file point = placed vertex in meters ÷ k, k = meters per drawing unit

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• 3D faces inside blocks

  Places the 3D faces a block holds once for every reference of that block.

  (x′, y′) = M·(x, y); z′ = z

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• 3D faces read from a file

  A file's loose 3D faces brought into the model in meters.

  corner in meters = file corner × k

  3D solids, faces and units in a drawing model3d.js

  • CadCraft
  64-bit

• Underlay shapes and text

  The underlay's paths as plotted segments and fills, and its text.

  B(t) = (1 − t)³·P₀ + 3(1 − t)²t·P₁ + 3(1 − t)t²·P₂ + t³·P₃, t = s ÷ 8; line width = max(0.05, w × √|det|); text height = h × √|det| ÷ FONT_EM

  PDF underlay in a plot pdf-emit.js

  • CadCraft
  64-bit

• Finding the paper

  The four corners of the sheet of paper in the photo.

  corner = (pixel + 0.5) × k on a copy reduced by the whole factor k; fill = bright pixels inside ÷ outline area

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Perspective matrix from four corners

  The 3 × 3 matrix that takes the straight sheet's corners to the corners found in the photo.

  8 equations in h₁ … h₈ with h₉ = 1, solved by Gauss–Jordan elimination

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Applying the perspective matrix

  Where a point of the straight sheet lies in the photo.

  w = h₇x + h₈y + h₉; x′ = (h₁x + h₂y + h₃) ÷ w; y′ = (h₄x + h₅y + h₆) ÷ w

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Sheet proportions

  The long-to-short side ratio of the straightened sheet.

  w = mean of top and bottom; h = mean of left and right; ratio = max ÷ min

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit

• Drawing units per pixel

  The size of one pixel of the straightened sheet in drawing units.

  units per px = long side (mm) ÷ long side (px) × scale ÷ mm per unit

  Sketch photo to drawing photosketch.js

  • CadCraft
  64-bit