Calculations

Calculations

• Cone

  A cone with a 32-sided base of radius R and its apex at height S.

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Sphere

  A sphere of radius R resting on z = 0, made of 12 rings of 24 faces.

  θ = π·i ÷ 12; φ = 2π·j ÷ 24; p = [R·sin θ·cos φ, R·sin θ·sin φ, R − R·cos θ]

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Roof

  A triangular prism: a triangle across x pulled along y from −R to +R.

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Align

  Moves each selected placement so the minimum, middle or maximum of its box on one axis meets that of the whole selection.

  target = lo, hi or (lo + hi) ÷ 2; move = target − own value; m[12 + axis] += move

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Drop to ground

  Moves each selected placement up or down so the bottom of its box is at z = 0.

  m[14] += −box.min.z

  Beginner shapes, align and drop to ground beginner.js

  • ModelCraft
  64-bit

• Rounding to 0.1 mm

  The rounding applied to every stored mesh coordinate, plan polygon point and item size.

  r1(v) = round(v × 10) ÷ 10

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Mesh point

  Adds one point to a part's mesh.

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Ring loft

  Trunks, pots, limbs and conifer tiers: rings of points round an axis joined into a capped solid.

  a = 2π·i ÷ seg; p = [cx + rx·cos a, cy + ry·sin a, z]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Crown and head blobs

  Tree crowns, shrubs, rocks and heads: an icosphere scaled to an ellipsoid with a seeded bump.

  t = (1 + √5) ÷ 2; unit point p; k = 1 + bump × (2·rand − 1); point = c + p × [rx, ry, rz] × k

  Trees, people and cars from numbers entourage.js

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  64-bit

• Wheel

  A car wheel: a 14-sided cylinder along y.

  a = 2π·i ÷ seg − π ÷ 2; p = [x + r·cos a, y, z + r·sin a]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Canopy plan symbol

  The plan outline of a tree: a 14-point irregular ring, a trunk circle and a cross.

  a = 2π·i ÷ 14; k = r × (0.86 + 0.18·rand); p = [k·cos a, k·sin a]

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Library item sizes

  The width, depth and height written into the library item.

  Trees, people and cars from numbers entourage.js

  • ModelCraft
  64-bit

• Signed polygon area

  The signed area of a plan polygon.

  area = ½ Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Room from a floor face

  The outline, holes, height and area of a room from a level face that looks up.

  floor if n_z > 0.999; plan = model point × 1000; area (m²) = (outer − Σ holes) ÷ 10⁶

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Doors and windows near the room

  Which starter doors and windows belong to the room, with their middle, width and sill height in millimeters.

  middle = box center × 1000; width = max(Δx, Δy) × 1000; near if distance to outline ≤ 400

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Placement matrix for a piece

  The matrix that lays a furniture piece on the planner's footprint.

  x axis = (fp₁ − fp₀) ÷ 1000 ÷ W; y axis = (fp₃ − fp₀) ÷ 1000 ÷ D; origin = fp₀ ÷ 1000 − x axis·min.x − y axis·min.y; z = floor − min.z

  Room outline and furniture placement furnish-mc.js

  • ModelCraft
  64-bit

• Reading position and turn

  The x, y, z and turn a control program reads for a placement.

  x, y, z = m[12], m[13], m[14]; rotZ = atan2(m[1], m[0]) × 180 ÷ π

  Control program host: placement position and turn plc-scene-mc.js

  • ModelCraft
  64-bit

• Writing position and turn

  Applies the values a control program set to the placement's matrix when the run is kept.

  sx = √(m₀² + m₁²); sy = √(m₄² + m₅²); a = rotZ × π ÷ 180; m₀ = sx·cos a, m₁ = sx·sin a, m₄ = −sy·sin a, m₅ = sy·cos a

  Control program host: placement position and turn plc-scene-mc.js

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  64-bit

• Triangle normals, areas and boxes

  The unit normal, area and bounding box of every triangle, and the size of the whole mesh.

  n = (b − a) × (c − a); area = |n| ÷ 2; unit normal = n ÷ |n|

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Nearest face along a ray

  The distance to the first triangle a ray meets between two limits, and whether it meets it from behind.

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

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Ray and box

  Whether a ray passes through a box within the current distance limits.

  t_near = (min − o) × (1 ÷ d); t_far = (max − o) × (1 ÷ d), per axis

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point to triangle distance

  The squared distance from a point to the closest point of one triangle.

  closest point q by the corner, edge and face regions; d² = |p − q|²

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Distance to the nearest face

  The distance from a point to the nearest triangle within a search radius.

  d = √(min over triangles of d²), limited to r

  Rays and distances on a printer mesh print-geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Inward offset of every corner

  Moves each corner inward so that every face around it moves by the wall thickness, and rejects the result where an offset would fold.

  q = p − wall × m, with n·m = 1 for every face direction n at the corner

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Corner move from the face directions

  The vector m with n·m = 1 for each face direction at a corner.

  one direction: m = n; two: m = (n₁ + n₂) ÷ (1 + n₁·n₂); more: (NᵀN) m = Nᵀ1

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Grid step of the distance field

  Chooses the grid step and size for the distance field.

  h = min(wall ÷ 2.5, largest size ÷ 24); h ← 1.15·h while Π(⌈sizeₖ ÷ h⌉ + 5) > maxCells; band B = wall + 2h

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Signed distance field

  The distance from every grid point to the outside surface, negative where the point is deeper inside than the wall.

  d² = min over triangles of |x − closest point|², capped at B²; φ = ±√d² + wall (− inside)

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  32-bit in part

• Cavity surface point on a grid edge

  Where the cavity surface crosses a grid edge between a point in the hollow and a point in the material.

  s = φₐ ÷ (φₐ − φ_b), clamped to [0.001, 0.999]; p = pₐ + (p_b − pₐ) × s

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  32-bit in part

• Signed volume of a mesh

  The volume enclosed by a triangle mesh, used for the body and for the cavity.

  V = Σ a·(b × c) ÷ 6

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Choosing the method and reporting the result

  Runs the offset, falls back to the field, and returns the hollow body, the cavity and their volumes.

  cavity volume = −V(cavity mesh)

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Drain hole cylinder

  A 24-sided cylinder along the inward normal of a picked face, long enough to reach the hollow.

  depth = first face behind the pick; top = p + n·extra; bottom = p − n·(depth + extra); ring = c + r·(cos θ·u + sin θ·w)

  Hollowing a body to a wall thickness print-hollow.js

  • ModelCraft
  64-bit

• Triangles of a placement

  Every triangle of a placement and what is nested inside it, in the open context's coordinates.

  p′ = M · p, M = product of the nested placement matrices

  Print tool results back in the model print-solid.js

  • ModelCraft
  64-bit

• Welding triangles into a mesh

  Joins corners that fall in the same cell of a 1e-7 m grid and marks edges sharper than a given angle.

  key = round(x ÷ weld), round(y ÷ weld), round(z ÷ weld); sharp if n₁·n₂ < cos(sharpDeg)

  Print tool results back in the model print-solid.js

  • ModelCraft
  64-bit

• Triangles into the boolean library

  Hands a part's corners to manifold-3d, measured from a nearby origin.

  q = p − o; corners welded where round(q × 10⁷) agree; stored as 32-bit

  Print tool results back in the model print-solid.js

  • ModelCraft
  32-bit in part

• Union and subtraction

  The triangles of the kept parts joined, less the cut parts.

  o = middle of the box of all parts; result corner = o + 32-bit corner from the library

  Print tool results back in the model print-solid.js

  • ModelCraft
  32-bit in part

• Volume of a triangle set

  The volume the triangles enclose, used to check that a solid kept the result's volume.

  V = Σ a·(b × c) ÷ 6

  Print tool results back in the model print-solid.js

  • ModelCraft
  64-bit

• Result as a solid on a 1 µm grid

  Turns the result's triangles into a solid group of flat faces, or reports why it stays a mesh.

  g(x) = round(x × 10⁶) ÷ 10⁶; keep if closed and |V_solid − V| ≤ |V| × 10⁻⁴ + 10⁻¹⁵

  Print tool results back in the model print-solid.js

  • ModelCraft
  64-bit

• Thickness behind a triangle

  The distance from a triangle's middle, straight in, to the far side of the wall.

  origin = centroid − n·eps; thickness = distance to the first face met from behind + eps

  Wall-thickness check print-walls.js

  • ModelCraft
  32-bit in part

• Values for unmeasured triangles

  On a decimated check, gives each unmeasured triangle the value of the nearest measured one across shared edges.

  Wall-thickness check print-walls.js

  • ModelCraft
  32-bit in part

• Thinnest wall and thin area

  The smallest thickness found and the total area of triangles under the limit.

  thinnest = min(values); thin area = Σ area of triangles with value < min

  Wall-thickness check print-walls.js

  • ModelCraft
  32-bit in part

• Thin regions

  Thin triangles joined across shared edges into regions, each with its area, its thinnest value and where that is.

  region area = Σ triangle areas; at = centroid of the thinnest triangle

  Wall-thickness check print-walls.js

  • ModelCraft
  32-bit in part

• Overhanging faces and their area

  Which triangles need support, and their total area.

  overhang if n_z < −cos((90 − angle)·π ÷ 180) and not on the bed; area = Σ triangle areas

  Supports preview and material estimate print-supports.js

  • ModelCraft
  64-bit

• Support points under the overhangs

  Grid points in plan that fall inside an overhanging triangle, with their height on it.

  s = max(spacing, √(overhang area ÷ maxPoints)); (x, y) = ((i + ½)·s, (j + ½)·s); z = λ₁z_a + λ₂z_b + λ₃z_c

  Supports preview and material estimate print-supports.js

  • ModelCraft
  64-bit

• Where a support lands

  The height below each support point where a straight line down meets the model, or the bed.

  bottom = z − eps − distance down to the first face, else bed z

  Supports preview and material estimate print-supports.js

  • ModelCraft
  64-bit

• Column volume

  The material one column adds to the estimate.

  columns: V = s² × density × h; tree style: V = π × tip² × h; drawn radius = s × √(density ÷ π)

  Supports preview and material estimate print-supports.js

  • ModelCraft
  64-bit

• Tree supports

  Joins nearby support points to one trunk and adds the branches' and trunk's volume.

  joint z = min(zᵢ − horizontal distanceᵢ ÷ tan(branch)); trunk radius = tip × √k; V = Σ π·tip²·lengthᵢ + π·r²·(joint z − base z)

  Supports preview and material estimate print-supports.js

  • ModelCraft
  64-bit

• Fewest cells that fit the bed

  How many equal cells along x, y and z make every piece fit the printer bed, allowing a piece to lie on its side.

  start: nₖ = ⌈sizeₖ ÷ largest bed side − 10⁻⁹⌉; cellₖ = sizeₖ ÷ nₖ (+ extra on a cut axis); fits if cellₖ ≤ bed_p(k) + 10⁻⁹

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Cut planes of the grid

  The planes that cut the model's box into the chosen number of cells.

  d = lo + (hi − lo) × i ÷ n

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Planes and triangles about the model's middle

  Normalizes the given planes and moves planes and triangles to coordinates measured from the middle of the model's box.

  n̂ = n ÷ |n|; d = d ÷ |n|; d_local = d − n̂·org; p_local = p − org

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  64-bit

• Model into the boolean library

  Hands the model's corners to manifold-3d, measured from the middle of its box.

  q = p − org; corners welded where round(q × 10⁵) agree; stored as 32-bit

  Splitting a model into printable pieces print-split.js

  • ModelCraft
  • PartCraft
  32-bit in part