Calculations

Calculations

• Reading a text file

  Parses every 'vertex x y z' line.

  STL import and export io/stl.js

  • ModelCraft
  32-bit in part

• Unit factors

  The factor from the file's unit to meters.

  mm 0.001; cm 0.01; m 1; inch 0.0254; foot 0.3048

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Welding corners and applying the unit

  Makes one vertex for every distinct corner and scales it to meters.

  same vertex when x, y and z are equal as text; stored as (x, y, z) × unit factor

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Triangles with no area

  Drops triangles whose corners lie on one line.

  dropped when |(b − a) × (c − a)| < 1e-18

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Exported triangles

  Writes each triangle's normal and corners in millimeters.

  n = (b − a) × (c − a), made unit; corner (mm) = world corner (m) × 1000

  STL import and export io/stl.js

  • ModelCraft
  64-bit

• Rounding of every parameter

  Rounds each number of a converted material.

  v → the number nearest v with 7 significant digits

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Film thickness and dispersion

  A thin film's thickness in micrometers and a glass's Abbe number.

  thickness (µm) = maximum thickness (nm) ÷ 1000; Abbe number = 20 ÷ dispersion

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Emission strength

  Splits an emissive color into a color with a peak of 1 and a luminance.

  peak = largest channel; color = channels ÷ peak; luminance = peak × strength

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Shininess to roughness

  A roughness from a Blinn–Phong shininess exponent.

  roughness = (2 ÷ (s + 2))^¼, clamped to [0, 1]

  Imported material parameters io/pbr.js

  • ModelCraft
  64-bit

• Knot span

  Finds the knot interval that holds a parameter.

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• de Boor's algorithm

  A point of a B-spline from its control points and knots.

  a = (t − k₀) ÷ (k₁ − k₀); d_j = (1 − a) d_(j−1) + a d_j, repeated degree times

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Rational point

  Divides a homogeneous point by its weight.

  (x, y, z) = (X, Y, Z) ÷ w

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a NURBS surface

  A surface point from its control net.

  de Boor along v for each active row, then along u, then the division by the weight

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a curve

  A point on a line, arc, polyline, NURBS curve or chain of curves at a parameter.

  arc: a = angle mapped from the parameter; p = o + r cos a × x + r sin a × y

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• A curve as a polyline

  Cuts a curve into chords that stay within a tolerance of it, keeping every knot and corner.

  a piece is halved while its middle or quarter points lie farther than tol from the chord, at most 10 times

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Rotation about an axis

  Turns a point about a line by an angle, for surfaces of revolution.

  p′ = o + v cos a + (z × v) sin a + z (z · v)(1 − cos a)

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Point on a surface

  A point on a plane, NURBS surface, surface of revolution or sum surface.

  plane: o + a x + b y; revolution: the profile point turned about the axis; sum: p(u) + q(v) + base

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• Surface normal

  The unit normal of a surface at a point.

  ∂u ≈ S(u + h, v) − S(u − h, v); ∂v likewise; n = ∂u × ∂v, made unit; h = 1e-5 × parameter range

  3DM curves and surfaces: points on them io/3dm/geom.js

  • ModelCraft
  • PartCraft
  64-bit

• A saved mesh as triangles

  Copies a mesh record's points and splits its quads into triangles.

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Joining points that coincide

  Makes one vertex of points that sit at the same place, so the faces of a solid close up.

  q = max(diagonal × 1e-9, 1e-12); same vertex when round(x ÷ q), round(y ÷ q) and round(z ÷ q) are equal

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Sampling a shared edge

  Samples an edge once on its 3D curve, for both faces that share it.

  fraction s = (t − t₀) ÷ (t₁ − t₀) along the edge's own direction

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Scaling the parameter plane

  Scales u and v so a step in each is about as long on the surface, before triangulating.

  L_u, L_v = lengths of the two mid-lines measured with 8 chords; k_u = L_u ÷ Δu; k_v = L_v ÷ Δv

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Halving an interior edge

  Decides whether a triangle edge needs a new point at its middle, and adds it.

  halve when the edge is longer than 2 tol and (|S(mid) − chord middle| > tol or n_a · n_c < cos 20°)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Grid step from the surface's bend

  How far apart interior points are placed, from how much the surface bends.

  k = angle between neighboring normals ÷ their distance (75th percentile of a 9 × 9 grid); h = min(√(8 tol ÷ k), 0.349 ÷ k), not below max(L) ÷ 400 or √(L_u L_v ÷ 40000)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Grid points inside a face

  Picks the grid points that lie inside the face's loops and not too near a side.

  inside by counting crossings of the row (even-odd); kept when at least half a grid step from every side

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Extrusions

  Walls along the path from each profile, and caps from the closed profiles with their holes.

  frame: T = path direction, X = up × T, Y = T × X; point = origin + x X + y Y; profile closed when its ends are within max(tol, 1e-9)

  3DM surfaces into triangles io/3dm/tess.js

  • ModelCraft
  • PartCraft
  64-bit

• Tessellation tolerance

  The distance triangles may stand off a surface, per object.

  base = given tolerance, else the file's, else 0.01; tol = max(base, 0.0015 × diagonal of the object's rough box)

  3DM surfaces into triangles io/3dm/scene.js

  • ModelCraft
  • PartCraft
  64-bit

• Rough box of an object

  A box round an object from its vertices, curve break points and control points, used only to set the tolerance.

  3DM surfaces into triangles io/3dm/scene.js

  • ModelCraft
  • PartCraft
  64-bit

• Preset curves

  The control points of a new curve: an S, a Bézier circle, a NURBS circle or a straight path.

  Bézier circle: handle length = r × 0.5522847498; NURBS circle: 8 points on an octagon of radius r ÷ 0.9254

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Automatic handles

  Places a Bézier point's handles when they are automatic or vector.

  auto: along (next − previous) made unit, a third of each neighbor's distance; vector: a third of the way to each neighbor

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Point on a cubic Bézier segment

  A point of a segment at a fraction t.

  p = (1 − t)³ a + 3 (1 − t)² t b + 3 (1 − t) t² c + t³ d

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Point on a NURBS spline

  A point of a NURBS spline at a parameter.

  de Boor on (x w, y w, z w, w), then (x, y, z) ÷ w

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Sampling a spline

  Points along a spline with tangents, the fraction of the length at each, and the radius and tilt there.

  resolution samples per segment; tangent = (next − previous) made unit; s = Σ chord lengths so far ÷ total

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Frames along the curve

  A side and an up direction at every sample that twist as little as possible, with the tilt applied.

  double reflection from one sample to the next; closed curve: the leftover angle is undone in proportion to s

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Taper along the curve

  The scale of the cross-section at a fraction of the length.

  Catmull-Rom through the taper values, not below 0

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Cross-section

  The outline swept along the curve: a circle, a stadium or a flat ribbon.

  circle: 4 (resolution + 1) points at radius depth; stadium: two half circles offset by ± extrude

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Sweeping the section

  Places the cross-section at every sample and joins the rings into faces.

  point = sample + side × (−x k) + up × (y k); k = radius × taper

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Offsetting a loop

  Moves each vertex of a flat outline outward by a distance, with mitered corners.

  m = (n₁ + n₂) made unit; p′ = p + m × d ÷ max(0.25, m · n₁)

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Filled shapes

  Sorts closed flat loops into outlines and holes and builds the top, bottom and edge of the filled shape.

  nesting: a loop inside an even number of others is an outline; direction from the sign of the shoelace area

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Adding a point in a segment

  Splits a segment at its middle; a Bézier segment keeps its shape.

  de Casteljau at t = ½

  Curve objects mesh/curve.js

  • ModelCraft
  64-bit

• Moving, scaling and turning points

  The new place of each selected point and handle while it is dragged.

  grab: p + (M⁻¹ w₁ − M⁻¹ w₀); scale: c + (p − c) × |a₁| ÷ max(1, |a₀|); turn: Rodrigues' formula about the view axis through c

  Curve objects mesh/curve-tool.js

  • ModelCraft
  64-bit

• A new point at the pointer

  Where a point added after the end of a spline goes.

  world point = pick ray ∩ plane through the last point; local = M⁻¹ × world; flat curve: z = 0

  Curve objects mesh/curve-tool.js

  • ModelCraft
  64-bit

• Selection center

  The mean position of the selected vertices: the pivot for rotate, scale and spin.

  c = (Σ pᵢ) × (1 ÷ n)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Point on an axis nearest the pointer

  The point of a line through w along dir that is nearest the pointer's ray; used for axis-locked moves and extrude distances.

  s = (b·e − c·d) ÷ (a·c − b²); point = w + d₁·s

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Grab, rotate and scale

  Builds the move applied to every affected vertex in world space.

  grab: p′ = p + d; rotate: p′ = R(k, θ)(p − P) + P; scale: p′ = P + f·(p − P), or along a locked axis a: p′ = p + a·((p − P)·a)·(f − 1)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Rotation about an axis

  Turns a point about the axis k through c by the angle a.

  q = p − c; p′ = c + q·cos a + (n × q)·sin a + n·(n·q)·(1 − cos a)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Extrude distance

  Moves the new vertices of an extrusion by the dragged or typed distance along the region's normal, or along each face's own normal.

  d = (line point now − line point at start)·dir; p′ = p + dirL·d, dirL = M⁻¹·dir

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Knife point on an edge

  Where along an edge a knife click lands, as a fraction of the edge from its start.

  s = clamp((q − a)·(b − a) ÷ L², 0, 1)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Bisect plane

  The cutting plane through two clicked points as seen from the camera, in the mesh's own space.

  n = unit((p₂ − p₁) × ray direction); plane = {M⁻¹·p₁, unit(M⁻¹ applied to n as a direction)}

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit

• Spin center, axis and angle

  The center, axis and total angle handed to the spin operator, in the mesh's own space.

  center = M⁻¹·c; axis = unit(M⁻¹ applied to the world axis); angle = typed × π ÷ 180, or −(a₁ − a₀)

  Mesh edit moves: grab, rotate, scale, extrude, spin, knife, bisect mesh/edit-tool.js

  • ModelCraft
  64-bit