Calculations

Calculations

• 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

• Proportional editing weights

  For every vertex, how much of the selected vertices' move it follows: 1 for the selected ones, falling to 0 at the radius.

  t = distance ÷ radius; smooth: x²(3 − 2x) with x = 1 − t; sphere: √(1 − t²); root: √x; linear: x; sharp: x²; constant: 1

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Set a crease

  Stores the crease (0 smooth to 1 sharp) on each listed edge for subdivision.

  v = clamp(value, 0, 1)

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Crease shared by edges

  The crease the selected edges have in common, shown as the starting value when a crease is edited.

  Proportional editing weights and edge creases mesh/ops.js

  • ModelCraft
  32-bit in part

• Applying a move with proportional weights

  Writes each affected vertex's new position while a grab, rotate or scale is in progress.

  q = T(p); r = p + (q − p)·w; stored position = M⁻¹·r

  Proportional editing weights and edge creases mesh/edit-tool.js

  • ModelCraft
  32-bit in part

• Crease from the pointer or the wheel

  Turns the pointer's distance from the selection's center, or a wheel step, into a crease between 0 and 1.

  pointer: crease = clamp(round(px ÷ 2) ÷ 100, 0, 1); wheel: crease = clamp(round((crease ± 0.1) × 10) ÷ 10, 0, 1)

  Proportional editing weights and edge creases mesh/edit-tool.js

  • ModelCraft
  32-bit in part

• Vertex normals

  A unit normal per vertex from the faces round it, weighted by face area.

  N(v) = unit(Σ Newell normal of each face at v)

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Inset corner

  Where a face corner goes when both of its edges move inward by d.

  bis = unit(in₁ + in₂); c = bis·in₁; k = d ÷ c; p′ = p + bis·k

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Inset of a region's border

  The new border vertices of a region inset, each from the two border edges that meet at it, raised by the depth.

  n = unit(n(face in) + n(face out)); p′ = insetCorner(p, a, b, n, thickness) + n·depth

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Bevel: slide along plain edges

  How far a new point slides along an edge that is not bevelled, so the bevel is the stated width inside the face.

  sin = |u_in × u_out|; slide = width ÷ sin; point = v + u·min(slide, cap)

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Bevel: where two bevelled edges meet

  The point inside a face where the offset lines of two bevelled edges cross.

  k = width ÷ |u_in × u_out|; point = v + (u_in + u_out)·k

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Bevel: profile points

  The points between the two sides of a bevelled edge when it has more than one segment.

  M = (p₀ + p₁) ÷ 2; Q = M + (p_c − M)·2·profile; B(t) = (1 − t)²·p₀ + 2t(1 − t)·Q + t²·p₁, t = k ÷ segments

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Shrink or fatten

  Moves each listed vertex along its normal by d.

  p′ = p + N(v)·d

  Inset, bevel and moves along normals mesh/ops.js

  • ModelCraft
  64-bit

• Even cuts along an edge

  The new vertices that divide an edge into cuts + 1 parts, shared by both faces on the edge.

  s = k ÷ (cuts + 1); with slide: s = clamp(s + slide ÷ (cuts + 1), 0.001, 0.999); p = A + (B − A)·s

  Subdivide, loop cut, bisect, knife and grid fill mesh/ops.js

  • ModelCraft
  64-bit

• Subdivide: interior points of a quad

  The points inside a quad whose four edges are all cut, placed so they follow the four sides.

  u = i ÷ N, w = j ÷ N; p = (1 − w)·S₀ + w·S₂ + (1 − u)·S₃ + u·S₁ − [(1 − u)(1 − w)·c₀₀ + u(1 − w)·c₁₀ + uw·c₁₁ + (1 − u)w·c₀₁]

  Subdivide, loop cut, bisect, knife and grid fill mesh/ops.js

  • ModelCraft
  64-bit

• Bisect by a plane

  Cuts every listed face's edges where the plane crosses them and splits the faces along the cut.

  dist(v) = P(v)·n − p·n; s = dₐ ÷ (dₐ − d_b); point = A + (B − A)·s

  Subdivide, loop cut, bisect, knife and grid fill mesh/ops.js

  • ModelCraft
  64-bit

• Knife: where the cut crosses an edge

  For each edge of the faces round the current point, whether the cutting plane crosses it and where.

  dₐ = (a − A)·n, d_b = (b − A)·n; s = dₐ ÷ (dₐ − d_b); X = a + (b − a)·s

  Subdivide, loop cut, bisect, knife and grid fill mesh/ops.js

  • ModelCraft
  64-bit

• Grid fill: interior points

  The points inside an open loop filled with a grid of quads, following the loop's four sides.

  u = i ÷ a, w = j ÷ b; p = ruled blend of the four side points − bilinear blend of the four corners

  Subdivide, loop cut, bisect, knife and grid fill mesh/ops.js

  • ModelCraft
  64-bit

• Merge at center

  Joins vertices into one at their mean position, at the first or last of them, or at a given point.

  p = (Σ pᵢ) × (1 ÷ n)

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Merge by distance

  Merges every vertex that lies within the given distance of a vertex already kept.

  cell = max(dist, 1e-9); merge v into w when |P(w) − P(v)| ≤ dist

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Spin

  Extrudes a region or open edges round an axis in a number of steps over a total angle.

  aᵢ = angle·i ÷ n; pᵢ = c + q·cos aᵢ + (k × q)·sin aᵢ + k·(k·q)·(1 − cos aᵢ), q = p₀ − c

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Source of a copied vertex

  Finds the vertex a fresh copy was made from, so spin can rotate from the starting position.

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Bridge: pairing two loops

  Chooses the rotation and direction that pair the vertices of two loops with the shortest total distance.

  d(r, dir) = Σᵢ |P(Aᵢ) − P(B₍r + dir·i₎)|; choose the smallest

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Slide along rails

  Puts each slid vertex a fraction t of the way from its starting place toward the far end of its rail.

  p′ = p + (P(w) − p)·|t|, w = rail end a for t ≥ 0, b for t < 0

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Smooth vertices

  Moves each vertex part of the way to the average of its neighbors, a number of times.

  c = (Σ neighbors) × (1 ÷ n); p′ = p + (c − p)·factor

  Merge, spin, bridge, slide and smooth mesh/ops.js

  • ModelCraft
  64-bit

• Scale

  Scales the mesh about its own origin by a factor for each axis.

  p′ = [x·f_x, y·f_y, z·f_z]

  Modifier stack: scale, mirror, array, solidify, displace, lattice, curve, shrinkwrap mesh/modifiers.js

  • ModelCraft
  64-bit

• Mirror

  Adds a mirrored copy across the x, y or z plane through the origin and merges the seam.

  copy: coordinate k → −k; seam: vertices with |coordinate k| ≤ dist merged within max(2·dist, 1e-12)

  Modifier stack: scale, mirror, array, solidify, displace, lattice, curve, shrinkwrap mesh/modifiers.js

  • ModelCraft
  64-bit

• Radial array

  Places copies turned round an axis through the origin.

  step = (angle·π ÷ 180) ÷ (n for a full circle, else n − 1); aᵢ = step·i; [u′, v′] = [u·cos aᵢ − v·sin aᵢ, u·sin aᵢ + v·cos aᵢ]

  Modifier stack: scale, mirror, array, solidify, displace, lattice, curve, shrinkwrap mesh/modifiers.js

  • ModelCraft
  64-bit