Calculations

Calculations

• 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

• Linear array

  Places copies at a step made of a fraction of the mesh's box size plus a fixed offset.

  d = size ∘ rel + abs; copy i: p′ = p + d·i

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

  • ModelCraft
  64-bit

• Frames along a polyline

  The point, tangent and two cross directions at a given length along a polyline; used by the curve modifier and the array along a curve.

  Lᵢ = Lᵢ₋₁ + |ptsᵢ − ptsᵢ₋₁|; f = (s − Lᵢ₋₁) ÷ (Lᵢ − Lᵢ₋₁); p = ptsᵢ₋₁ + (ptsᵢ − ptsᵢ₋₁)·f; b = unit(t × up), n = b × t

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

  • ModelCraft
  64-bit

• Curve modifier

  Bends the mesh along a curve: each vertex's coordinate on the chosen axis becomes a length along the curve.

  frame = F.at(q_k); p′ = frame.p − b·q_i1 + n·q_i2

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

  • ModelCraft
  64-bit

• Solidify

  Gives a surface thickness by making two copies moved along the vertex normals.

  o₁ = t·(offset + 1) ÷ 2, o₂ = t·(offset − 1) ÷ 2; p′ = p + N(v)·o

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

  • ModelCraft
  64-bit

• Displace noise

  A seeded smooth noise value between −1 and 1 at a point; the displace modifier multiplies it by the strength.

  value = 2 × trilinear blend of 8 lattice values, with smoothstep weights t²(3 − 2t), − 1

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

  • ModelCraft
  64-bit

• Lattice displacement

  The move of a point inside a cage whose boundary points were moved.

  u = (x − min_x) ÷ (max_x − min_x) (same for v, w); d = Σ faces − Σ edges + Σ corners (transfinite interpolation)

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

  • ModelCraft
  64-bit

• Closest point on a triangle

  The point of a triangle nearest a given point; used by shrinkwrap and by snapping onto a retopology target.

  vertex, edge or interior region chosen by the signs of six dot products; interior: a + ab·(v_b ÷ (v_a + v_b + v_c)) + ac·(v_c ÷ (v_a + v_b + v_c))

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

  • ModelCraft
  64-bit

• Nearest point on a triangle mesh

  Finds the nearest point of a target's triangles to a point, with its distance.

  r = size ÷ 64, doubled up to 30 times; best = min over triangles in the box p ± r of |closest − p|; accepted when best ≤ r

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

  • ModelCraft
  64-bit

• Vertex quadrics

  Adds each triangle's plane to the error quadric of its three vertices.

  n = unit((b − a) × (c − a)), d = −n·a; Q(v) += [a², ab, ac, ad, b², bc, bd, c², cd, d²]

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Border planes

  Adds a plane square to the face through each open border edge, so borders stay in place.

  n = unit((B − A) × face normal), d = −n·A; added four times with n and d multiplied by 10

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Collapse cost

  The error of collapsing an edge to its midpoint.

  p = (P_a + P_b) ÷ 2; e = pᵀ·(Q_a + Q_b)·p = q₀x² + 2q₁xy + 2q₂xz + 2q₃x + q₄y² + 2q₅yz + 2q₆y + q₇z² + 2q₈z + q₉

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Collapse test

  Decides whether the cheapest edge is collapsed now, put back with a new cost, or skipped.

  put back when e > e₀·(1 + 1e-9) + 1e-18; skip when (new normal)·(old normal) ≤ 0 for any triangle that stays

  Decimate modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Boolean of two meshes

  Runs the union, intersection or difference and copies the result's vertex positions.

  Boolean modifier mesh/modifiers.js

  • ModelCraft
  32-bit in part

• Mesh from triangles

  Builds a mesh from triangle positions and indices, welds coincident vertices and marks sharp edges; used after a boolean and after decimate.

  weld within 1e-9; sharp when n₁·n₂ < cos(30°)

  Boolean modifier mesh/modifiers.js

  • ModelCraft
  64-bit

• Grid

  A flat grid of nx by ny quads, size s, centered on the origin.

  p(i, j) = [(i ÷ nx − 0.5)·s, (j ÷ ny − 0.5)·s, 0]

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• Cube

  A cube of side s standing on z = 0, centered in x and y.

  h = s ÷ 2; corners (±h, ±h, 0) and (±h, ±h, s)

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• Cylinder and cone

  A ring of n points of radius r at z = 0, with a second ring at height h for a cylinder or one apex for a cone.

  a = 2π·i ÷ n; p = [r·cos a, r·sin a, 0 or h]; apex = [0, 0, h]

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• UV sphere

  A sphere of radius r resting on z = 0, made of k rings of n points between two poles.

  θ = π·j ÷ k, φ = 2π·i ÷ n; p = [r·sin θ·cos φ, r·sin θ·sin φ, r − r·cos θ]

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• Ico sphere

  A sphere of radius r resting on z = 0, from an icosahedron whose triangles are split into four a number of times.

  g = (1 + √5) ÷ 2; corners = unit(±1, ±g, 0) and its rotations; midpoint = unit((a + b) ÷ 2); p = unit point·r + [0, 0, r]

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• Torus

  A torus of ring radius R and tube radius r resting on z = 0.

  a = 2π·i ÷ n, b = 2π·j ÷ k; w = R + r·cos b; p = [w·cos a, w·sin a, r + r·sin b]

  Mesh primitives mesh/primitives.js

  • ModelCraft
  64-bit

• Squared distance to a triangle

  The squared distance from a point to the nearest point of a triangle.

  q = nearest point by region (vertex, edge or interior); d² = |p − q|²

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  64-bit

• Distance samples near the surface

  For every grid point within two voxels of a triangle, the distance to the nearest triangle.

  g = voxel ÷ 2; d(i, j, k) = √(min over triangles of d²), capped at 2·voxel

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  32-bit in part

• Inside or outside

  Marks each grid point as inside or outside the mesh by counting the faces a ray along +x crosses.

  crossing when w₀, w₁, w₂ share a sign; x = (w₀·x_A + w₁·x_B + w₂·x_C) ÷ (w₀ + w₁ + w₂); inside when the winding count before x is above 0

  Voxel remesh and snapping onto a surface mesh/remesh.js

  • ModelCraft
  64-bit