Calculations

Calculations

• Level set of the samples

  The distance at any point by blending the eight samples round it, handed to the library that builds the new mesh.

  f(x, y, z) = −trilinear blend of the 8 samples; new mesh = level set of f at 0 with edge length = voxel

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

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• Put vertices on a surface

  Moves each listed vertex to the nearest point of a target surface.

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

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• Brush falloff

  How strongly a vertex follows the brush, from 1 at its middle to 0 at its edge.

  w(x) = 1 − 3x² + 2x³ for 0 < x < 1; 1 for x ≤ 0; 0 for x ≥ 1

  Sculpt brushes mesh/sculpt.js

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• Vertex normals for sculpting

  A unit normal per vertex from the triangles round it, weighted by triangle area.

  N(v) = unit(Σ (b − a) × (c − a))

  Sculpt brushes mesh/sculpt.js

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• Vertices under the brush

  The vertices within the brush radius of its center, each with its falloff weight.

  d² = |P(v) − p|² ≤ r²; w = falloff(√d² ÷ r)

  Sculpt brushes mesh/sculpt.js

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• Brush plane

  The normal and center of the surface under the brush, and the size of one step.

  n = unit(Σ wᵢ·Nᵢ); c = Σ wᵢ·Pᵢ ÷ Σ wᵢ; step = 0.1·r·k, k = strength × pressure

  Sculpt brushes mesh/sculpt.js

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• Brush moves

  The new position of each vertex under the brush for draw, inflate, flatten, clay, smooth, pinch and crease.

  f = w·(1 − mask); draw: p + n·(±step·f); inflate: p + N(v)·(±step·f); flatten: p − n·dd·min(1, f·max(0.1, k)), dd = (p − plane)·n; smooth: p + (mean of neighbors − p)·min(1, f·max(0.1, k))

  Sculpt brushes mesh/sculpt.js

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• Grab

  Moves the vertices that were under the brush at the start of the stroke by the pointer's move.

  p = p₀ + δ·w·(1 − mask)

  Sculpt brushes mesh/sculpt.js

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• Painting the mask

  Raises or lowers the mask of the vertices under the brush, between 0 and 1.

  mask = clamp(mask ± w·k, 0, 1)

  Sculpt brushes mesh/sculpt.js

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• Split an edge at its middle

  The new vertex when dynamic detail splits a long triangle edge.

  mid = (P_a + P_b) ÷ 2

  Sculpt brushes mesh/sculpt.js

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• Brush radius in mesh units

  Turns the brush's radius in pixels into a length in the mesh's own space at the point under the pointer.

  r = px × pixel size at w ÷ scale, scale = |first column of the placement matrix|

  Sculpt brushes mesh/sculpt-tool.js

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• Dabs along a stroke

  Places dabs a quarter of the brush radius apart between the last dab and the pointer.

  step = 0.25·r; k = min(32, floor(d ÷ step)); pᵢ = last + (hit − last)·i ÷ k

  Sculpt brushes mesh/sculpt-tool.js

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• Sparse least squares (CGLS)

  Finds the unknowns that minimize |Ax − b| for a sparse set of rows.

  columns scaled by 1 ÷ |column|; stop when |Aᵀr|² ≤ tol²·max(1, |Aᵀr₀|²), tol = 1e-12

  UV unwrap solvers mesh/uv.js

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• Conformal map (LSCM)

  UV positions for the corners of an island from the 2D shape of each of its triangles, with two points pinned.

  per triangle, with d = twice its signed area and s = 1 ÷ √|d|: Σⱼ Wⱼ·(uⱼ + i·vⱼ) = 0, W = s × the opposite edge vectors

  UV unwrap solvers mesh/uv.js

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• Angle based flattening

  Corner angles close to the surface's own that a flat triangulation can have.

  minimize Σ (αᵢ − βᵢ)² ÷ βᵢ² subject to: each triangle sums to π, each inner wedge to 2π, Σ [ln sin α(next) − ln sin α(prev)] = 0 round each inner wedge

  UV unwrap solvers mesh/uv.js

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• Preconditioned conjugate gradients

  Solves the constraint system of each Newton step of the angle based method.

  stop when r·r < 1e-14 × r₀·r₀

  UV unwrap solvers mesh/uv.js

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• Corner angles of a triangle

  The three corner angles of a triangle in 3D.

  angle = acos(clamp(u·v ÷ (|u|·|v|), −1, 1))

  UV unwrap solvers mesh/uv.js

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• A triangle's own 2D shape

  A triangle laid flat: A at the origin, B on the x axis, C where its side lengths put it.

  B = (|AB|, 0); C = (AC·x, AC·y), x = AB ÷ |AB|, y = unit((AB × AC) × AB)

  UV unwrap solvers mesh/uv.js

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• A triangle's shape from its angles

  A flat triangle with the given corner angles, sized by the law of sines.

  A = (0, 0); B = (sin a₂, 0); C = (sin a₁·cos a₀, sin a₁·sin a₀)

  UV unwrap solvers mesh/uv.js

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• Pinned points and first guess

  Chooses the two corners to pin and a starting layout for the solver.

  pins: the two ends along the longest box axis at (0, 0) and (d, 0), d = their distance in 3D

  UV unwrap solvers mesh/uv.js

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• Storing an unwrap

  Checks which way up the flattened island came out and writes its UVs to the mesh.

  signed = Σ over triangles of (u_b − u_a)(v_c − v_a) − (u_c − u_a)(v_b − v_a); u is negated when signed < 0

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Are two UVs the same

  Decides whether two face corners share a UV point, which decides where UV islands end.

  |u_a − u_b| ≤ 1e-7 and |v_a − v_b| ≤ 1e-7

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Box projection

  Maps each face onto the side of a box its normal faces most.

  u = (pᵢ − loᵢ) ÷ s, v = (pⱼ − loⱼ) ÷ s; u → 1 − u on the sides seen from behind

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Cylinder projection

  u is the angle round an axis through the faces' center, v the height along it.

  u = (atan2(p_j − c_j, p_i − c_i) ÷ 2π + 1) mod 1; v = (p_k − lo_k) ÷ height

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Sphere projection

  u is the angle round the axis, v the latitude, seen from the faces' center.

  d = p − c; u = (atan2(d_j, d_i) ÷ 2π + 1) mod 1; v = 1 − acos(clamp(d_k ÷ |d|, −1, 1)) ÷ π

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• View projection

  UVs from where each corner appears on screen, fitted to the unit square.

  s = max(1e-12, width, height); u = (x − x₀) ÷ s; v = 1 − (y − y₀) ÷ s

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Smallest bounding rectangle

  The rotation that makes an island's bounding rectangle smallest.

  for each hull edge: θ = −atan2(Δy, Δx); area = (x₁ − x₀)·(y₁ − y₀) of the rotated hull; keep the smallest

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Packing: island scale

  Scales each island so that its UV area matches its area on the mesh.

  A₃ = Σ |(b − a) × (c − a)| ÷ 2; A₂ = Σ |UV triangle area|; k = √(A₃ ÷ A₂)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Packing: placing the islands

  Places the islands' rectangles on shelves and scales the whole layout into the unit square.

  pad = margin·√(total area)·1.1; widths tried: max(widest, √(sum of areas)·(0.7 + 0.05·k)), k = 0…23; uv = (offset + pad + point) ÷ side

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Stretch per face

  How much each face's UVs distort it, from 0 for none to 1 for a lot.

  σ₁,₂ = √(tr ÷ 2 ± √(tr² ÷ 4 − det)); angle: 1 − σ₂ ÷ σ₁; area: min(1, |ln(σ₁·σ₂)| ÷ ln 4); per face: area-weighted mean over its triangles

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv.js

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• Move, rotate and scale UVs

  Moves, turns or scales the selected faces' UVs about their mean as the pointer moves in the UV editor.

  move: (u + Δu, v + Δv); rotate by a = atan2(v₁ − c_v, u₁ − c_u) − atan2(v₀ − c_v, u₀ − c_u); scale by k = |p₁ − c| ÷ max(1e-9, |p₀ − c|)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/uv-editor.js

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• UVs and sharp edges of a primitive

  Stores a new primitive's positions, its UVs per face corner and the flags of its sharp edges.

  sharp when n₁·n₂ < cos(sharpDeg·π ÷ 180)

  UV coordinates: unwrap result, projections, packing, stretch and editing mesh/primitives.js

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• Rotation matrix

  The matrix that turns geometry by an angle about an axis through a point.

  R = c·I + s·[axis]ₓ + (1 − c)·axis·axisᵀ, c = cos(angle·π ÷ 180), s = sin(angle·π ÷ 180); t = p − R·p

  Node graph functions nodes/library.js

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• Scale matrix

  The matrix that scales geometry by a factor about a center.

  p′ = f·p + c·(1 − f)

  Node graph functions nodes/library.js

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• Applying a matrix to geometry

  Moves a point, a curve, a surface, a mesh or a list of them by a 4 × 4 matrix.

  Node graph functions nodes/library.js

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• Divide, power and remainder

  The Divide, Power, Smaller, Larger and Remainder nodes.

  a ÷ b; aᵇ; remainder = a − b·floor(a ÷ b)

  Node graph functions nodes/library.js

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• Sine, cosine, tangent and angle

  Trigonometry with angles in degrees.

  sin(x·π ÷ 180), cos(x·π ÷ 180), tan(x·π ÷ 180); angle = atan2(y, x) ÷ (π ÷ 180)

  Node graph functions nodes/library.js

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• Series and range

  A list of evenly stepped numbers, by step or between two ends.

  series: start + k·step; range: start + (end − start)·k ÷ (n − 1)

  Node graph functions nodes/library.js

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• Sum of a list

  Adds the numbers of a list.

  Σ xᵢ

  Node graph functions nodes/library.js

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• Vector length and distance

  The length of a vector and the distance between two points.

  √(Δx² + Δy² + Δz²)

  Node graph functions nodes/library.js

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• Point on a circle

  A point at a radius and an angle from a center, in the plane z = center z.

  [c_x + r·cos(a·π ÷ 180), c_y + r·sin(a·π ÷ 180), c_z]

  Node graph functions nodes/library.js

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• Point on curve and divide curve

  A point at a fraction of a curve, and the points that divide a curve into a number of parts.

  u = u₀ + (u₁ − u₀)·t; divide: u = u₀ + (u₁ − u₀)·k ÷ n

  Node graph functions nodes/library.js

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• Array around an axis

  Copies of geometry turned round an axis in equal steps of a full circle.

  copy k: rotate by 360·k ÷ n degrees

  Node graph functions nodes/library.js

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• Surface to mesh

  A triangle mesh of a surface within a tolerance.

  Node graph functions nodes/library.js

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• Point on a curve

  The point of a NURBS curve at parameter u.

  C(u) = Σ Nᵢ(u)·wᵢ·Pᵢ ÷ Σ Nᵢ(u)·wᵢ

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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• Derivatives of a curve

  The point and the first d derivatives of a curve at u.

  C⁽ʳ⁾ = (A⁽ʳ⁾ − Σᵢ₌₁..ᵣ C(r, i)·w⁽ⁱ⁾·C⁽ʳ⁻ⁱ⁾) ÷ w

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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• Curvature of a curve

  The curvature at u.

  κ = |C′ × C″| ÷ |C′|³

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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• Curvature vector

  The vector that points to the center of curvature with length equal to the curvature.

  κN = (C″ − t·(C″·t)) ÷ |C′|², t = C′ ÷ |C′|

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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• Torsion

  The torsion of a curve at u.

  τ = (C′ × C″)·C‴ ÷ |C′ × C″|²

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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• From homogeneous points

  Turns weighted (homogeneous) control points back into control points and weights.

  P = (wx, wy, wz) ÷ w

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs/curve.js

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