Calculations

Calculations

• 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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• Arc and circle

  A circular arc, or a full circle, as an exact rational quadratic curve.

  pieces of at most 90°; Δθ = θ ÷ pieces; middle weight = cos(Δθ ÷ 2); P₁ = P₀ + T₀·r·tan(Δθ ÷ 2)

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

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• Polyline as a curve

  A degree 1 curve through given points, with knots in proportion to the lengths of its segments.

  Lᵢ = Lᵢ₋₁ + max(1e-12, |Pᵢ − Pᵢ₋₁|); knotᵢ = Lᵢ ÷ L(last)

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

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• Transforming a curve

  Moves a curve by a 4 × 4 matrix.

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

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• Point on a surface

  The point of a NURBS surface at (u, v).

  S(u, v) = Σᵢ Σⱼ Nᵢ(u)·Nⱼ(v)·wᵢⱼ·Pᵢⱼ ÷ Σᵢ Σⱼ Nᵢ(u)·Nⱼ(v)·wᵢⱼ

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

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• Derivatives of a surface

  The point and the partial derivatives up to total order d at (u, v).

  S⁽ᵏˡ⁾ = (A⁽ᵏˡ⁾ − lower-order terms weighted by binomials and derivatives of w) ÷ w

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

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• Surface normal

  The unit normal at (u, v), including at poles and collapsed edges.

  n = unit(S_u × S_v)

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

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• Moving off a collapsed edge

  Chooses where curvature is evaluated when the point asked for lies on a collapsed edge.

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

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• Surface curvature

  Gaussian and mean curvature, the two principal curvatures and their directions.

  K = (LN − M²) ÷ (EG − F²); H = (EN − 2FM + GL) ÷ (2(EG − F²)); k₁,₂ = H ± √(H² − K)

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

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• Normal curvature in a direction

  The curvature of the surface along a given tangent direction.

  a, b from t = a·S_u + b·S_v; κₙ = (L·a² + 2M·ab + N·b²) ÷ (E·a² + 2F·ab + G·b²)

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

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• Iso-curve of a surface

  The curve on a surface at a fixed u or v, as an exact NURBS curve.

  row j = Σₐ Nₐ(t)·Pwₐⱼ in homogeneous coordinates

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

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• Angle between two vectors

  The angle between two vectors, 0 to π.

  angle = atan2(|a × b|, a·b)

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

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• Applying a 4 × 4 matrix

  A point or a direction multiplied by a row-major affine matrix.

  p′ᵢ = mᵢ₀·x + mᵢ₁·y + mᵢ₂·z + mᵢ₃·w

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

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• Curvature maps

  Gaussian, mean and largest principal curvature at every vertex of a surface's tessellation, with their ranges.

  k_max = max(|H + d|, |H − d|), d = √(max(0, H² − K))

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

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• Curvature comb

  Teeth along a curve whose length shows the curvature, and the largest curvature found.

  u = a + (b − a)·i ÷ samples; q = p − scale·κN; max = largest |κN| at the samples

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

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• Curvature range for the legend

  The Gaussian curvature range that the curvature colors and their legend span.

  max = largest |K| over all shown surfaces; range = [−max, max]

  NURBS curves and surfaces: points, derivatives, normals and curvature nurbs-ui/display.js

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• Knot insertion

  Adds a knot to a curve r times without changing its shape.

  α = (u − U[L + i]) ÷ (U[i + k + 1] − U[L + i]); Q = α·R[i + 1] + (1 − α)·R[i]

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Tolerance in homogeneous space

  Converts a distance tolerance into one for weighted control points.

  TOL = tol·w_min ÷ (1 + max |P|)

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Knot removal

  Removes copies of an interior knot while the curve moves no more than a tolerance.

  candidates: (P[i] − (1 − α)·T[i − 1]) ÷ α and (P[j] − α·T[j + 1]) ÷ (1 − α); removed when they agree within TOL

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Bézier pieces

  The curve as one Bézier piece per knot span.

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Degree elevation

  Raises the degree of a curve by t.

  Qᵢ = Σⱼ C(p, j)·C(t, i − j) ÷ C(p + t, i) · Pⱼ on each Bézier piece

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Degree reduction

  Lowers the degree by one if the curve moves no more than a tolerance, and reports a bound on the change.

  left and right reduced points blended by λ = i ÷ q; error = largest distance between the re-elevated and the original control points

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Splitting a curve

  Cuts a curve in two at a parameter.

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Rescaling the parameter

  Maps a curve's knots onto a new interval.

  knot′ = a + (knot − c_a)·(b − a) ÷ (c_b − c_a)

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Reversing a curve

  The same curve run the other way.

  knot′ = a + b − knot, in reverse order

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Joining curves

  Makes one curve from curves that meet end to end.

  joint point = (end of one + start of the next) ÷ 2 in homogeneous coordinates; knot ranges in proportion to arc length

  NURBS knot insertion and removal, degree change, split and join nurbs/curve.js

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• Rows back into a surface

  Rebuilds a surface from its rows after a curve operation was applied to each row.

  NURBS knot insertion and removal, degree change, split and join nurbs/surface.js

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• Degree elevation of a surface

  Raises the degree of a surface in one direction.

  NURBS knot insertion and removal, degree change, split and join nurbs/surface.js

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