Calculations

Calculations

• 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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• Rescaling a surface's parameters

  Maps the knots of one direction onto a new interval.

  knot′ = a + (knot − k_a)·(b − a) ÷ (k_b − k_a)

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

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• Uniform knots for a curve from control points

  The knot vector given to a curve drawn by its control points when none is supplied.

  p + 1 zeros; i ÷ (n − p) for i = 1 … n − p − 1; p + 1 ones

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

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• Parameter from a fraction

  The curve parameter for a command's position given as a fraction 0 to 1.

  u = a + (b − a)·f

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

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• Closest point on a curve

  The parameter, point and distance of the point of a curve nearest a given point.

  f(u) = (C − P)·C′; u ← u − f ÷ (C″·(C − P) + C′·C′)

  NURBS lengths, areas, closest points and tessellation nurbs/curve.js

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• Length of a curve

  The length of a curve, or of a part of it between two parameters.

  s = Σ pieces Σ₈ wᵍ·h·|C′(x₀ + h·(1 + xᵍ))|, 4 pieces per knot span

  NURBS lengths, areas, closest points and tessellation nurbs/curve.js

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• Parameter at a length

  The parameter at which the curve has run a given length from its start.

  f(u) = length(a … u) − s; u ← u − f ÷ |C′(u)|, or the middle of the bracket

  NURBS lengths, areas, closest points and tessellation nurbs/curve.js

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• Curve to polyline

  Points along a curve such that the curve stays within a tolerance of the chords between them.

  a segment is kept when the curve at 1/4, 1/2 and 3/4 of its parameter range lies within tol of the chord; else it is halved

  NURBS lengths, areas, closest points and tessellation nurbs/curve.js

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• Are two curves the same

  Decides whether one curve lies on another within a tolerance.

  33 points of a, each within tol of its closest point on b

  NURBS lengths, areas, closest points and tessellation nurbs/curve.js

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• Area of a surface

  The area of an untrimmed surface.

  A = Σ cells Σ₅ₓ₅ wₐ·w_b·h_u·h_v·|S_u × S_v|, 2 × 2 pieces per knot-span cell

  NURBS lengths, areas, closest points and tessellation nurbs/surface.js

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• Closest point on a surface

  The (u, v), point and distance of the point of a surface nearest a given point.

  f = (S − P)·S_u, g = (S − P)·S_v; Newton on the 2 × 2 system with J = [S_u·S_u + r·S_uu, S_u·S_v + r·S_uv; ·, S_v·S_v + r·S_vv]

  NURBS lengths, areas, closest points and tessellation nurbs/surface.js

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• Tessellation steps per knot span

  How many parameter steps each knot span gets so that the grid stays within the tolerance.

  n = ceil(h·√(M ÷ (4·tol))), 2 ≤ n ≤ maxPerSpan, M = largest |second derivative| at 5 × 7 sample points

  NURBS lengths, areas, closest points and tessellation nurbs/surface.js

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• Surface to triangle grid

  A grid of points, normals and parameters on the surface with two triangles per cell.

  NURBS lengths, areas, closest points and tessellation nurbs/surface.js

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• Deviation of points from a surface

  The largest, mean and root-mean-square distance of a set of points from a surface.

  dᵢ = closest-point distance; max; mean = Σ dᵢ ÷ n; rms = √(Σ dᵢ² ÷ n)

  NURBS lengths, areas, closest points and tessellation nurbs/analysis.js

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• Merging knot lists

  The list of distinct knots, each with its highest multiplicity, over several curves or surfaces.

  key = round(u × 10¹²) ÷ 10¹²

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Making curves compatible

  Brings curves to one degree and one knot vector on 0…1.

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Extrude

  A surface from a curve pulled along a direction.

  second row = P·w + dir·w in homogeneous coordinates

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Revolve

  A surface from a profile turned about an axis by an angle.

  o = axis point + Z·((P − axis point)·Z); r = |P − o|; row = arc(o, X, Z × X, r, 0, angle)·w

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Loft parameters

  The v parameter of each section of a loft.

  tₖ = mean over control-point columns of the chord-length parameter of section k; t₀ = 0, t(last) = 1

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Loft

  A surface that passes through each section curve at its parameter.

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Sum of control nets

  The control net a + b − c used by the Gordon and Coons surfaces.

  Pᵢⱼ = aᵢⱼ + bᵢⱼ − cᵢⱼ

  Surfaces from curves: extrude, ruled, revolve and loft nurbs/construct.js

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• Parameters of the points

  A parameter between 0 and 1 for each point to fit.

  dᵢ = dᵢ₋₁ + |Qᵢ − Qᵢ₋₁| (or its square root, or 1); tᵢ = dᵢ ÷ d(last)

  Curves through points nurbs/fit.js

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• Knots by averaging

  A clamped knot vector for a given number of control points from the points' parameters.

  equal counts: U(j + p) = (t(j) + … + t(j + p − 1)) ÷ p; more points than control points: U = (1 − α)·t(i − 1) + α·t(i), i = floor(j·d), α = j·d − i

  Curves through points nurbs/fit.js

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• Interpolating curve

  The curve of a given degree through every point.

  Σⱼ Nⱼ(tᵢ)·Pⱼ = Qᵢ for every point; Σⱼ N′ⱼ(t)·Pⱼ = D at an end with a given derivative

  Curves through points nurbs/fit.js

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