Calculations

Calculations

• 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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• Interpolating many lists at once

  Control points for several point lists that share parameters, used for the columns of a loft or a curve network.

  Curves through points nurbs/fit.js

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• Linear solve

  Solves A·x = B for several right-hand sides.

  Gaussian elimination with partial pivoting, then back-substitution

  Curves through points nurbs/vec.js

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• Least-squares curve with given knots

  The curve with given knots that is closest to a set of points, with its ends on the first and last point.

  (NᵀN)·P = NᵀR, R = Q − N₀·Q₀ − Nₙ·Q(last)

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Least squares for many lists

  The same fit for several point lists that share parameters and knots.

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Fitting a curve to a function

  A curve within a tolerance of a parametric function, with as few control points as the refinement finds.

  deviation = max |C(s) − f(s)| at the midpoints between fitting samples; count ← ceil(1.5·count) + 1 until deviation ≤ tol

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Fitting a surface to a function

  A surface within a tolerance of a function of (u, v).

  deviation = max of three sampled sets (between samples along u, along v, and at cell middles); the direction with the larger error gets ceil(1.5·n) + 1 control points

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Rebuilding a curve

  A curve with a chosen number of control points and uniform knots that follows the original, with the deviation measured.

  samples = max(4·count, 32); deviation = closest-point distance of the points midway between samples

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Deviation of points from a curve

  The largest and mean distance of a set of points from a curve.

  dᵢ = closest-point distance; max; mean = Σ dᵢ ÷ n

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/fit.js

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• Frames along a rail

  A frame (origin, tangent and two cross directions) at each requested parameter of a rail, turning as little as possible.

  double reflection: r reflected in the chord, then in the difference of tangents; then made square to the tangent

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Placing a profile in a frame

  The profile's control points carried from the start frame to another frame, turned and scaled.

  (a, b, z) = (d·r₀, d·s₀, d·t₀); (a, b) rotated by the twist; P′ = o + r·a·scale + s·b·scale + t·z·scale

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Sweep refinement

  Lofts through placed sections and adds sections until the surface matches the true placement between them.

  deviation = max |S(u, v_mid) − section(u)| at 13 points on each middle section; sections doubled until deviation ≤ tol

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Frame between two rails

  The frame and the gap between two rails at a parameter, which place and scale the profile of a two-rail sweep.

  X = (B − A) ÷ d, d = |B − A|; Z = unit(T₁ + T₂ − X·((T₁ + T₂)·X)); scale = d ÷ d₀

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Where two curves come closest

  The parameters on two curves where they are nearest each other, and the distance there.

  f = (A − B)·A′, g = −(A − B)·B′; Newton on the 2 × 2 system

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Monotone cubic interpolation

  A smooth increasing map between a curve's own crossing parameters and the shared ones of a network.

  interior slope = (w₁ + w₂) ÷ (w₁ ÷ δ₁ + w₂ ÷ δ₂), 0 where δ changes sign; cubic Hermite between nodes

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Surface through a network of curves

  A Gordon surface through a grid of profile and guide curves.

  S = L_u + L_v − T

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Coons patch

  A surface filling four boundary curves that meet at their corners.

  S = ruled(a, b) + ruled(l, r) − bilinear(corners)

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Offset surface

  A surface at a distance from another along its normals.

  target(u, v) = S(u, v) + n(u, v)·d, fitted within tol

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Rebuilding a surface

  A surface with a chosen number of control points and uniform knots that follows the original, with the deviation measured.

  samples = max(4·n, 16) each way; deviation = max |S′ − S| at the cell middles

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Extending a surface

  Carries a surface on past an edge by a length measured across the edge at its middle.

  the last span's polynomial evaluated by de Casteljau at t = (1 + du − last) ÷ (1 − last) > 1; du by secant steps until |length found − length| ≤ 1e-10·length

  Fitted curves and surfaces: sweeps, networks, Coons patches, offsets, rebuilds and extensions nurbs/construct.js

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• Continuity class

  Turns the measured gap, angle and curvature difference into G2, G1, G0 or none.

  gap ≤ g0 → G0; and angle ≤ g1 → G1; and (dk ≤ g2 or rel ≤ g2rel) → G2

  Continuity between curves and between surfaces nurbs/continuity.js

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• Join of two curves

  The gap, the angle between tangents and the curvature difference where two curve ends meet.

  gap = |P_a − P_b|; angle = atan2(|t_a × t_b|, t_a·t_b)·180 ÷ π; dk = |κN_a − κN_b|; rel = dk ÷ max(κ_a, κ_b)

  Continuity between curves and between surfaces nurbs/continuity.js

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• Shared edge of two surfaces

  Gap, normal angle and curvature difference at sample points along a shared edge, with the largest of each.

  per sample: gap = |p_A − p_B|; angle between n_A and n_B; dk = |κ_A − κ_B| in the direction n_A × edge tangent

  Continuity between curves and between surfaces nurbs/continuity.js

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• Finding shared edges

  Finds pairs of surface edges that lie on each other.

  |L_A − L_B| ≤ max(10·tol, 0.05·max(L_A, L_B)); 7 points of A within tol of B

  Continuity between curves and between surfaces nurbs/continuity.js

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• Values written into a problem

  Rounds the measured gap, angle or curvature difference for a problem's found value and its text.

  r4(x) = x rounded to 4 significant digits

  Continuity between curves and between surfaces nurbs/continuity.js

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• Signed area of a loop

  The signed area of a loop in the (u, v) plane; its sign tells which way the loop runs.

  a = Σ (uᵢ·vᵢ₊₁ − uᵢ₊₁·vᵢ) ÷ 2

  Trimmed surfaces: loops, cut tessellation and area nurbs/trim.js

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• Point inside a loop

  Decides whether a (u, v) point is inside a loop.

  count edges with (v_a > v) ≠ (v_b > v) and u < (u_b − u_a)·(v − v_a) ÷ (v_b − v_a) + u_a

  Trimmed surfaces: loops, cut tessellation and area nurbs/trim.js

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• Do two segments cross

  Decides whether two loop segments properly cross.

  t, u = cross products ÷ d; crossing when 1e-12 < t < 1 − 1e-12 and 1e-12 < u < 1 − 1e-12

  Trimmed surfaces: loops, cut tessellation and area nurbs/trim.js

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