Calculations

Calculations

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• Bézier pieces

  The curve as one Bézier piece per knot span.

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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• Splitting a curve

  Cuts a curve in two at a parameter.

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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• Loft

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

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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• 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

  • ModelCraft
  • PartCraft
  64-bit

• Placement matrix (classic files)

  A component's placement as a 4 × 4 matrix from the file's 13 doubles.

  rows of the 3 × 3 become columns; when w ≠ 1 every entry ÷ w

  SKP import: placements, texture coordinates and face loops skp/legacy.js

  • ModelCraft
  64-bit

• Texture coordinates (classic files)

  The texture coordinates of each loop point of a textured face.

  x_r = (−n_y, n_x, 0) ÷ |(n_x, n_y)|, y_r = n × x_r; (p_x, p_y) = (p·x_r, p·y_r); uv = (p_x, p_y, 1)·M⁻¹ ÷ q ÷ tile size

  SKP import: placements, texture coordinates and face loops skp/legacy.js

  • ModelCraft
  64-bit

• Loop normal

  The Newell normal of a face loop, used to check which way round the loop runs.

  n = Σ [(y_a − y_b)(z_a + z_b), (z_a − z_b)(x_a + x_b), (x_a − x_b)(y_a + y_b)]

  SKP import: placements, texture coordinates and face loops skp/legacy.js

  • ModelCraft
  64-bit

• Placement matrix (2021 and later files)

  A component's placement as a 4 × 4 matrix from the file's 13 doubles.

  k = 1 ÷ w; every rotation and translation entry × k

  SKP import: placements, texture coordinates and face loops skp/vff.js

  • ModelCraft
  64-bit

• Texture coordinates (2021 and later files)

  The texture coordinates of a point on a textured face.

  (p_x, p_y) = (p·x_r, p·y_r); without a placement: uv = (p_x ÷ tile width, p_y ÷ tile height); with one: (p_x, p_y, 1)·M⁻¹ ÷ q ÷ tile size

  SKP import: placements, texture coordinates and face loops skp/vff.js

  • ModelCraft
  64-bit

• Outer loop and loop direction

  Chooses a face's outer loop and turns loops so that the outer runs counter-clockwise about the normal and holes clockwise.

  area(loop) = Newell(loop)·unit normal ÷ 2; outer = loop with the largest |area|

  SKP import: placements, texture coordinates and face loops skp/vff.js

  • ModelCraft
  64-bit

• Inverse of a rigid transform

  Inverts a rotation-plus-translation matrix.

  R⁻¹ = Rᵀ; t⁻¹ = −Rᵀ t

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Mate frame on a face

  The frame a joint end uses: the face's axes, with the origin moved to where the chosen axis crosses the face.

  o = a + d × ((f − a) · n) ÷ (d · n)

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Joint value inside its limits

  A joint's angle or distance, held between its minimum and maximum.

  value = min(max, max(min, v))

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Motion of a joint

  The transform of part B relative to part A's mate frame, from the joint's angles and distances.

  revolute, slider, cylindrical, rigid: T(0, 0, offset) × Rz(angle); planar: T(dx, dy, 0) × Rz; ball: Rx(tiltX) × Ry(tiltY) × Rz; pin-slot: T(s cos δ, s sin δ, 0) × Rz

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit

• Placement walk and loop mismatch

  Places each part from one already placed through their joint, and measures how far a loop-closing joint is from closing.

  W_B = W_A × F_A × motion × FLIP × F_B⁻¹; off = max |want − have| (rotation entries × 100)

  Assembly joints and part placement assembly.js

  • PartCraft
  64-bit