Calculations

Calculations

• 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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• Cleaning a loop

  Checks a loop's points, clamps them to the patch and drops repeats.

  refused when |area| < 1e-14·(u range)·(v range)

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

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• Circle loop

  A circular trim loop given by a center and radius as shares of the parameter range.

  pointᵢ = [u_a + (u_b − u_a)·(u + r·cos aᵢ), v_a + (v_b − v_a)·(v + r·sin aᵢ)], aᵢ = 2π·i ÷ n

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

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• Loop from a curve on the surface

  Projects a closed 3D curve into the surface's (u, v) plane as a trim loop.

  256 points of the curve, each → closest point (u, v) on the surface

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

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• Moving loop points off grid lines

  Shifts loop points that fall on a tessellation grid line, and keeps all of them inside the patch.

  d = 1e-6 × parameter range; a point within d ÷ 2 of a grid line moves to line + d; every point is clamped to [start + d, end − d]

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

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• Where a loop crosses the grid

  The points where a loop segment crosses the tessellation's parameter lines.

  t = (line − p) ÷ (q − p); point = p + t·(q − p)

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

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• Triangulating a cut cell

  Triangles for a polygon left in a grid cell after cutting.

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

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• Area of the shown part

  The area of a trimmed surface.

  A = Σ |(b − a) × (c − a)| ÷ 2 over the triangles of the cut tessellation

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

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• Area in the parameter plane

  The area of the shown part in (u, v).

  outer loop's area (or the whole domain) + Σ holes' signed areas

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

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• Line buffer

  The positions of a polyline in the host's units, for drawing.

  Drawing NURBS curves and surfaces nurbs-ui/display.js

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  32-bit, display only

• Surface mesh buffers

  The positions, normals, parameters and colors of a surface's triangles, for drawing.

  Drawing NURBS curves and surfaces nurbs-ui/display.js

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  32-bit, display only

• Finding the sheet

  The four corners of the paper in the photo.

  corners = the pixels of the largest bright region with the smallest and largest x + y and x − y; photo position = (pixel + 0.5) × reduction factor

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Perspective transform from four points

  The 3 × 3 matrix that takes four points to four others.

  8 equations in h₁…h₈ with h₉ = 1, solved by Gauss–Jordan elimination

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Applying the perspective transform

  A point taken through the 3 × 3 matrix.

  w = h₆x + h₇y + h₈; x′ = (h₀x + h₁y + h₂) ÷ w; y′ = (h₃x + h₄y + h₅) ÷ w

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Sheet proportions

  The long-to-short ratio of the straightened sheet.

  w = mean of top and bottom lengths, h = mean of left and right; r = max ÷ min; replaced by the named paper's ratio, or by the nearest standard ratio within 8 %

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Drawing units per pixel

  The size of one pixel of the straightened sheet in drawing units, and the mapping from pixels to drawing coordinates.

  mm per px = long side in mm ÷ long side in px; units per px = mm per px × scale ÷ unit in mm; x = origin_x + px·k, y = origin_y + (height − py)·k

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Arcs and circles into the drawing

  Converts each traced arc to drawing units as a circle or as an arc through its start, middle and end.

  r = r_px × units per px; circle when a₁ − a₀ ≥ 2π − 1e-6; else arc through start, middle (at (a₀ + a₁) ÷ 2) and end

  Sketch photo to drawing: sheet, perspective and scale photosketch/photosketch.js

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• Arc through three points

  The center, radius and end angles of the arc through three points.

  d = 2·(a_x(b_y − c_y) + b_x(c_y − a_y) + c_x(a_y − b_y)); center from the sums of |p|²·(differences) ÷ d; r = |a − center|

  Sketch photo to drawing: sheet, perspective and scale photosketch/paper.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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