Calculations

Calculations

• Gamma function and factorial

  Γ(x), and factorial(n) = Γ(n + 1).

  whole x < 171: (x − 1)! by product; x < 0.5: π ÷ (sin(πx) Γ(1 − x)); else √(2π) t^(x − ½) e^(−t) Σ c_i ÷ (x − 1 + i), t = x + 6.5

  CraftScript core math functions cs-lib-core.js

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• Binomial coefficient

  nchoosek(n, k): the number of ways to choose k of n.

  r ← r × (n − m + i) ÷ i for i = 1 … m, m = min(k, n − k); result = round(r)

  CraftScript core math functions cs-lib-core.js

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• Spacing of numbers

  eps(x): the gap between x and the next larger 64-bit number.

  2^(floor(log₂|x|) − 52); eps(0) = 5e-324

  CraftScript core math functions cs-lib-core.js

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• Evenly spaced values

  linspace(a, b, n): n values from a to b.

  value i = a + (b − a) × i ÷ (n − 1)

  CraftScript core math functions cs-lib-core.js

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• Logarithmically spaced values

  logspace(a, b, n): n values from 10^a to 10^b.

  value i = 10 ^ (a + (b − a) × i ÷ (n − 1))

  CraftScript core math functions cs-lib-core.js

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• Running sum and product

  cumsum and cumprod along a dimension.

  acc ← acc + x_k (or acc × x_k), stored at each step

  CraftScript core math functions cs-lib-core.js

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• Variance and standard deviation

  var and std of a set of values.

  m = Σx ÷ n; var = Σ(x − m)² ÷ (n − 1), or ÷ n with weight 1; std = √var

  CraftScript core math functions cs-lib-core.js

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• Median

  The middle value of a set, or the mean of the two middle values.

  sorted s; n odd: s[(n − 1) ÷ 2]; n even: (s[n ÷ 2 − 1] + s[n ÷ 2]) ÷ 2

  CraftScript core math functions cs-lib-core.js

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• Dot product

  dot(a, b): the sum of products of two vectors.

  Σ a_i b_i

  CraftScript core math functions cs-lib-core.js

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• Trapezoid integration

  trapz(y) or trapz(x, y): the area under sampled data.

  Σ (x_i − x_{i−1}) (y_i + y_{i−1}) ÷ 2

  CraftScript core math functions cs-lib-core.js

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• One-dimensional interpolation

  interp1(x, y, xi): linear, nearest or previous-value interpolation in a table.

  y = y_lo + (y_hi − y_lo)(q − x_lo) ÷ (x_hi − x_lo)

  CraftScript core math functions cs-lib-core.js

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• Polynomial evaluation

  polyval(p, x): the value of a polynomial.

  s ← s × x + c for each coefficient (Horner)

  CraftScript core math functions cs-lib-core.js

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• Polynomial fit

  polyfit(x, y, n): the least-squares polynomial of degree n through data.

  V_ij = x_i^(n − j); solve V p ≈ y

  CraftScript core math functions cs-lib-core.js

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• Matrix to text

  mat2str(A): text that can be typed back in to recreate a matrix.

  each value written with toPrecision(15) by default

  CraftScript core math functions cs-lib-core.js

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• Number to text

  num2str(x): a short text form of a number.

  whole numbers in full; otherwise max(ceil(log₁₀|x|), 1) + 4 significant digits

  CraftScript core math functions cs-lib-core.js

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• Text to number

  str2double(s): reads a number from text.

  Number(text) for text matching a decimal pattern, else NaN

  CraftScript core math functions cs-lib-core.js

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• Read an STL file

  Reads the triangles of a binary or text STL file into a model's node coordinates.

  CraftScript file readers and writers cs-lib-io.js

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• Weld STL vertices

  Merges the repeated corners of an STL triangle list into shared nodes.

  CraftScript file readers and writers cs-lib-io.js

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• Write a binary STL file

  Writes a triangle surface with a unit normal per triangle.

  n = (b − a) × (c − a) ÷ |(b − a) × (c − a)|

  CraftScript file readers and writers cs-lib-io.js

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• Write an OBJ file

  Writes node coordinates and triangle corner numbers as text.

  CraftScript file readers and writers cs-lib-io.js

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• Read an OBJ file

  Reads vertex lines and face lines into node coordinates and triangles.

  CraftScript file readers and writers cs-lib-io.js

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• Read a CSV file

  Reads a table of numbers into a matrix.

  CraftScript file readers and writers cs-lib-io.js

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• Write a CSV file

  Writes a matrix as rows of text.

  CraftScript file readers and writers cs-lib-io.js

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• Values to JSON

  Prepares a script value for a JSON file.

  CraftScript file readers and writers cs-lib-io.js

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• Read an NPY file

  Reads a NumPy array file into a tensor.

  CraftScript file readers and writers cs-lib-io.js

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• Write an NPY file

  Writes a tensor as a NumPy array file.

  CraftScript file readers and writers cs-lib-io.js

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• Write a tensor to HDF5

  Writes a tensor as an HDF5 dataset in row-major order.

  CraftScript file readers and writers cs-lib-io.js

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• Read an HDF5 dataset

  Reads a dataset into a number, a logical value or a tensor.

  CraftScript file readers and writers cs-lib-io.js

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• Plot data as CSV

  Writes the x and y values of a script plot as a CSV file.

  CraftScript file readers and writers cs-plot-export.js

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• One CSV cell

  Turns one value into the text of a CSV cell.

  CraftScript file readers and writers cs-plot-export.js

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• Preconditioned conjugate gradients

  Solves a symmetric positive definite sparse system with the pcg builtin.

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Jacobi preconditioner

  Scales a residual by the inverse of the matrix diagonal.

  zᵢ = rᵢ ÷ A_ii

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Incomplete LU factorization

  The ILU(0) preconditioner used by BiCGSTAB.

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• BiCGSTAB

  Solves a general sparse system by iteration.

  stop when ‖r‖ ÷ ‖b‖ < tol

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Rigid-body vectors for AMG

  Builds the translation and rotation vectors the multigrid preconditioner uses.

  rotation about the centroid: (−(y − c_y), x − c_x)

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Smallest eigenvalues of a sparse problem

  The k smallest eigenvalues and vectors of K v = λ M v.

  stop when |λⱼ − λⱼ(previous)| ≤ tol × |λⱼ| for every j ≤ k

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Unit factors

  The factors that turn the given force, length and torque units into newtons, millimeters and newton millimeters.

  Design-it sizing and hand checks design-it.js

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• Arguments to working units

  Converts the given force, lengths and torque to N, mm and N·mm.

  value × unit factor

  Design-it sizing and hand checks design-it.js

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• Rounding to stock sizes and to reported decimals

  Rounds a thickness up to a stock plate or a 0.5 mm step, and rounds reported values.

  r2(v) = round(100 v) ÷ 100; r3(v) = round(1000 v) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Bolt positions in the flange

  The heights of the two bolts and the flange height.

  y_b = t + 1.5 × hole; spacing = clamp(round(a ÷ 10) × 5, 30, 100); H = y_t + 1.5 × hole

  Design-it sizing and hand checks design-it.js

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• Bracket checks

  Bending and shear stress, tip deflection and bolt loads of the bracket, each with its limit and safety factor.

  σ = 6 F a ÷ (b t²); τ = 1.5 F ÷ (b t); δ = F a³ ÷ (3 E I), I = b t³ ÷ 12; T = M y₂ ÷ (y₁² + y₂²); (T ÷ T_all)² + (V ÷ V_all)² ≤ 1

  Design-it sizing and hand checks design-it.js

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• Bracket sizing, volume and mass

  The thickness for strength and for the deflection limit, the bolt size, and the bracket's volume and mass.

  t = √(6 M ÷ (b σ_allow)); t = ∛(4 F a³ ÷ (E b δ)); V = b (L t + (H − t) t) − 2 (π ÷ 4) hole² t; mass = V × density × 1e-6

  Design-it sizing and hand checks design-it.js

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• Shaft checks

  Von Mises stress, torsional shear and twist of a round shaft.

  σ' = 32 ÷ (π d³) × √(M² + ¾T²); τ = 16 T ÷ (π d³); θ = T L ÷ (G J), G = E ÷ (2(1 + ν)), J = π d⁴ ÷ 32

  Design-it sizing and hand checks design-it.js

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• Shaft sizing, volume and mass

  The required diameter, the standard size above it, and the shaft's volume and mass.

  d = ∛(32 SF √(M² + ¾T²) ÷ (π S_y)); V = (π ÷ 4) d² L

  Design-it sizing and hand checks design-it.js

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• The built part

  Sketches and extrudes the sized bracket or shaft as part features.

  density (g/cm³) = density (kg/m³) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Edges and facet normals

  Finds the mesh's edges, each facet's unit normal, and which edges are sharp.

  sharp when n₁ · n₂ < cos 20°

  Drawing sheet views, dimensions and PDF drawing.js

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• One projected view with hidden lines

  Projects the mesh onto a view plane, finds the view's extents, and splits each drawn edge into visible and hidden parts.

  (u, v, w) = (p · right, p · up, p · dir)

  Drawing sheet views, dimensions and PDF drawing.js

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• Merging collinear segments

  Joins touching or overlapping segments on the same line and removes hidden lines under visible ones.

  Drawing sheet views, dimensions and PDF drawing.js

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• Section hatching

  Hatch lines at an angle inside the cut outline.

  Drawing sheet views, dimensions and PDF drawing.js

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• Clip to the detail circle

  Keeps the parts of segments inside the detail view's circle.

  A t² + B t + C = 0; t = (−B ± √(B² − 4AC)) ÷ 2A

  Drawing sheet views, dimensions and PDF drawing.js

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