Calculations

Calculations

• Series coefficients

  The factors 1/n! of the sine and cosine series.

  cₙ = ±1 ÷ n!

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Angle reduction

  Brings an angle into [−π, π].

  n = floor(x ÷ 2π + ½); r = (x − n × 2π_hi) − n × 2π_lo

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine series

  The sine of a reduced angle.

  sin r = r(c₀ + c₁r² + … + c₁₂r²⁴)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine series

  The cosine of a reduced angle.

  cos r = c₀ + c₁r² + … + c₁₂r²⁴

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sine

  sin x for any finite x.

  sin r = sin(π − r) for r > π/2; sin r = sin(−π − r) for r < −π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cosine

  cos x for any finite x.

  cos r = −cos(π − |r|) for |r| > π/2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Exponential

  The exponential eˣ of a number.

  k = floor(x ÷ ln 2 + ½); r = (x − k × ln2_hi) − k × ln2_lo; eˣ = 2ᵏ × Σ rⁿ ÷ n! (n = 0 … 18)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Natural logarithm

  The natural logarithm ln x of a number.

  x = m × 2ᵉ, m in [√2/2, √2]; ln m = 2(s + s³/3 + … + s²⁹/29), s = (m − 1) ÷ (m + 1); ln x = e × ln2_hi + (ln m + e × ln2_lo)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-2 logarithm

  The base-2 logarithm log₂ x of a number.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Base-10 logarithm

  The base-10 logarithm log₁₀ x of a number.

  log₁₀ x = ln x ÷ 2.302585092994046

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Whole-number power

  xⁿ for a whole n ≥ 0.

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Power

  A number x raised to a power y.

  xʸ = exp(y × ln x) for a fractional y

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arctangent

  The arctangent atan x of a number.

  atan z = 2·atan(z ÷ (1 + √(1 + z²))), twice; then 4(z − z³/3 + … + z⁴¹/41)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Two-argument arctangent

  The angle of the point (x, y).

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arcsine

  The arcsine asin x of a number in [−1, 1].

  asin x = atan2(x, √((1 − x)(1 + x)))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Arccosine

  The arccosine acos x of a number in [−1, 1].

  acos x = 2·atan2(√(1 − x), √(1 + x))

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Tangent

  The tangent tan x of an angle.

  tan x = sin x ÷ cos x

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic sine

  The hyperbolic sine sinh x of a number.

  sinh x = (eˣ − e⁻ˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic cosine

  The hyperbolic cosine cosh x of a number.

  cosh x = (eˣ + 1 ÷ eˣ) ÷ 2

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Hyperbolic tangent

  The hyperbolic tangent tanh x of a number.

  tanh x = (e²ˣ − 1) ÷ (e²ˣ + 1)

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Cube root

  The cube root ∛x of a number.

  r₀ = exp(ln|x| ÷ 3); r ← r − (r³ − |x|) ÷ (3r²), twice

  Deterministic sine, cosine, exp, log and power field/detmath.js

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• Sphere

  Signed distance to a sphere.

  d = √((x − cx)² + (y − cy)² + (z − cz)²) − r

  Field shapes: signed distance functions field/field.js

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• Box with rounded edges

  Signed distance to a box, optionally with rounded edges and corners.

  q = |p − c| − (half − round); d = |max(q, 0)| + min(max(q_x, q_y, q_z), 0) − round

  Field shapes: signed distance functions field/field.js

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• Cylinder

  Signed distance to a cylinder between two points.

  t = (p − m)·u; d_r = |p − m − t·u| − r; d_t = |t| − L ÷ 2; d = |max(d_r, d_t, 0)| + min(max(d_r, d_t), 0)

  Field shapes: signed distance functions field/field.js

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• Torus

  Signed distance to a torus about the z axis.

  d = √((√(x² + y²) − R)² + z²) − r

  Field shapes: signed distance functions field/field.js

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• Half space

  Signed distance to a plane.

  d = (n·p − d₀) ÷ |n|

  Field shapes: signed distance functions field/field.js

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• Solid of revolution

  Signed distance to the solid under a profile r(x) turned about the x axis.

  ρ = √(y² + z²); d = least distance from (x, ρ) to the profile's segments; inside when ρ ≤ r(x)

  Field shapes: signed distance functions field/field.js

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• Union, intersection and subtraction

  Combines shapes by union, intersection or subtraction.

  union = min(a, b); intersection = max(a, b); subtraction = max(a, −b)

  Field shapes: signed distance functions field/field.js

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• Smooth minimum

  Blends two shapes with a radius k.

  h = clamp(½ + (b − a) ÷ 2k, 0, 1); s = b + (a − b)h − k·h(1 − h)

  Field shapes: signed distance functions field/field.js

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• Offset and shell

  Grows or shrinks a shape, or hollows it to a wall thickness.

  offset: d − δ; shell inward: max(d, −d − t); shell centered: |d| − t ÷ 2

  Field shapes: signed distance functions field/field.js

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• Move, turn and scale

  A shape under a rigid motion and a uniform scale.

  q = Rᵀ(p − move) ÷ s; d = d₀(q) × s

  Field shapes: signed distance functions field/field.js

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• Rotation matrix

  The 3 × 3 matrix of a turn about an axis by an angle.

  R = c·I + s·[u]ₓ + (1 − c)·u uᵀ, c = cos θ, s = sin θ

  Field shapes: signed distance functions field/field.js

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• Beam or round cone

  Signed distance to a beam whose radius changes from one end to the other.

  t = clamp((p − a)·(b − a) ÷ |b − a|², 0, 1); d = |p − a − t(b − a)| − (r₀ + (r₁ − r₀)t)

  Field shapes: signed distance functions field/field.js

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• Lattice of beams

  The least distance to many beams.

  Field shapes: signed distance functions field/field.js

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• Ramped cell size and wall

  A number that changes linearly along an axis or with radius.

  v = v₀ + (v₁ − v₀) × clamp((p − a) ÷ (b − a), 0, 1)

  Field shapes: signed distance functions field/field.js

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• Phase of a graded lattice

  Maps a point to the lattice's own period coordinates when the cell size changes.

  fixed cell: X = 2π·p ÷ cell; ramp: X = ∫k dp with k linear in p

  Field shapes: signed distance functions field/field.js

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• Gyroid, Schwarz P and diamond

  Signed distance to a triply periodic minimal surface, as a sheet or a network.

  d ≈ g ÷ |∇g|; near the surface: two projections q ← q − g∇g ÷ |∇g|², d = ±|p − q|

  Field shapes: signed distance functions field/field.js

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• Brick classification

  Decides whether a ball of space is wholly outside, wholly inside or near the surface.

  outside when f − L·r > band; inside when f + L·r < −band

  Field shapes: signed distance functions field/field.js

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• Bricks to compute

  Which bricks of the lattice can hold the surface.

  brick range = floor((box ∓ band × h) ÷ h ÷ 8); ball radius = 3.5√3 × h

  Distance grid: baking, sampling and volume field/bake.js

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• Node values of a brick

  The signed distance at each of a brick's 512 nodes.

  value = clamp(f(i·h, j·h, k·h), −band × h, band × h)

  Distance grid: baking, sampling and volume field/bake.js

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• Brick key

  One whole number that names a brick.

  key = ((bx + 65536) × 131072 + (by + 65536)) × 131072 + (bz + 65536)

  Distance grid: baking, sampling and volume field/grid.js

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• Value at a point

  The signed distance at any point, between the lattice nodes.

  trilinear: a + (b − a) × t along x, then y, then z

  Distance grid: baking, sampling and volume field/grid.js

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• Gradient at a point

  The direction and rate of change of the distance.

  g = (f(p + h/2) − f(p − h/2)) ÷ h, per axis

  Distance grid: baking, sampling and volume field/grid.js

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• Box of the stored bricks

  The world box that holds every stored node.

  lo = brick × 8 × h; hi = (brick × 8 + 7) × h

  Distance grid: baking, sampling and volume field/grid.js

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• Enclosed volume

  The volume inside the surface.

  V = h³ × Σ clamp(½ − d ÷ h, 0, 1)

  Distance grid: baking, sampling and volume field/grid.js

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• Welding corners

  Merges mesh corners that sit at the same position.

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Angle-weighted normals

  The pseudo-normal at each corner and edge, used for the sign.

  angle = acos(e₁·e₂ ÷ (|e₁||e₂|)); N_vertex = Σ angle × n_face

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Signed distance at a node

  The distance from a lattice node to the mesh, negative inside.

  d = √(least squared distance); sign = sign((p − closest)·N)

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Closest point on a triangle

  The nearest point of a triangle to a point, and which face, edge or corner it lies on.

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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• Inside or outside far from the surface

  Marks bricks no triangle comes near as inside or outside.

  x_cross = (w₀a_x + w₁b_x + w₂c_x) ÷ (w₀ + w₁ + w₂); inside when an odd number of crossings lie before the brick

  Distance grid: baking, sampling and volume field/mesh-sdf.js

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