Calculations

Calculations

• Convert between two units

  A value in one unit as a value in another unit of the same kind.

  result = v × (n₁ × d₂) ÷ (d₁ × n₂)

  Unit conversion unit-table.js Every app
  Exact

• Typed text to a unit

  What was typed, with its unit, as a value in the unit the field wants.

  3 yd typed into a feet field = convert(3, yd, ft) = 9

  Unit conversion unit-table.js Every app
  Exact

• A value to the base unit

  A number in any unit as meters, square meters, cubic meters, radians or rise over run.

  base = v × n ÷ d (v × π × n ÷ d for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• A base value to a unit

  A stored base value as a number in the unit shown.

  value = base × d ÷ n (base ÷ π × d ÷ n for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• Add, subtract, multiply, divide

  The four operations on values that carry units.

  a + b and a − b need the same dimensions; a × b and a ÷ b combine them

  Calc sheets calc.js Every app
  64-bit

• Comparisons

  Pass or fail checks such as M <= 40 kN m.

  a ≤ b is true when a ≤ b + 1e-12 × max(1, |a|, |b|)

  Calc sheets calc.js Every app
  64-bit

• Number shown

  A result as text in the sheet.

  6 significant digits; exponent form from 1e9 up and below 1e-4

  Calc sheets calc.js Every app
  64-bit

• Knot span

  Which span of the knot vector a parameter falls in.

  i such that U[i] ≤ u < U[i + 1], by bisection

  B-spline basis functions nurbs/basis.js

  • ModelCraft
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  64-bit

• Basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N(i, p)(u) by the Cox-de Boor recurrence (A2.2)

  B-spline basis functions nurbs/basis.js

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  64-bit

• Basis function derivatives

  The basis functions and their derivatives up to a chosen order.

  N⁽ᵏ⁾(i, p)(u) for k = 0 … d (A2.3)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
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  64-bit

• Conjugate gradient solve

  The displacement of every node under the loads.

  repeat until ‖r‖ ÷ ‖b‖ < 1e-8

  Stress analysis fea.js

  • PartCraft
  64-bit

• Von Mises stress

  One equivalent stress from the six stress components.

  σᵥ = √(½[(σ_x − σ_y)² + (σ_y − σ_z)² + (σ_z − σ_x)²] + 3(τ_xy² + τ_yz² + τ_zx²))

  Stress analysis fea.js

  • PartCraft
  64-bit

• Stress and displacement at each node, and their maximum

  The values the plot colors by, and the maximum the report states.

  σ at a node = mean of its corners; |u| = √(u_x² + u_y² + u_z²); maximum over the nodes

  Stress analysis fea.js

  • PartCraft
  32-bit in part

• Volume, surface area and bounding box

  The measured properties of the finished body.

  volume, area, genus and bounding box, asked of the kernel

  Solid modeling kernel.js

  • PartCraft
  64-bit

• Vertex positions of the body

  The mesh that faces, edges and the display are built from.

  positions = 32-bit copy of the kernel's vertices

  Solid modeling kernel.js

  • PartCraft
  32-bit in part

• Key times to one millisecond

  The rounding applied to every key time, clip start and the animation length.

  t = round(t × 1000) ÷ 1000

  Animation keys, curves and poses render/anim.js

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  64-bit

• Stretching an animation to a new length

  Every key time and clip start when the animation is made longer or shorter.

  k = new length ÷ old length; t′ = round₀.₀₀₁(t × k)

  Animation keys, curves and poses render/anim.js

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• Spreading chosen keys about a time

  The new times of selected keys scaled away from or toward a pivot time.

  t′ = round₀.₀₀₁(pivot + (t − pivot) × f)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Frame count and frame times

  The number of frames a render of the animation has, and the time of each frame.

  frames = max(1, round(length × fps) + 1); time of frame i = i ÷ fps

  Animation keys, curves and poses render/anim.js

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• Position between two keys

  Which pair of keys a time falls between and how far along that stretch it is.

  s = (t − tᵢ) ÷ (tᵢ₊₁ − tᵢ)

  Animation keys, curves and poses render/anim.js

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• Time on a repeating track

  The time inside the keys' span that a repeating, ping-pong or offset track plays at.

  x = (t − t₀) ÷ P; k = floor(x); f = x − k

  Animation keys, curves and poses render/anim.js

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  64-bit

• Speed through a key

  The rate of change of a value at a key, from its two neighbors.

  m = (pᵢ₊₁ − pᵢ₋₁) ÷ (tᵢ₊₁ − tᵢ₋₁)

  Animation keys, curves and poses render/anim.js

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• Value between two keys

  A position or other value between two keys: held, straight, or on a smooth curve.

  p = h₀₀ p₀ + h₀₁ p₁ + h₁₀ Δt m₀ + h₁₁ Δt m₁

  Animation keys, curves and poses render/anim.js

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  64-bit

• Bézier handles of a property key

  The two handle points of a key on a property curve.

  automatic: handle = key ± (Δt ÷ 3, m × Δt ÷ 3)

  Animation keys, curves and poses render/anim.js

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• Property value on a Bézier curve

  The value of an animated property at a time between two keys with Bézier handles.

  solve x(u) = t by bisection; v = B(v₀, p₁, p₂, v₃; u)

  Animation keys, curves and poses render/anim.js

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• Property value at a time

  A property track's value at any time, with its cycles followed.

  offset cycle: v + n × (last key − first key)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Camera at a time

  The camera's eye, target, view angle and view height between camera keys.

  eye, target, fov on the curve; height = exp(curve of ln height)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Matrix to position, rotation and scale

  A placement matrix split into its position, a unit quaternion and three scale factors.

  s = column lengths; q from the rotation matrix (Shepperd)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Position, rotation and scale to matrix

  The placement matrix built from a position, a quaternion and a scale.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Blend of two rotations

  The rotation part of a pose between two keys.

  q = (sin((1 − k)θ) a + sin(kθ) b) ÷ sin θ, θ = acos(a · b)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Angles to quaternion

  XYZ rotation angles in degrees turned into a unit quaternion.

  q = q_z × q_y × q_x, each from sin and cos of half the angle

  Animation keys, curves and poses render/anim.js

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  64-bit

• Quaternion to angles

  XYZ rotation angles in degrees from a unit quaternion, chosen close to the key before.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Object placement at a time

  A moving object's placement matrix between two matrix keys.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Time inside a clip

  The time of a saved action that a clip plays at a given timeline time.

  x = (t − start) ÷ scale; local = from + (x mod span)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Mixing a clip into a pose

  An object's placement when a clip is blended in at less than full influence or added on top.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Driver expression

  The value of a property worked out from other properties by a typed expression.

  Animation keys, curves and poses render/anim.js

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  64-bit

• Camera keys from saved scenes

  The key times of a walkthrough built from saved scenes with a move time and a hold time.

  t accumulates move and hold; key time = round₀.₀₀₁(t)

  Animation keys, curves and poses render/anim.js

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  64-bit

• Default gravity for rigid bodies

  The gravity a physics bake starts from when the file sets none.

  g = (0, 0, −9.81) m/s²

  Animation keys, curves and poses render/anim.js

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  64-bit

• Axis placement

  The origin and three unit axes of an AXIS2_PLACEMENT_3D.

  z = axis ÷ |axis|; x = (ref − z (ref·z)) ÷ |…|; y = z × x

  STEP curves and surfaces: points from parameters step/geom.js

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• Circle

  The point at an angle on a circle, and the angle of a point.

  P(t) = o + r cos t · x + r sin t · y; t = atan2(l_y, l_x)

  STEP curves and surfaces: points from parameters step/geom.js

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• Polyline

  The point at a parameter along a polyline, and the parameter nearest a point.

  P(t) = (1 − f) Pᵢ + f Pᵢ₊₁, i = ⌊t⌋, f = t − i

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Knot span

  Which knot interval a B-spline parameter falls in.

  STEP curves and surfaces: points from parameters step/geom.js

  • BimCraft
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  Exact

• B-spline basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N_{i,0} = 1; N_{i,j} built from (t − u) ÷ (u_b − u_a) terms in the Cox–de Boor triangular table

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• B-spline curve point

  The point on a B-spline or rational B-spline curve at a parameter.

  C(t) = Σ Nᵢ wᵢ Pᵢ ÷ Σ Nᵢ wᵢ

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit

• Is a B-spline curve closed

  Decides whether a B-spline curve's two ends meet, so it can be treated as periodic.

  |C(t₀) − C(t₁)| < 1e-9 × max(1, max |Pᵢ|)

  STEP curves and surfaces: points from parameters step/geom.js

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• B-spline surface point

  The point on a B-spline or rational B-spline surface at (u, v).

  S(u, v) = Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ Pᵢⱼ ÷ Σ Σ Nᵢ(u) Nⱼ(v) wᵢⱼ

  STEP curves and surfaces: points from parameters step/geom.js

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• Seams and poles of a B-spline surface

  Decides which direction of a B-spline surface closes on itself and which boundary rows collapse to a point.

  two points are the same when |a − b| < 1e-7 × max(1, |a|)

  STEP curves and surfaces: points from parameters step/geom.js

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• Cylinder

  The point at (angle, height) on a cylinder, and the angle and height of a point.

  S(u, v) = o + r cos u · x + r sin u · y + v · z

  STEP curves and surfaces: points from parameters step/geom.js

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• Cone

  The point at (angle, distance along the axis) on a cone, and the inverse.

  r(v) = R + v tan α; v = ((ρ − R) tan α + l_z) ÷ (tan² α + 1)

  STEP curves and surfaces: points from parameters step/geom.js

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• Sphere

  The point at (longitude, latitude) on a sphere, and the inverse.

  S(u, v) = o + r cos v cos u · x + r cos v sin u · y + r sin v · z

  STEP curves and surfaces: points from parameters step/geom.js

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  64-bit