Calculations

Calculations

• Supplied sizes written to CSV

  Container volume, height, spread, girth and clear stem of each supplied size as one CSV cell.

  Plant sizes, spacing and symbols plants/plants.js

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• Supplied sizes read from CSV

  The same cell read back into numbers.

  Plant sizes, spacing and symbols plants/plants.js

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• 2D planting symbol

  The circles, lines, arcs and outlines of a plant's plan symbol, sized to its spread.

  R = spread ÷ 2 × scale; point = (r cos a, r sin a)

  Plant sizes, spacing and symbols plants/plants.js

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• Library item size

  The width, depth and height of the library item placed for a plant.

  mm = round(m × 1000)

  Plant sizes, spacing and symbols plants/plants.js

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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• Rounding of manifest numbers

  The rounding every number in the manifest goes through.

  round(x, k) = round(x × 10ᵏ) ÷ 10ᵏ, k = 6 unless stated

  Render hand-off manifest render/handoff.js

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• Color to linear values

  A hex color turned into the linear Rec.709 triple the manifest carries.

  c ≤ 0.04045: c ÷ 12.92; else ((c + 0.055) ÷ 1.055)^2.4

  Render hand-off manifest render/handoff.js

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• Points and directions on the stage

  A point or a direction of the scene moved to the USD stage's coordinates.

  p′ = M p; d′ = (M d) ÷ |M d|

  Render hand-off manifest render/handoff.js

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• Camera entry

  The camera's matrices, position, view direction, aspect and view angle or height.

  fovY = 2 atan(1 ÷ |P₅|) × 180 ÷ π; ortho height = 2 ÷ |P₅|

  Render hand-off manifest render/handoff.js

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• Sun and lamp entries

  The sun's direction and strength, and each lamp's lumens, color, position, aim and size.

  Render hand-off manifest render/handoff.js

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• Scene, exposure and white balance entries

  The ground height, sample count, exposure in stops, shutter time and white-balance gains.

  Render hand-off manifest render/handoff.js

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• Animation frames entry

  The start and end time, frame rate, frame count and the camera matrix of each frame.

  Render hand-off manifest render/handoff.js

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• Material values

  The OpenPBR parameters written for each material, where they differ from the defaults.

  Render hand-off manifest render/handoff.js

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• Exposure value at ISO 100

  The EV figure the render dialog shows for the chosen f-stop, shutter time and ISO.

  EV = log₂(N² ÷ t) − log₂(S ÷ 100)

  Camera exposure and white balance render/pt-camera.js

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• Exposure in stops

  The stops added to the picture's exposure from the brightness slider and, in camera mode, the f-stop, shutter and ISO.

  stops = slider + log₂(t × S × 256 ÷ N²)

  Camera exposure and white balance render/pt-camera.js

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• Color temperature to chromaticity

  The CIE xy color of light of a given temperature in kelvin.

  x = a ÷ T³ + b ÷ T² + c ÷ T + d; y = polynomial in x

  Camera exposure and white balance render/pt-camera.js

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• White-balance gains

  The three color gains that make light of the chosen temperature show as white.

  gain = (ref ÷ w) × (Y(w) ÷ Y(ref))

  Camera exposure and white balance render/pt-camera.js

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• Angle of a point on an ellipse

  The ellipse parameter of the point on the ellipse nearest a given point.

  f(t) = (E(t) − q)·E′(t) = 0, solved by t ← t − f ÷ f′

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

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• Parameter of a point on a B-spline curve

  The curve parameter nearest a given point.

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

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• Parameters of a point on a B-spline surface

  The (u, v) of the surface point nearest a given point.

  [S_u·S_u S_u·S_v; S_u·S_v S_v·S_v] d = −[r·S_u; r·S_v], r = S(u, v) − q

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

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• Surface of linear extrusion

  The point of a curve swept along a vector, and the (u, v) of a point.

  S(u, v) = C(u) + v · vec; v = (q − C(u))·d ÷ |vec|, u = project(q − v · vec)

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

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• Offset surface

  The point of a surface moved a distance along its normal.

  S_off(u, v) = S(u, v) + d · n, n = (ΔS_u × ΔS_v) ÷ |…|

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

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• Points along an edge

  The points of an edge from its start vertex to its end vertex along its curve.

  STEP import: triangles for curved faces step/tess.js

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• How finely a curve is divided

  The parameters at which an edge curve is sampled.

  circle step = min(π ÷ 24, 2 acos(1 − tol ÷ r)); else halve while the middle is more than tol from the chord or the turn exceeds π ÷ 24

  STEP import: triangles for curved faces step/tess.js

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• Turn and crossing tests

  Whether three points turn left, right or run straight, and whether two segments cross.

  x = (a − o) × (b − o); left when x > 1e-9 × |ab| × |bc|, right when x < −1e-9 × |ab| × |bc|

  STEP import: triangles for curved faces step/tess.js

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• Holes joined to the outer ring

  One ring made from a face's outer loop and its holes by bridges, ready for ear clipping.

  STEP import: triangles for curved faces step/tess.js

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• Ear clipping

  Triangles of a simple polygon in the plane.

  STEP import: triangles for curved faces step/tess.js

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• A loop in the surface's (u, v)

  The parameters of each boundary point, with periodic directions unwrapped and poles split.

  uᵢ ← uᵢ + round((uⱼ − uᵢ) ÷ period) × period

  STEP import: triangles for curved faces step/tess.js

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