Calculations

Calculations

• Matrix product

  Combines two transforms into one.

  o[c][r] = Σₖ a[k][r] × b[c][k]

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Transforming a point

  Applies a transform to a point.

  p′ = (M × [x, y, z, 1]) ÷ w

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Rotation about an axis

  The matrix of a rotation by an angle in degrees about X, Y or Z.

  a = deg × π ÷ 180; entries cos a, ± sin a

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Quaternion to rotation matrix

  The rotation matrix of an orientation stored as a quaternion.

  q ÷ |q|; R₀₀ = 1 − 2(y² + z²), R₀₁ = 2(xy + wz), …

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Value at a time

  An animated value at a given time, between its stored samples.

  v = a + (b − a) × (t − t₀) ÷ (t₁ − t₀)

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• An object's own transform

  Builds the transform of one object from its list of translate, rotate, scale and matrix operations.

  M = op₁ × op₂ × … in the listed order; an inverted operation uses its inverse

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• World transform

  The transform of an object in the scene, through all its parents.

  W = M_root × … × M_parent × M_own

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Scene units and up axis

  The transform from the scene's own units and up axis to meters with Z up.

  scale by metersPerUnit; Y-up scenes: rotate 90° about X

  USD transforms and placement usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Placement of instanced copies

  The transform of each copy of a point instancer from its position, orientation and scale.

  M = translate(p) × rotate(q) × scale(s)

  USD transforms and placement usd/scene.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Writing an object's transform

  Stores an object's placement in the exported scene.

  USD transforms and placement usd/build.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Attribute value to its storage type

  Converts a list of numbers to the number type its USD attribute type calls for.

  USD mesh coordinates in saved and exported files usd/build.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Mesh points and extent

  Stores the points of a mesh and its bounding box in the scene.

  extent = [min x, min y, min z, max x, max y, max z] of the points

  USD mesh coordinates in saved and exported files usd/build.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Lamp intensity

  Converts a lamp's lumens to the intensity a renderer reads.

  intensity = lumens ÷ (683 × area × π)

  USD mesh coordinates in saved and exported files usd/build.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit, display only

• Single values in the binary file

  Writes one number, tuple or matrix into the binary USD file.

  USD mesh coordinates in saved and exported files usd/usdc-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Arrays in the binary file

  Writes an array of numbers into the binary USD file.

  USD mesh coordinates in saved and exported files usd/usdc-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Reading a mesh

  The points, faces and attached values of a mesh read from a USD scene.

  USD mesh coordinates in saved and exported files usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  32-bit in part

• Built-in shapes as meshes

  A cube, sphere, cylinder, cone, capsule or plane of a USD scene as a mesh.

  ring point = (r·cos θ, r·sin θ, z), θ = 2π·j ÷ 24

  USD mesh coordinates in saved and exported files usd/geom.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Raw double, read

  A 64-bit value read from eight bytes of the bit stream.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Bit double, read

  A value stored as a two-bit code: a full 64-bit value, exactly 1.0 or exactly 0.0.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Double with default, read

  A value stored as the bytes in which it differs from a default value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• 64-bit whole number, read

  An unsigned 64-bit whole number from two 32-bit halves.

  value = high × 2³² + low

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Bit long long, read

  A whole number of up to seven bytes, the count given by three bits.

  value = Σ byteᵢ × 256ⁱ

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Handle reference, read

  An object handle, absolute or as an offset from another handle.

  value = Σ byte × 256ᵏ; codes 6, 8, A, C: ref + 1, ref − 1, ref + value, ref − value

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Modular char, read

  A signed whole number stored seven bits per byte.

  value = Σ (byte & 0x7F) × 2^(7i); the last byte carries 6 bits and the sign

  Numbers in a DWG file, read and written bit for bit dwg/bits-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Raw double, written

  A 64-bit value written as its eight bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Bit double, written

  A value written as a two-bit code for exactly 0.0 or 1.0, or as a full 64-bit value.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Double with default, written

  A value written as a two-bit code when it equals its default, or in full.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Thickness, written

  A thickness written as one bit when it is zero, or as a bit double.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Bit long, written

  A 32-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  Exact

• Bit short, written

  A 16-bit whole number in its short or full form.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  Exact

• Bit long long, written

  A 64-bit whole number as a three-bit byte count and its bytes.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  Exact

• Point, written

  The three coordinates of a point as bit doubles.

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Handle bytes, written

  A handle as its bytes, most significant first.

  byte = v mod 256; v = ⌊v ÷ 256⌋

  Numbers in a DWG file, read and written bit for bit dwg/bits-write.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  Exact

• Lineweight to stored index

  The index written for a lineweight, or the codes for by layer, by block and default.

  index of the list value with the smallest |value − lineweight|

  Lineweights in a DWG file dwg/model.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  Exact

• Stored index to lineweight

  The lineweight for an index read from a file.

  Lineweights in a DWG file dwg/model.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  Exact

• Triangles of a polygon with holes

  The triangles of a face as triples of vertex indexes.

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Orientation of three points

  Twice the signed area of a triangle, whose sign says which way three points turn.

  (q_y − p_y)(r_x − q_x) − (q_x − p_x)(r_y − q_y)

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Direction of a ring

  Whether a ring of points runs clockwise or counterclockwise.

  Σ (x_j − x_i)(y_i + y_j)

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Is a corner an ear

  Whether the triangle at a corner is convex and holds no other point of the polygon.

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Point in a triangle

  Whether a point lies in or on a triangle.

  three comparisons (c − p) × (a − p) ≥ 0, each written as product ≥ product

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Duplicate and collinear points removed

  Drops points that repeat their neighbor or lie exactly on the line between their neighbors.

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Bridge from a hole to the outer ring

  The outer-ring point a hole is connected to, found by a ray to the left from the hole's leftmost point.

  x = p_x + (h_y − p_y)(n_x − p_x) ÷ (n_y − p_y); tan = |h_y − p_y| ÷ (h_x − p_x)

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Do two edges cross

  Whether two segments intersect, used to keep a diagonal from crossing the polygon.

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Is a diagonal's midpoint inside

  Whether the middle of a candidate diagonal lies inside the polygon.

  crossing count of a ray from ((a_x + b_x) ÷ 2, (a_y + b_y) ÷ 2)

  Triangulation of faces with holes m3d/earcut.js

  • CadCraft
  • ModelCraft
  • TakeoffCraft
  64-bit

• Triangulate a polygon with holes

  The triangles of a polygon, as indexes into its point list.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Winding of a ring

  Whether a ring runs clockwise or counterclockwise.

  sum = Σ (xⱼ − xᵢ) × (yᵢ + yⱼ)

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Orientation of three points

  On which side of the line through two points a third point lies.

  area = (q.y − p.y) × (r.x − q.x) − (q.x − p.x) × (r.y − q.y)

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Removing duplicate and collinear points

  Drops points that repeat their neighbor or lie on the line between their neighbors.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Ear test

  Whether a corner can be cut off as a triangle: it is convex and no other point of the ring lies inside it.

  Polygon triangulation (ear clipping) usd/earcut.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit